diff --git "a/AtAzT4oBgHgl3EQfv_4W/content/tmp_files/load_file.txt" "b/AtAzT4oBgHgl3EQfv_4W/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/AtAzT4oBgHgl3EQfv_4W/content/tmp_files/load_file.txt" @@ -0,0 +1,512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf,len=511 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='01714v1 [quant-ph] 1 Jan 2023 Canonical steering ellipsoids of pure symmetric multiqubit states with two distinct spinors and volume monogamy of steering B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Divyamani,1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Reena,2 Prasanta K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Panigrahi,3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Usha Devi,2, 4 and Sudha5, 4, ∗ 1Tunga Mahavidyalaya, Thirthahalli-577432, Karnataka, India 2Department of Physics, Bangalore University, Bangalore-560 056, India 3Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur-741246, West Bengal, India 4Inspire Institute Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', Alexandria, Virginia, 22303, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 5Department of Physics, Kuvempu University, Shankaraghatta-577 451, Karnataka, India (Dated: January 5, 2023) Quantum steering ellipsoid formalism provides a faithful representation of all two-qubit states and helps in obtaining correlation properties of the state through the steering ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The steering ellipsoids corresponding to the two-qubit subsystems of permutation symmetric N-qubit states is analysed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The steering ellipsoids of two-qubit states that have undergone local operations on both the qubits so as to bring the state to its canonical form are the so-called canonical steering ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We construct and analyze the geometric features of the canonical steering ellipsoids corresponding to pure permutation symmetric N-qubit states with two distinct spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Depending on the degeneracy of the two spinors in the pure symmetric N-qubit state, there arise several families which cannot be converted into one another through Stochastic Local Operations and Classical Communications (SLOCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The canonical steering ellipsoids of the two-qubit states drawn from the pure symmetric N-qubit states with two distinct spinors allow for a geometric visualization of the SLOCC-inequivalent class of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We show that the states belonging to the W-class correspond to oblate spheroid centered at (0, 0, 1/(N −1)) with fixed semiaxes lengths 1/ √ N − 1 and 1/(N −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The states belonging to all other SLOCC inequivalent families correspond to ellipsoids centered at the origin of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We also explore volume monogamy relations of states belonging to these families, mainly the W-class of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' PACS numbers: 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='Ud, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='Bg I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' INTRODUCTION The Bloch sphere representation of a single qubit contains valuable geometric information needed for quantum information processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' A natural generalization and an analogous picture for a two-qubit system is provided by the quantum steering ellipsoid [1–3] and is helpful in understanding correlation properties such as quantum discord [4, 5], volume monogamy of steering [2, 3] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', Quantum steering ellipsoid is the set of all Bloch vectors to which one party’s qubit could be ‘steered’ when all possible measurements are carried out on the qubit belonging to other party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The volume of the steering ellipsoids [1] corresponding to the two-qubit subsystems of an N-qubit state, N > 3, capture monogamy properties of the state effectively [2, 3] and provides insightful information about two-qubit entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' While the quantum steering ellipsoid [1–3] is the set of all Bloch vectors of first qubit steered by local operations on second qubit, the so-called canonical steering ellipsoid [6–8] is the steering ellipsoid of a two-qubit state that has attained a canonical form under suitable SLOCC operations on both the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' It has been shown that the SLOCC canonical forms of a two-qubit state can either be a Bell diagonal form or a nondiagonal one (when the two-qubit state is rank-deficient) [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The canonical steering ellipsoids corresponding to the two-qubit states can thus have only two distinct forms and provide a much simpler geometric picture representing the set of all SLOCC equivalent two-qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The canonical steering ellipsoids corresponding to the two-qubit subsystems of pure three-qubit permutation sym- metric states are analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' It has been shown that [9] the two SLOCC inequivalent families of pure three-qubit permutation symmetric states, the W-class of states (with two distinct spinors) and the GHZ class of states (with three distinct spinors) correspond to distinct canonical steering ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' While an ellipsoid centered at the origin of the Bloch sphere is the canonical steering ellipsoid for the GHZ class of states, an oblate spheroid with its center shifted along the z-axis is the one for W-class of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Using these, the volume monogamy relations are established and the obesity of the steering ellipsoids is made use of to obtain expressions for concurrence of states belonging to these two SLOCC inequivalent families in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' ∗ tthdrs@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='com 2 In this paper, we continue with the work in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [9], authored by some of us, and extend the analysis to a class of N-qubit pure states which are symmetric under exchange of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Through the SLOCC canonical forms of the two-qubit reduced state, extracted from pure symmetric multiqubit states with two distinct spinors and the Lorentz canonical forms of their real representative, we examine the features of canonical steering ellipsoids associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We identify the special features of the canonical steering ellipsoid representing N-qubit states of the W-class and these features distinguish this class from all other SLOCC inequivalent families of pure symmetric N-qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We discuss the volume monogamy of steering for pure permutation symmetric N-qubit states and obtain the volume monogamy relation satisfied by W-class of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' An expression for obesity of the steering ellipsoid and thereby an expression for concurrence of two-qubit subsystems of N-qubit states belonging to the W-class is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Contents of this paper are organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='II, we give a brief review on SLOCC classification of pure permutation symmetric multiqubit states based on Majorana representation [10, 12, 13] and obtain the two-qubit subsystems of the states belonging to SLOCC inequivalent families of pure symmetric multiqubit states with two distinct spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' III provides an outline of the real matrix representation of a two-qubit density matrix and their Lorentz canonical forms under SLOCC transformation of the two-qubit density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We also obtain the Lorentz canonical forms of two-qubit subsystems corresponding to SLOCC inequivalent families, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='IV, we analyse the nature of steering ellipsoids associated with the distinct Lorentz canonical forms obtained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The volume monogamy of steering for pure symmetric multiqubit states with two distinct spinors is discussed along with illustration for W-class of states, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Summary of our results is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' MAJORANA GEOMETRIC REPRESENTATION OF PURE SYMMETRIC N-QUBIT STATES WITH TWO DISTINCT SPINORS Ettore Majorana, in his novel 1932 paper [10] proposed that a pure spin j = N 2 quantum state can be represented as a symmetrized combination of N constituent spinors as follows: |Ψsym⟩ = N � P ˆP {|ǫ1, ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' ǫN⟩}, (1) where |ǫl⟩ = (cos(αl/2) |0⟩ + sin(αl/2) |1⟩) eiβl/2, l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (2) The symbol ˆP corresponds to the set of all N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' permutations of the spinors (qubits) and N corresponds to an overall normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The name Majorana geometric representation is owing to the fact that it leads to an intrinsic geometric picture of the state in terms of N-points on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In fact, the spinors |ǫl⟩, l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , N of (2) correspond geometrically to N points on the unit sphere S2, with the pair of angles (αl, βl) determining the orientation of each point on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The pure symmetric N-qubit states characterized by two distinct qubits are given by [11–13], |DN−k,k⟩ = N � P ˆP {| ǫ1, ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , ǫ1 � �� � N−k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' ǫ2, ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , ǫ2 � �� � k ⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (3) Here one of the spinors say |ǫ1⟩ occurs N − k times whereas the other spinor |ǫ2⟩ occurs k times in each term of the symmetrized combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Under identical local unitary transformations, the pure symmetric N qubit states with two distinct spinors can be brought to the canonical form [13], |DN−k,k⟩ ≡ k � r=0 β(k) r ���� N 2 , N 2 − r � , k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' �N 2 � (4) β(k) r = N � N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (N − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' ak−r br (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (k − r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', 0 ≤ a < 1, b = � 1 − a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (5) Notice that �� N 2 , N 2 − r � , r = 0, 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' k are the Dicke states, which are common eigenstates of the collective angular momentum operators J2 and Jz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' They are the basis states of the N + 1 dimensional symmetric subspace of collective angular momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The states |DN−k,k⟩ (see (4), (5)) are characterized by only one real parameter ‘a’ and thus form one parameter family of states {DN−k,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' It is important to notice that [13] in the family {DN−k,k}, different values of k, (k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' � N 2 � ), correspond to different SLOCC inequivalent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' That is, a state |DN−k,k⟩ cannot be converted into |DN−k′,k′⟩, k ̸= k′ through 3 any choice of local unitary (identical) transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In fact, different values of k lead to different degeneracy configurations [13] of the two spinors |ǫ1⟩, |ǫ2⟩ in the state |DN−k,k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' When k = 1, one gets the W-class of states {DN−1,1} where one of the qubits say |ǫ1⟩ repeats only once in each term of the symmetrized combination (see (3)) and the other qubit |ǫ2⟩ repeating N − 1 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The N-qubit W-state |WN⟩ = 1 √ N [|000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='1⟩ + |000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='10⟩ + · · · + |100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='00⟩] ≡ ���� N 2 , N 2 − 1 � (6) belongs to the family {DN−1,1} and hence the name W-class of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Two-qubit reduced density matrices of the states |DN−k, k⟩ The two-qubit marginal ρ(k) corresponding to any random pair of qubits in the pure symmetric N-qubit state |DN−k, k⟩ ∈ {DN−k,k} is obtained by tracing over the remaining N − 2 qubits in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [15], it has been shown, using the algebra of addition of angular momenta, j1 = 1 (corresponding to two-qubit marginal) and j2 = (N − 2)/2, that the two-qubit reduced density matrix ρ(k) has the form ρ(k) = \uf8eb \uf8ec \uf8ec \uf8ed A(k) B(k) B(k) C(k) B(k) D(k) D(k) E(k) B(k) D(k) D(k) E(k) C(k) E(k) E(k) F (k) \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (7) The elements A(k), B(k), C(k), D(k), E(k) and F (k) are real and are explicitly given by [15] A(k) = k � r=0 � βk r �2 � c(r) 1 �2 , B(k) = 1 √ 2 k−1 � r=0 β(k) r β(k) r+1 c(r) 1 c(r+1) 0 C(k) = k−2 � r=0 β(k) r β(k) r+2 c(r) 1 c(r+2) −1 , D(k) = 1 2 k � r=1 � β(k) r �2 � c(r) 0 �2 (8) E(k) = 1 √ 2 k−1 � r=0 β(k) r β(k) r+1 c(r) 0 c(r+1) −1 , F (k) = k � r=0 � β(k) r �2 � c(r) −1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' where, β(k) r are given as functions of the parameter ‘a’ in (5) and c(r) 1 = � (N − r)(N − r − 1) N(N − 1) , c(r) −1 = � r (r − 1) N(N − 1), c(r) 0 = � 2r (N − r) N(N − 1) (9) are the Clebsch-Gordan coefficients c(r) m2 = C � N 2 − 1, 1, N 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' m − m2, m2, m � , m = N 2 − r, m2 = 1, 0, −1 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In particular, for W-class of states i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', when k = 1, we have ρ(1) = TrN−2 (|DN−1, 1⟩⟨DN−1, 1|) = �� β(1) 0 �2 + � β(1) 1 c(1) 1 �2� |1, 1⟩⟨1, 1| + � β(1) 1 c(1) 0 �2 |1, 0⟩⟨1, 0| + β(1) 0 β(1) 1 c(1) 0 |1, 1⟩⟨1, 0| +β(1) 0 β(1) 1 c(1) 0 |1, 0⟩⟨1, 1| (10) Here (see (5)) we have β(1) 0 = NN a, β(1) 1 = N � N(1 − a2) with N = 1 √ N 2 a2+N(1−a2) and the associated non-zero Clebsch-Gordan coefficients (see (9)) are given by c(1) 1 = � N − 2 N , c(1) 0 = � 2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (11) 4 In the standard two-qubit basis {|0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 0B⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' |0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 1B⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' |1A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 0B⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' |1A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 1B⟩},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' the two-qubit density matrix ρ(1) drawn from the states |DN−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='1⟩ take the form ρ(1) = \uf8eb \uf8ec \uf8ec \uf8ed A(1) B(1) B(1) 0 B(1) D(1) D(1) 0 B(1) D(1) D(1) 0 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (12) where A(1) = N 2a2 + (N − 2)(1 − a2) N 2 a2 + N(1 − a2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' B(1) = a √ 1 − a2 1 + a2(N − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' D(1) = 1 − a2 N 2 a2 + N(1 − a2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (13) In a similar manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' the two-qubit subsystems of pure symmetric N-qubit states |DN−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='k⟩ belonging to each SLOCC inequivalent family {DN−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' k},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � can be obtained as a function of N and ‘a’ using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (7), (8), (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' As is shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [8, 9], the real representative Λ(k) of the two-qubit subsystem ρ(k) and its Lorentz canonical form �Λ(k) are essential in obtaining the geometric visualization of the states |DN−k,k⟩ for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We thus proceed to obtain Λ(k) and its Lorentz canonical form �Λ(k) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' THE REAL REPRESENTATION OF ρ(k) AND ITS LORENTZ CANONICAL FORMS The real representative Λ(k) of the two-qubit state ρ(k) is a 4 × 4 real matrix with its elements given by Λ(k) µ ν = Tr � ρ(k) (σµ ⊗ σν) � (14) That is, Λ(k) µ ν, µ, ν = 0, 1, 2, 3 are the coefficients of expansion of ρ(k), expanded in the Hilbert-Schmidt basis {σµ⊗σν}: ρ(k) = 1 4 3 � µ, ν=0 Λ(k) µ ν (σµ ⊗ σν) , (15) Here, σi, i = 1, 2, 3 are the Pauli spin matrices and σ0 is the 2 × 2 identity matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' σ0 = � 1 0 0 1 � , σ1 = � 0 1 1 0 � , σ2 = � 0 −i i 0 � , σ3 = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (16) It can be readily seen that (see (14), (15)) the real 4 × 4 matrix Λ(k) has the form Λ(k) = \uf8eb \uf8ec \uf8ed 1 r1 r2 r3 s1 t11 t12 t13 s2 t21 t22 t23 s3 t31 t32 t33 \uf8f6 \uf8f7 \uf8f8 , (17) where r = (r1, r2, r3)T , s = (s1, s2, s3)T are Bloch vectors of the individual qubits and T = (tij) is the correlation matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' ri = Λ(k) i 0 = Tr � ρ(k) (σi ⊗ σ0) � (18) sj = Λ(k) 0 j = Tr � ρ(k) (σ0 ⊗ σj) � (19) tij = Λ(k) i j = Tr � ρ(k) (σi ⊗ σj) � , i, j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (20) For a symmetric two-qubit density matrix, the Bloch vectors r and s are identical and hence ri = si, i = 1, 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' From the structure of ρ(k) in (7) and using (18), (19), (20) we obtain the general form of the real matrix Λ(k) as Λ(k) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 2(B(k)+E(k)) A(k)+2D(k)+F (k) 0 A(k)−F (k) A(k)+2D(k)+F (k) 2(B(k)+E(k)) A(k)+2D(k)+F (k) 2(C(k)+D(k)) A(k)+2D(k)+F (k) 0 2(B(k)−E(k)) A(k)+2D(k)+F (k) 0 0 2(D(k)−C(k)) A(k)+2D(k)+F (k) 0 A(k)−F (k) A(k)+2D(k)+F (k) 2(B(k)−E(k)) A(k)+2D(k)+F (k) 0 1 − 4D(k) A(k)+2D(k)+F (k) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (21) The elements of Λ(k), for different k, can be evaluated using (8), (9)): 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Lorentz canonical forms of Λ(k) Under SLOCC transformation, the two-qubit density matrix ρ(k) transforms to �ρ(k), ρ(k) −→ �ρ(k) = (A ⊗ B) ρ(k) (A† ⊗ B†) Tr � ρ(k) (A† A ⊗ B† B) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (22) Here, A, B ∈ SL(2, C) denote 2 × 2 complex matrices with unit determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' A suitable choice of A and B takes the two-qubit density matrix ρ(k) to its canonical form �ρ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Under the transformation ρ(k) −→ �ρ(k) (22) of the two-qubit state, its real representative Λ(k) transforms as [8, 9] Λ(k) −→ �Λ(k) = LA Λ(k) LT B � LA Λ(k) LT B � 00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (23) Here LA, LB ∈ SO(3, 1) are 4 × 4 proper orthochronous Lorentz transformation matrices [17] corresponding respec- tively to A, B ∈ SL(2, C) and the superscript ‘T ’ denotes transpose operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The Lorentz canonical form �Λ(k) of Λ(k) and thereby the SLOCC canonical form of the two-qubit density matrix ρ(k) (see (22)) can be obtained by constructing the 4 × 4 real symmetric matrix Ω(k) = Λ(k) G � Λ(k)�T , where G = diag (1, −1, −1, −1) denotes the Lorentz metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Using the defining property [17] LT G L = G of Lorentz transformation L, it can be seen that Ω(k) undergoes a Lorentz congruent transformation under SLOCC (upto an overall factor) [8] as Ω(k) → �Ω(k) A = �Λ(k) G � �Λ(k)�T = LA Λ(k) LT B G LB Λ(k)T LT A = LA Ω(k) LT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (24) It has been shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [8] that �Λ(k) can either be a real 4 × 4 diagonal matrix or a nondiagonal matrix with only one off-diagonal element, depending on the eigenvalues, eigenvectors of G Ω(k) = G � Λ(k) G � Λ(k)�T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (i) The diagonal canonical form �Λ(k) Ic results when the eigenvector X0 associated with the highest eigenvalue λ0 of G Ω(k) obeys the Lorentz invariant condition XT 0 G X0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The diagonal canonical form �Λ(k) Ic is explicitly given by Λ(k) −→ �Λ(k) Ic = LA1 Λ(k) LT B1 � LA1 Λ(k) LT B1 � 00 = diag � 1, � λ1 λ0 , � λ2 λ0 , ± � λ3 λ0 � , (25) where λ0 ≥ λ1 ≥ λ2 ≥ λ3 > 0 are the non-negative eigenvalues of G Ω(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The Lorentz transformations LA1, LB1 ∈ SO(3, 1) in (25) respectively correspond to SL(2, C) transformation matrices A1, B1 which take the two-qubit density matrix ρ(k) to its SLOCC canonical form �ρ(k) Ic through the transformation (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The diagonal form of �Λ(k) Ic readily leads, on using (15), to Bell-diagonal form �ρ(k) Ic = 1 4 \uf8eb \uf8edσ0 ⊗ σ0 + � i=1,2 � λi λ0 σi ⊗ σi ± � λ3 λ0 σ3 ⊗ σ3 \uf8f6 \uf8f8 (26) as the canonical form of the two-qubit state ρ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (ii) The Lorentz canonical form of Λ(k) turns out to be a nondiagonal matrix (with only one nondiagonal element) given by Λ(k) −→ �Λ(k) IIc = LA2 Λ(k) LT B2 � LA2 Λ(k) LT B2 � 00 = \uf8eb \uf8ec \uf8ed 1 0 0 0 0 a1 0 0 0 0 −a1 0 1 − a0 0 0 a0 \uf8f6 \uf8f7 \uf8f8 (27) 6 when the non-negative eigenvalues of GΩ(k) are doubly degenerate with λ0 ≥ λ1 and the eigenvector X0 belonging to the highest eigenvalue λ0 satisfies the Lorentz invariant condition XT 0 G X0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [8], it has been shown that when the maximum amongst the doubly degenerate eigenvalues of GΩ(k) possesses an eigenvector X0 satisfying the condition XT 0 G X0 = 0, the real symmetric matrix Ω(k) = Λ(k)G � Λ(k)�T attains the nondiagonal Lorentz canonical form given by Ω(k) IIc = �Λ(k) IIc G � �Λ(k) IIc �T = LA2 Ω(k) LT A2 = \uf8eb \uf8ec \uf8ed φ0 0 0 φ0 − λ0 0 −λ1 0 0 0 0 −λ1 0 φ0 − λ0 0 0 φ0 − 2λ0 \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (28) The parameters a0, a1 in (27) are related to the eigenvalues λ0, λ1 of GΩ(k) and the 00th element of �Λ(k) IIc, the canonical form of Ω(k) (see (28)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' It can be seen that [8] a0 = λ0 φ0 , a1 = � λ1 φ0 , where φ0 = � Ω(k) IIc � 00 = �� LA2 Λ(k) LT B2 � 00 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (29) The Lorentz matrices LA2, LB2 ∈ SO(3, 1) correspond to the SL(2,C) transformations A2, B2 which take the density matrix ρ(k) to its SLOCC canonical form ρ(k) IIc through the transformation (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The nondiagonl canonical form �Λ(k) IIc leads to the SLOCC canonical form �ρ(k) IIc of the two-qubit density matrix ρ(k) on using (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' �ρ(k) IIc = 1 2 \uf8eb \uf8ec \uf8ed 1 0 0 a1 0 1 − a0 0 0 0 0 0 0 a1 0 0 a0 \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (30) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Lorentz canonical form of Λ(1) corresponding to W-class of states Using the explicit structure of the two-qubit state ρ(1) given in (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' its real representative Λ(1) is obtained as (see (14)) Λ(1) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 2a √ 1−a2 1+a2(N−1) 0 1 + 2a2 1+a2(N−1) − 2 N 2a √ 1−a2 1+a2(N−1) 2(1−a2) N(1+a2(N−1)) 0 2a √ 1−a2 1+a2(N−1) 0 0 2(1−a2) N(1+a2(N−1)) 0 1 + 2a2 1+a2(N−1) − 2 N 2a √ 1−a2 1+a2(N−1) 0 1 + 4a2 1+a2(N−1) − 4 N \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = � Λ(1)�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (31) We now construct the 4 × 4 symmetric matrix Ω(1) and obtain Ω(1) = Λ(1) G � Λ(1)�T = Λ(1) G Λ(1) = χ \uf8eb \uf8ec \uf8ed N − 1 0 0 N − 2 0 −1 0 0 0 0 −1 0 N − 2 0 0 N − 3 \uf8f6 \uf8f7 \uf8f8 , χ = � 2(1 − a2) N (1 + a2(N − 1)) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (32) The eigenvalues of the matrix G Ω(1), G = diag (1, −1, −1, −1) are readily seen to be four-fold degenerate and are given by λ0 = λ1 = λ2 = λ3 = χ = � 2(1 − a2) N (1 + a2(N − 1)) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (33) 7 It can be seen that X0 = (1, 0, 0, −1) is an eigenvector of G Ω(1) belonging to the four-fold degenerate eigenvalue λ0 and obeys the Lorentz invariant condition XT 0 G X0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We notice here that Ω(1) is already in the canonical form (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' On comparing (32) with (28), we get φ0 = (Ω(1))00 = (N − 1)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (34) On substituting the parameters a0, a1 (See (29), (33), (34) in (27), we arrive at the Lorentz canonical form of the real matrix Λ(1) as �Λ(1) = \uf8eb \uf8ec \uf8ec \uf8ed 1 0 0 0 0 1 √N−1 0 0 0 0 − 1 √N−1 0 N−2 N−1 0 0 1 N−1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (35) It can be readily seen that �Λ(1), the Lorentz canonical form corresponding to the W-class of states, is independent of the parameter ‘a’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Lorentz canonical form of Λ(k), k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � The real representative Λ(k) given in (21) can readily be evaluated for different values of k (k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � ) on using (8), (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We then construct the real symmetric matrix Ω(k) = Λ(k) G � Λ(k)�T for k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � and observe that GΩ(k) = GΛ(k) G (Λ(k)) T has non-degenerate eigenvalues λ0 ̸= λ1 ̸= λ2 ̸= λ3 when k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � and the highest eigenvalue λ0 possesses a positive eigenvector X0 satisfying the relation XT 0 G X0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The Lorentz canonical form �Λ(k), k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � , is thus given by the diagonal matrix (see (25)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' �Λ(k) = diag � 1, � λ1/λ0, � λ2/λ0, ± � λ3/λ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The eigenvalues λµ, (µ = 0, 1, 2, 3) of GΩ(k) are dependent on the parameters ‘a’, k and N characterizing the state |DN−k, k⟩, when k takes any of the integral values greater than 1 and less than � N 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Hence the canonical form �Λ(k), k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � is different for different states |DN−k, k⟩ unlike in the case of �Λ(1), the canonical form of W-class of states, which depends only on the number of qubits N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' GEOMETRIC REPRESENTATION OF THE STATES |DN−k,k⟩ In this section, based on the two different canonical forms of Λ(k) obtained in Section III, we find the nature of canonical steering ellipsoids associated with the pure symmetric multiqubit states |DN−k,k⟩ belonging to SLOCC inequivalent families {DN−k, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' To begin with, we give a brief outline [8, 9] of obtaining the steering ellipsoids of a two-qubit density matrix ρ(k) based on the form of its real representative Λ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In the two-qubit state ρ(k), local projective valued measurements (PVM) Q > 0, Q = �3 µ=0 qµ σµ, q0 = 1, �3 i=1 q2 i = 1 on Bob’s qubit leads to collapsed state of Alice’s qubit characterized by its Bloch-vector pA = (p1, p2, p3)T through the transformation [8] (1, p1, p2, p3)T = Λ(k) (1, q1, q2, q3)T , q2 1 + q2 2 + q2 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (36) Notice that the vector qB = (q1, q2, q3)T , q2 1 + q2 2 + q2 3 = 1 represents the entire Bloch sphere and the steered Bloch vectors pA of Alice’s qubit constitute an ellipsoidal surface EA| B enclosed within the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' When Bob employs convex combinations of PVMs i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', positive operator valued measures (POVMs), to steer Alice’s qubit, he can access the points inside the steering ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Similar will be the case when Bob’s qubit is steered by Alice through local operations on her qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' For the Lorentz canonical form �Λ(k) Ic (see (25)) of the two-qubit state �ρ(k) Ic , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (36) leads to p1 = � λ1 λ0 q1, p2 = � λ2 λ0 q2, p3 = ± � λ3 λ0 q3, (37) 8 as steered Bloch points pA of Alice’s qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' They are seen to obey the equation λ0 p2 1 λ1 + λ0 p2 2 λ2 + λ0 p2 3 λ3 = 1 (38) of an ellipsoid with semiaxes ( � λ1/λ0, � λ2/λ0, � λ3/λ0) and center (0, 0, 0) inside the Bloch sphere q2 1 +q2 2 +q2 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We refer to this as the canonical steering ellipsoid representing the set of all two-qubit density matrices which are on the SLOCC orbit of the state �ρ(k) Ic (see (22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' For the second Lorentz canonical form �ΛIIc (see (27)) we get the coordinates of steered Alice’s Bloch vector pA, on using (36);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' p1 = a1 q1, p2 = −a1q2, p3 = (1 − a0) + a0q3, q2 1 + q2 2 + q2 3 = 1 (39) and they satisfy the equation p2 1 a2 1 + p2 2 a2 1 + (p3 − (1 − a0))2 a2 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (40) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (40) represents the canonical steering spheroid (traced by Alice’s Bloch vector pA) inside the Bloch sphere with its center at (0, 0, 1 − a0) and lengths of the semiaxes given by a1 = � λ1/φ0, a0 = � ��2/φ0 (see (29)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In other words, a shifted spheroid inscribed within the Bloch sphere, represents two-qubit states that are SLOCC equivalent to �ρ(k) IIc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Canonical steering ellipsoids of W-class of states We have seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' III B that the Lorentz canonical form of Λ(1), the real representative of the symmetric two- qubit state ρ(1) drawn from the W-class of states |DN−1,1⟩ has a nondiagonal form given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' On comparing (35) with the canonical form in (27), we get a1 = 1 √ N − 1, a0 = 1 N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (41) From (40) and the discussions prior to it, it can be readily seen that the quantum steering ellipsoid associated with �Λ(1) in (35) is a spheroid centered at (0, 0, 1 N−1) inside the Bloch sphere, with fixed semiaxes lengths ( 1 √N−1, 1 √N−1, 1 N−1) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' It is interesting to note that the Lorentz canonical form �Λ(1) is not dependent on the state parameter ‘a’, 0 ≤ a < 1 and hence all states |DN−1, 1⟩ in the family {DN−1, 1} are represented by an oblate spheroid, all its parameters such as center, semiaxes, volume etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', dependent only on the number of qubits N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (Colour online) Steering spheroids inscribed within the Bloch sphere representing the Lorentz canonical form �Λ(1) (see (35)) of W-class of states |DN−1,1⟩ for N = 4 and N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The spheroids are centered (0, 0, 1 N−1) and the length of the semi-axes are given by ( 1 √N−1, 1 √N−1, 1 N−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 9 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Canonical steering ellipsoids of the states |DN−k,k⟩, k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � As is seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' III C, the Lorentz canonical form of Λ(k), k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � , the real representative of the two- qubit states ρ(k) drawn from the pure symmetric N-qubit states |DN−k,k⟩, has the diagonal form (see (25)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The values of λ0, λ1, λ2, λ3, the eigenvalues of the matrix G Ωk can be evaluated for each value of k, k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � for a chosen N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' From (38) and the discussions therein, it follows that the canonical steering ellipsoids of the states |DN−k,k⟩, k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � is an ellipsoid centered at the origin of the Bloch sphere with lengths of the semiaxes given by � λ1/λ0, � λ2/λ0, � λ3/λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The eigenvalues λµ, µ = 0, 1, 2, 3 of GΩ(k) depend on the parameter ‘a’ also, unlike in the case of W-class of states where they depend only on N, the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Thus each state |DN−k,k⟩ belonging to the family {DN−k, k}, k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � is represented by an ellipsoid whose semiaxes depend on the values of k, N and ‘a’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 3 the canonical steering ellipsoids for some chosen values of k, N and ‘a’ are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (Colour online) Steering ellipsoids centered at the origin of the Bloch sphere representing the Lorentz canonical form of pure symmetric multiqubit states |DN−k,k⟩ (see (3)) when (i) N = 10, k = 2, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='2 and (ii) N = 10, k = 5, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (Colour online) Steering ellipsoids representing the Lorentz canonical form of pure symmetric multiqubit states |DN−k,k⟩ (see (3)) when (i) N = 100, k = 2, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='2 and (ii) N = 100, k = 5, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' VOLUME MONOGAMY RELATIONS FOR PURE SYMMETRIC MULTIQUBIT STATES |DN−k,k⟩ Monogamy relations restrict shareability of quantum correlations in a multipartite state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' They find potential ap- plications in ensuring security in quantum key distribution [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Milne et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [2, 3] introduced a geometrically intuitive monogamy relation for the volumes of the steering ellipsoids representing the two-qubit subsystems of mul- tiqubit pure states, which is stronger than the well-known Coffman-Kundu-Wootters monogamy relation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' In this 10 section we explore how volume monogamy relation [2] imposes limits on the volumes of the quantum steering ellip- soids representing the two-qubit subsystems ρ(k) = TrN−2 [|DN−k,k⟩⟨DN−k,k|] of pure symmetric multiqubit states |DN−k,k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' For the two-qubit state ρAB(= ρ(k)) (see (15)), we denote by EA| B, the quantum steering ellipsoid containing all steered Bloch vectors of Alice when Bob carries out local operations on his qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The volume of EA| B is given by [1] VB|A = �4π 3 � | det Λ| (1 − r2)2 , (42) where r2 = r · r = r2 1 + r2 2 + r2 3 (see (18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' As the steering ellipsoid is constrained to lie within the Bloch sphere, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', VB|A ≤ Vunit = (4π/3), one can choose to work with the normalized volumes vA|B = VA|B 4π/3 , the ratio of the volume of the steering ellipsoid to the volume of a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The volume monogamy relation satisfied by a pure three-qubit state shared by Alice, Bob and Charlie is given by [1–3] � VA|B + � VC|B ≤ � 4π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (43) where VA|B, VC|B are respectively the volumes of the ellipsoids corresponding to steered states of Alice and Charlie when Bob performs all possible local measurements on his qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The normalized form of the volume monogmay relation (43) turns out to be √vA|B + √vC|B ≤ 1, (44) where vA|B = VA|B 4π/3 are the normalized volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The monogamy relation (44) is not, in general, satisfied by mixed three-qubit states [3] and it has been shown that � vA|B � 2 3 + � vC|B � 2 3 ≤ 1, (45) is the volume monogamy relation for pure as well as mixed three-qubit states [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' As there are 1 2(N − 2)(N − 1) three qubit subsystems in a N-qubit state, each of which obey monogamy relation (45), on adding these relations and simplifying, one gets [3] � vA|B � 2 3 + � vC|B � 2 3 + � vD|B � 2 3 + · · · ≤ N − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (46) The relation (46) is the volume monogamy relation satisfied by pure as well as mixed N-qubit states .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' For N = 3, it reduces to (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' For multiqubit states that are invariant under exchange of qubits, vA|B = vC|B = vD|B = · · · = vN where vN denotes the normalized volume of the steering ellipsoid corresponding to any of the N − 1 qubits, the steering performed by, say Nth qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (46) thus reduces to (N − 1) (vN) 2 3 ≤ N − 1 2 =⇒ (vN) 2 3 ≤ 1 2 (47) implying that (vN) 2 3 ≤ 1 2 is the volume monogamy relation for permutation symmetric multiqubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Volume monogamy relations governing the W-class of states {DN−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='1} On denoting the normalized volume of a steering ellipsoid corresponding to the states |DN−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='1⟩ by v(1) N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' we have (see (42)) v(1) N = | det Λ(1)| (1 − r2)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (48) where Λ(1) is given in (31) and r1 = 2a √ 1 − a2 1 + a2(N − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' r2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' r3 = 1 + 2a2 1 + a2(N − 1) − 2 N (49) 11 Under suitable Lorentz transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' the real matrix Λ(1) (see (31)) associated with the state ρ(1) 2 gets transformed to its Lorentz canonical form �Λ(1) (see (35)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' It follows that (see (29), (33)) � LA Λ(1) LT B � 00 = � φ0 = 2 √ N − 1 � 1 − a2 N(1 + (N − 1) a2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (50) Using the property det LA = det LB = 1 of orthochronous proper Lorentz transformations [17] and substituting | det �Λ(1)| = 1 (N−1)2 in (23), we obtain | det �Λ(1)| = 1 (N − 1)2 = | det LA| | det LB| ����det � Λ(1) √φ0 ����� = | det Λ(1)| φ2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (51) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (51) leads to | det Λ(1)| = φ2 0| det �Λ(1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The normalized volume v(1) N of the steering ellipsoid corresponding to W-class of states thus becomes (see (48)) v(1) N = | det �Λ(1)| φ2 0 (1 − r2)2 (52) From (49) and (50) it readily follows that φ2 0 = (1 − r2)2 and hence (see (52)) the simple form for the normalized volume of the corresponding steering ellipsoid associated with the two-qubit state ρ(1) turns out to be v(1) N = φ2 0 (N − 1)2 (1 − r2)2 = 1 (N − 1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (53) The volume monogamy relation � v(1) N � 2 3 ≤ 1 2 (see (47)) takes the form � 1 (N − 1)2 �2/3 ≤ 1 2 =⇒ 2(N − 1) −4 3 ≤ 1 (54) and is readily satisfied for any N ≥ 3 as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content='4 N (N-1) 4 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (Colour online) The LHS of the monogamy relation 2(N − 1) −4 3 ≤ 1 is seen to be less than 1 for the states |DN−1, 1⟩ for any N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Relation between obesity of steering ellipsoids and concurrence We recall here that the obesity O(ρAB) = | det Λ|1/4 of the quantum steering ellipsoid [2] depicting a two-qubit state ρAB is an upper bound for the concurrence C(ρAB): C(ρAB) ≤ O(ρAB) = | det Λ|1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (55) 12 Furthermore, if ρAB −→ �ρAB = (A ⊗ B)ρAB (A† ⊗ B†)/(Tr(A† A ⊗ B†B)ρAB], A, B ∈ SL(2, C) it follows that [2] O(ρAB) C(ρAB) = O(�ρAB) C(�ρAB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (56) We make use of the relation (56) to obtain a relation for concurrence [21] of a pair of qubits in the symmetric N-qubit pure states |DN−k,k⟩, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' , � N 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' For the states |DN−1,1⟩ belonging to W-class, we readily get (see (31), (35)) det Λ(1) = � 2(1 − a2) N(1 + a2(N − 1)) �4 , det �Λ(1) = � 1 N − 1 �2 (57) and thereby the obesities O(ρ(1)), O(�ρ(1)): O(ρ(1)) = 2(1 − a2) N(1 + a2(N − 1)), O(�ρ(1)) = 1 √ N − 1 (58) As the concurrence of the state �ρ(1) turns out to be C(�ρ(1)) = O(�ρ(1)) = 1 √ N − 1 (59) we obtain (see (56),(59)) C(ρ(1)) = O(ρ(1)) = 2(1 − a2) N(1 + a2(N − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' (60) The value of concurrence in (60) matches exactly with that obtained [21] using C(ρ(1)) = max(0, µ1 − µ2 − µ3 − µ4) where µ1 ≥ µ2 ≥ µ3 ≥ µ4 are square-roots of the eigenvalues of the matrix R = ρ(1) (σ2 ⊗ σ2) ρ(1)∗ (σ2 ⊗ σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We have seen that the state |DN−1, 1⟩ reduces to W-state when a = 0 and hence for the N-qubit W-state, concurrence of any pair of qubits is given by C(ρ(1) W ) = 2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' SUMMARY In this work we have analyzed the canonical steering ellipsoids and volume monogamy relations of the pure symmetric N-qubit states characterized by two distinct Majorana spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' We have shown that the entire W-class of states has a geometric representation in terms of a shifted spheroid inscribed inside the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' The center of the spheroid, the length of its semiaxes and its volume are shown to be dependent only on the number of qubits N and hence all states in the N-qubit W-class are characterized by a single spheroid, shifted along the polar axis of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' All other families of pure symmetric N-qubit states with two distinct spinors which are SLOCC inequivalent to the W-class are geometrically represented by ellipsoids centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' A discussion on volume monogamy relations applicable to identical subsystems of a pure N-qubit symmetric state is given here and a volume monogamy relation applicable for W-class of states is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' A relation connecting concurrence of the two-qubit state and obesity of the associated quantum steering ellipsoid with its canonical counterparts is made use of to obtain concurrence of the states belonging to W-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' It would be interesting to examine the features of canonical steering ellipsoids and volume monogamy relations for the SLOCC inequivalent families of pure symmetric multiqubit states with more than two distinct spinors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' in particular, the class of pure symmetric N-qubit states belonging to GHZ-class (with three distinct spinors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' ACKNOWLEDGEMENTS BGD thanks IASC-INSA-NASI for the award of Summer Research Fellowship-2022, during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' Sudha, ARU and IR are supported by the Department of Science and Technology (DST), India through Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' DST/ICPS/QUST/2018/107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' [1] Jevtic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=', Pusey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfv_4W/content/2301.01714v1.pdf'} 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