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d6b09ed685b5fe316a680948618a4563a0d8bdc0 | subsection | 79 | 89 | Type | For r\in R, letD_{1,r}=\bigotimes _{2|n_{i}}(Q_{i,1}^{r_{i,1}}\otimes Q_{i,2}^{r_{i,2}}),\,\, D_{2,r}=\bigotimes _{2\nmid n_{i}}P_{i}^{r_{i}}, \text{ and } D_{r}=D_{1,r}\otimes D_{2,r}.Let L be the function field of the product \prod _{r\in R}\operatorname{SB}(D_{r}) of Severi-Brauer varieties \operatorname{SB}(D_{r}) ... | {
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6f3719e4a841a7582821d24515327e8e4c5abe51 | subsection | 80 | 89 | Type | Then, by and together with (REF ) we obtainC(A_{i},\sigma _{i})={\left\lbrace \begin{array}{ll} M_{2^{n_{i}-2}}(Q_{i,1})\times M_{2^{n_{i}-2}}(Q_{i,2}) & \text{ if } n_{i} \text{ even,}\\ M_{2^{n_{i}-3}}(P_{i})\times M_{2^{n_{i}-3}}(P_{i})^{^{\mathrm {op}}} & \text{ if } n_{i} \text{ odd},\end{array}\right.}thus by a t... | {
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8d31f326fea73d282846360e718d53852499cd5b | subsection | 81 | 89 | Type | Therefore, the Arason invariant induces a normalized invariant {\mathrm {e}}_{3}(\phi [r]) of order dividing 2 that sends an m-tuple in (REF ) to {\mathrm {e}}_{3}(\phi [r])\in H^{3}(K).Let r\in \bar{R}_{1}^{\prime \prime }+\bar{R}_{2}^{\prime \prime }, where \bar{R}_{1}^{\prime \prime }=\langle \bar{e}_{i}\in R^{\prim... | {
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abc270561a360a61417eed537864a6489879473a | subsection | 82 | 89 | Type | Then, by the proof of , there exists a G^{\prime }_{\operatorname{red}}(L^{\prime })-torsor \eta ^{\prime }:=(A_{i}^{\prime }) for some power series field L^{\prime }:=K((z)), whereA_{i}^{\prime }={\left\lbrace \begin{array}{ll} P_{i}^{\prime } & \text{ if } n_{i}\equiv 1 \mod {2},\\
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391342caed2aa0b9ef523369e44b81da331739f4 | subsection | 83 | 89 | Type | Let(A_{i}, \sigma _{i})={\left\lbrace \begin{array}{ll} \big (M_{n_{i}}(Q_{i}^{\prime }), \sigma _{\psi _{i}} \big ) & \text{ if } n_{i}\equiv 1 \mod {2},\\
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8635ae8f0631e4bc009c90d582c4db4ebc69981c | subsection | 84 | 89 | Type | By a direct calculation we have\phi _{i}={\left\lbrace \begin{array}{ll} T_{\gamma _{i}}\perp h & \text{ if } n_{i}\equiv 1 \mod {2},\\
T^{+}_{\gamma _{i1}\otimes \gamma _{i2}}\perp h\,\, (\text{resp.}\,\, T^{-}_{\gamma _{i1}\otimes \gamma _{i2}}\perp h) & \text{ if } n_{i}\equiv 2 \mod {8} \,\,(\text{resp.}\,\, n_{i}\... | {
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6065600df4c5e720fa8fdb5d846dbb4d1516b625 | subsection | 85 | 89 | Type | Let \alpha be a normalized invariant in \operatorname{Inv}^{3}(G_{\operatorname{red}}) be written as in (REF ) for some subsets I_{1}^{\prime }\subseteq I_{1}, I_{2}^{\prime }\subseteq I_{2} and R^{\prime \prime }\subseteq R^{\prime }.First, assume that there exists j\in I^{\prime }_{1}. Let Q=(x, y) be a division quat... | {
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d1b477120ba13bd4a0dbf6c9e78110997f2ab7a1 | subsection | 86 | 89 | Type | Then, we have either\partial _{x_{j,1}}\big (\alpha (\eta )\big )=\partial _{x_{j,1}}\big (\Delta _{j}^{\prime }(\eta )\big )\ne 0 \text{ or } \partial _{x_{j,2}}\big (\alpha (\eta )\big )=\partial _{x_{j,2}}\big (\Delta _{j}^{\prime }(\eta )\big )\ne 0,thus the invariant \alpha ramifies.Now we may assume that I_{1}^{\... | {
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4fdb169db521e6c52555356067fbb571b5b417e6 | subsection | 87 | 89 | The subgroup | In this section we will compute the subgroup \operatorname{Dec}(G) of decomposable elements of G for types B, C, and D. In this section we shall denote by T and T^{*} the maximal split torus of G and its character group, respectively and we denote by \Lambda and \Lambda _{r} the weight lattice and the root lattice of G... | {
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4185ee38be07658420652e31da93861349512bd4 | subsection | 88 | 89 | Unramified invariants for semisimple groups | In this section, we first describe torsors for the corresponding reductive groups in Lemmas REF , REF , and REF . Then, using this together with Theorems REF , REF , and REF , we present a complete description of the corresponding cohomological invariants in Propositions REF , REF , and REF . Finally, using such descri... | {
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773a23d36ce1f546f598e87224ee28a64c57a62c | abstract | 0 | 50 | Abstract | Numerical investigations are an important research tool in quantum
information theory. There already exists a wide range of computational tools
for quantum information theory implemented in various programming languages.
However, there is little effort in implementing this kind of tools in the Julia
language. Julia is ... | {
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47321a072202362a71377819f14f7338b089aaca | subsection | 1 | 50 | Abstract | Numerical investigations are an important research tool in
quantum information theory. There already exists a wide range of computational
tools for quantum information theory implemented in various programming
languages. However, there is little effort in implementing this kind of tools in
the Julia language. Julia is ... | {
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} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
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f40369ec2479b16e603ad5db9cd1f11526d8f8c3 | subsection | 2 | 50 | Introduction | Numerical investigations are prevalent in quantum information theory. Numerical
experiments can be used to find counter examples for theorems, to test
hypotheses or to gain insight about quantum objects and operations.The variety of software that supports investigations in quantum information
theory is very large. Yet ... | {
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2f62081173b9fedfd5a6ecd83e11a746ba4e2a88 | subsection | 3 | 50 | Related work | A comprehensive collection of software related to
quantum mechanics, computation and information can be found at Quantiki
—an on-line resource for quantum information research
community. There exist several notable libraries aimed at numerical and symbolic
computation for quantum information theory. Two Mathemetica
lib... | {
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5c30ac00bf2d71ca61e2304eee5c51dd990deb83 | subsection | 4 | 50 | Design principles | Our goal while designing QuantumInformation.jl library
was to follow the principles presented in the book “Geometry of Quantum
States”. We work with column vectors representing kets and row
vectors representing bras. We fix our basis to the computational one. Density
matrices and quantum channels are represented as two... | {
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4317984085114aa686826cf40cdc702998dc80e1 | subsection | 5 | 50 | Testing | The QuantumInformation.jl library was tested
using standard Julia framework. Tests where performed using three
distinct approaches. In case of most of the functions the basic properties, such
as e.g. dimensions, norms, hermititicty, positivity, trace are tested, where it was
appropriate. Additionally some test cases wh... | {
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8fb3755212c77ccf0b20a0a95be0426cda5ed214 | subsection | 6 | 50 | Organization of the paper | In the section Linear algebra in
Julia, we describe briefly how the linear algebra routines are implemented in Julia.
Next, in the section States and channels, we
introduce the notions of quantum states and quantum channels
and we discuss how we implement these concepts in Julia.
Subsequently, the section Functionals f... | {
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d877224ffd266f7b759b5b3753956cefd230c75e | subsection | 7 | 50 | Linear algebra in Julia | A basic construction of vector in Julia creates a full one-index array
containing elements of a number type as presented below.
julia> x = [0.0, 1.0im]
2-element ArrayComplexFloat64,1:
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0.0+1.0im
A transposition of a column vector returns an object of type
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0.008077801205217838,
... |
56801359343e4eaa8e6d66590003dd89769f741b | subsection | 8 | 50 | States and channels | In this and the following sections we will denote complex Euclidean spaces
d with \mathcal {X}, \mathcal {Y}, \mathcal {Z} etc. When needed the dimension of a space \mathcal {X}
will be denoted \mathrm {dim}(\mathcal {X}). The set of matrices transforming vectors
from \mathcal {X} to \mathcal {Y} will be denoted \mathr... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.022831853479146957,
0.0018629097612574697,
-0.012446107342839241,
-0.0005031668697483838,
-0.005440879613161087,
-0.009302148595452309,
-0.014018085785210133,
-0.0029455532785505056,
0.030233891680836678,
0.00722652580589056,
0.0027261630166321993,
0.039314743131399155,
0.0179022476077079... |
62d9433320f45dba77bff5024c81c7effc4d77de | subsection | 9 | 50 | States | By |\psi \rangle \in \mathcal {X} we denote a normed column
vector. Notice that any |\psi \rangle can be expressed as
|\psi \rangle =\sum _{i=1}^{n} \alpha _i |i\rangle , where \sum _{i=1}^{n}
|\alpha _i|^2=1 and the set \lbrace |i\rangle \rbrace _{i=1}^{n} is the computational
basis.julia> ket(1,2)
2-element ArrayComp... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.03536827117204666,
0.028761515393853188,
-0.06255820393562317,
0.04650668054819107,
0.008269889280200005,
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0.039793118834495544,
0.008895470760762691,
0.06329058855772018,
-0.013923014514148235,
0.002706405008211732,
0.036710985004901886,
0.017... |
78877f0573bb96a858545965422707dcddf48d6d | subsection | 10 | 50 | States | Generally, any quantum state \rho can be
expressed as \rho =\sum _{i=1}^{n} q_i |\psi _i\rangle \langle \psi _i|, where
\sum _{i=1}^{n}
q_i=1 and |\psi _i\rangle \langle \psi _i| are rank-one projectors.
Notice that \rho is a trace-one positive semi-definite linear operator
i.e.: \rho =\rho ^\dagger , \rho \ge 0 and \m... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.024995382875204086,
0.020509032532572746,
-0.011925453320145607,
-0.015587780624628067,
-0.010262145660817623,
0.01757916994392872,
0.016923002898693085,
0.06384657323360443,
0.0011034668423235416,
0.05404984578490257,
0.03839339688420296,
0.010430002585053444,
0.0003586028760764748,
-0... |
0087fbcf9b377e380e1e051ede85559a24be3ede | subsection | 11 | 50 | Non-standard matrix transformations | We will now introduce
reshaping operators, which map matrices to vectors and vice versa. We start with
the mapping
\mathrm {res}:\mathrm {L}(\mathcal {X,Y})\rightarrow \mathcal {Y}\otimes \mathcal {X}, which
transforms the matrix \rho into a vector row by row. More precisely, for
dyadic operators |\psi \rangle \langle ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.008977188728749752,
0.038868099451065063,
-0.03285788372159004,
0.049210552126169205,
0.015307745896279812,
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0.006517423316836357,
0.0425291433930397,
0.025444265455007553,
0.04075963795185089,
-0.027701910585165024,
-0.011852634139358997,
0.05482415482401848,
0.01... |
d4e2a47a9524bc0e036f485cc076fe90723865d8 | subsection | 12 | 50 | Non-standard matrix transformations | As this is a
linear map, it may be uniquely extended to the case of operators which are not
in a tensor product form.
julia> ρ = [0.25 0.25im; -0.25im 0.75]
2×2 ArrayComplexFloat64,2:
0.25+0.0im 0.0+0.25im
-0.0-0.25im 0.75+0.0imjulia> σ = [0.4 0.1im; -0.1im 0.6]
2×2 ArrayComplexFloat64,2:
0.4+0.0im 0.0+0.1im
-0.0-0.1i... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.057173773646354675,
0.005803611129522324,
-0.06623976677656174,
0.03528717905282974,
0.0029323308262974024,
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0.0011818991042673588,
0.07173431664705276,
0.03678291290998459,
0.02548857592046261,
-0.011385914869606495,
-0.024664394557476044,
0.026953786611557007,
0.... |
cdb5aaf33352c03b68f93e0a35cede76ccea2abd | subsection | 13 | 50 | Channels | Physical transformations of quantum states into quantum
states are called quantum channels i.e. linear Completely Positive
Trace Preserving (CP-TP) transformations. Probabilistic transformations of
quantum states are called quantum operations and mathematically they are defined
as linear Completely Positive Trace Non-i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/9781316848142",
"end": 779,
"openalex_id": "https://openalex.org/W2804172638",
"raw": "Watrous J. The Theory of Quantum Information. Cambridge University Press; 2018.",
"source_ref_id": "d8120052ecc11027b96ae0bac3292756f2f2097... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.03460258990526199,
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0.03460258990526199,
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0.058769479393959045,
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-0.022030925378203392,
0.005423818714916706,
0.01... |
2b12fba8d520f8a494e163ebae7c3757cc4be212 | subsection | 14 | 50 | Channels | For the operators that are not in a tensor product form this notion can be
uniquely extended from linearity.According to Kraus' theorem, any completely positive trace-preserving (CP-TP)
map \Phi can always be written as \Phi (\rho )=\sum _{i=1}^r K_i \rho K_i^\dagger for some set of operators \lbrace K_i\rbrace _{i=1}^... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/9781316848142",
"end": 1931,
"openalex_id": "https://openalex.org/W2804172638",
"raw": "Watrous J. The Theory of Quantum Information. Cambridge University Press; 2018.",
"source_ref_id": "d8120052ecc11027b96ae0bac3292756f2f209... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.0665261298418045,
-0.007625328842550516,
-0.020385071635246277,
0.0039023070130497217,
0.004722439683973789,
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0.046812426298856735,
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0.008926098234951496,
0.008544640615582466,
... |
9ed68196f32ba0bbafce96483c90e6b57308c5e4 | subsection | 15 | 50 | Channels | Let \Phi \in \mathrm {T}(\mathcal {X}, \mathcal {Y})
be a quantum channel which can be written in the Kraus representation as\Phi (\rho )=\sum _{i=1}^r K_i \rho K_i^\dagger ,where \lbrace K_i\rbrace _{i=1}^r are Kraus operators satisfying \sum _{i=1}^r K_i^\dagger K_i =
\mathbb {I}_\mathcal {X}. According to this assum... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.06391111016273499,
0.005043606273829937,
-0.015184230171144009,
0.002794584957882762,
-0.011765871196985245,
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-0.0031474849674850702,
0.007588299922645092,
-0.019777650013566017,
0.032779622822999954,
0.0033210734836757183,
0.003782704472541809,
-0.00697787851095199... |
fe84bd145aaea55b2190b249a1d7358db484c0f9 | subsection | 16 | 50 | Constructors | Channel objects can be constructed from matrices that represent them, as shown
in the following listing
julia> γ=0.4
0.4julia> K0 = Matrix([1 0; 0 sqrt(1-γ)])
2×2 ArrayFloat64,2:
1.0 0.0
0.0 0.774597julia> K1 = Matrix([0 sqrt(γ); 0 0])
2×2 ArrayFloat64,2:
0.0 0.632456
0.0 0.0julia> Φ = KrausOperators([K0,K1])
KrausOpe... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.0324631929397583,
0.026315322145819664,
-0.03786355257034302,
0.01916060596704483,
-0.012051654979586601,
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0.019313158467411995,
0.023691417649388313,
-0.02597970701754093,
0.03301238268613815,
0.022654060274362564,
-0.01978607103228569,
0.008619221858680248,
0.003... |
8594a0c6ccafa695057c2814077e1a0c9f83ffdd | subsection | 17 | 50 | Conversion | Conversions between all quantum channel types,
i.e. these that derive from |AbstractQuantumOperationT|
are implemented. The users are not limited by any single channel representation
and can transform between representations they find the most efficient or
suitable for their purpose.julia> Ψ1 = convert(SuperOperatorMat... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.06176120787858963,
0.01187774259597063,
-0.032253753393888474,
-0.010840252041816711,
-0.0013531242730095983,
0.004363563843071461,
0.0034424092154949903,
0.014006123878061771,
0.007895609363913536,
0.03030083142220974,
0.022595936432480812,
-0.011534455232322216,
0.031338322907686234,
... |
005c576c549f45c410c547ebd6f9f8977934b7df | subsection | 18 | 50 | Application | Channels can act on pure and mixed states
represented by vectors and matrices respectively. Channels are callable and therefore mimic
application of a function on a quantum state.
julia> ρ1=ψ * ψ'
2×2 ArrayComplexFloat64,2:
0.5+0.0im 0.5+0.0im
0.5+0.0im 0.5+0.0imjulia> Φ(ρ1)
2×2 ArrayComplexFloat64,2:
0.7+0.0im 0.3872... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.03396681323647499,
0.011947895400226116,
-0.036743976175785065,
0.007229010574519634,
-0.012016561813652515,
0.0333564467728138,
0.00933095533400774,
0.066163569688797,
-0.037873148918151855,
0.027191761881113052,
0.00304228812456131,
-0.02070663310587406,
0.0033951555378735065,
-0.0043... |
1c2bd62ad7be3bbea852e82723bb0c44a4336b8f | subsection | 19 | 50 | Composition | Channels can be composed in parallel or in
sequence. Composition in parallel is done using |kron()| function or
the overloaded \otimes operator. Composition in sequence can be done in two
ways either by using Julia built-in function composition operator
(f\circ g)(\cdot )=f(g)(\cdot ) or by using multiplication of obje... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.05406051129102707,
0.038156770169734955,
-0.02019256353378296,
0.027518663555383682,
0.020970961079001427,
0.029304400086402893,
-0.0013145006960257888,
0.045513395220041275,
-0.04508604109287262,
0.04133141413331032,
0.004388028755784035,
-0.037027329206466675,
0.007387150544673204,
0.... |
f84894551d198487a9e56b0a755c6ad64f06c035 | subsection | 20 | 50 | Trace norm and distance | Let \rho , \sigma \in \mathrm {L}(\mathcal {X}). The trace norm is defined as \Vert \rho \Vert _1 =
\mathrm {Tr} \sqrt{\rho \rho ^\dagger } and the trace distance is defined as
D_1(\rho ,\sigma )=\frac{1}{2}\Vert \rho -\sigma \Vert _1.
julia> ψ=(1/sqrt(2)) * (ket(1,2) + ket(2,2))
2-element ArrayComplexFloat64,1:
0.707... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.05351049825549126,
-0.013316573575139046,
-0.05003063380718231,
0.0015720425872132182,
-0.008211251348257065,
-0.017841920256614685,
-0.04667287319898605,
0.023382224142551422,
-0.004807702731341124,
0.026175271719694138,
0.012309245765209198,
0.018025070428848267,
0.003813729388639331,
... |
57a6fa1cf26caf20f2f123dde83b92cb944b8755 | subsection | 21 | 50 | Hilbert–Schmidt norm and distance | The Hilbert–Schmidt norm and distance defined by
\Vert \rho \Vert _{HS}=\sqrt{\mathrm {Tr}\rho ^\dagger \rho } and
D_{HS}(\rho ,\sigma )=\frac{1}{2}\Vert \rho -\sigma \Vert _{HS}, respectively, can be used as follows
julia> normhs(ρ)
0.9999999999999998julia> hsdistance(ρ, σ)
0.36602540378443854 | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.0699000209569931,
0.009966094978153706,
-0.04868582263588905,
0.016650857403874397,
-0.019169090315699577,
-0.016101425513625145,
-0.03150079771876335,
-0.009950833395123482,
0.025044964626431465,
0.0008885353454388678,
-0.015544361434876919,
0.02773107960820198,
0.026326974853873253,
-... |
6af5f287c3d8041cbbb9ff4370457a56aea14359 | subsection | 22 | 50 | Fidelity and superfidelity | Fidelity is a measure of distance of quantum states. It is an example of
a
distance measure which is not a metric on the space of quantum states. The
fidelity of two quantum states \rho , \sigma \in \mathrm {L}(\mathcal {X}) is
given by
F(\rho ,\sigma )=\Vert \sqrt{\rho }\sqrt{\sigma }\Vert _1
julia> fidelitysqrt(ρ, σ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.06201339513063431,
-0.001608892111107707,
-0.03744608163833618,
-0.025543903931975365,
-0.031647585332393646,
-0.03418061137199402,
0.030609959736466408,
0.02075251378118992,
-0.006542382761836052,
0.02072199620306492,
0.0061837914399802685,
0.015320238657295704,
-0.001933150109834969,
... |
355893be8c9667d596a72b5e5ed20cf5126fe98e | subsection | 23 | 50 | Diamond norm | In order to introduce the diamond norm, we first introduce the notion of
the
induced trace norm. Given \Phi \in \mathrm {T}(\mathcal {X}, \mathcal {Y}) we
define its
induced trace
norm as \Vert \Phi \Vert _1 = \mathrm {max} \left\lbrace \Vert \Phi (X) \Vert _1: X \in L(\mathcal {X}), \Vert X
\Vert _1 \le 1 \right\rbrac... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1109/hptcdl.2014.5",
"end": 1627,
"openalex_id": "https://openalex.org/W2001728872",
"raw": "Udell M, Mohan K, Zeng D, Hong J, Diamond S, Boyd S. Convex Optimization in Julia. SC14 Workshop on High Performance Technical Computing in Dyna... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.0748286098241806,
0.02255234308540821,
-0.0723261833190918,
0.009765560738742352,
-0.031677037477493286,
-0.03243997320532799,
-0.0379025824368,
0.016418348997831345,
0.024810628965497017,
0.04342623054981232,
-0.003461815183982253,
0.04306001961231232,
-0.035461194813251495,
0.03573584... |
621a478ef1d8b86c8427e1dc7b496d9954e95389 | subsection | 24 | 50 | Shannon entropy and von Neumann entropy | Shannon entropy
is defined for a probability vector p as H(\mathrm {p})=-\sum _{i=1}^n
p_i\log _2 p_i. We also provide an implementation for the point Shannon entropy.
It is defined as h(a) = -a \log a - (1-a)\log (1-a).
julia> p = [0.3, 0.2, 0.5]
3-element ArrayFloat64,1:
0.3
0.2
0.5julia> shannonentropy(p)
1.0296530... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.02034049481153488,
-0.005157604813575745,
-0.010383876040577888,
-0.001297984505072236,
-0.02783275581896305,
-0.0370187871158123,
0.05652002617716789,
0.05673365667462349,
-0.004417534451931715,
0.022919299080967903,
0.03326502814888954,
-0.005405566655099392,
-0.008026331663131714,
-0... |
7b29f714d73385ae5d5b22535301c195748a6610 | subsection | 25 | 50 | Distinguishability between two quantum states | One of the
measure of distinguishability between two quantum states is the quantum
relative entropy, called also Kullback–Leibler divergence, defined as
S(\rho \Vert \sigma )=-\mathrm {Tr}\rho \log \sigma + \mathrm {Tr}\rho \log \rho
julia> relativeentropy(ρ, σ)
0.11273751829075163julia> kldivergence(ρ, σ)
0.11273751... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.06402426958084106,
0.03747464343905449,
-0.019011972472071648,
-0.000051139497372787446,
-0.039977021515369415,
0.006202541757375002,
0.02778555639088154,
0.000407208688557148,
0.020781947299838066,
0.0574936717748642,
0.014060620218515396,
-0.002803731942549348,
-0.012008365243673325,
... |
d07f3bfd978fac5ab371918ab74c9cb88cb165bb | subsection | 26 | 50 | Quantum entanglement | One of the entanglement measures is negativity defined as
\mathrm {N}(\rho )=\frac{\Vert \rho ^{T_A}\Vert _1-1}{2}.
julia> negativity(ρ ⊗ σ, [2, 2], 2)
-0.0julia> negativity(proj((1/sqrt(2)*(ket(1,2) ⊗ ket(1,2)-ket(2,2) ⊗ ket(2,2)))), [2, 2], 2)
0.4999999999999999julia> lognegativity(ρ ⊗ σ, [2, 2], 2)
-1.1102230246251... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1103/physrevlett.78.5022",
"end": 780,
"openalex_id": "https://openalex.org/W2073781628",
"raw": "Hill S, Wootters WK. Entanglement of a pair of quantum bits. Physical Review Letters. 1997;78(26):5022.",
"source_ref_id": "f77aad08e... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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... |
2df5fcc1cd3522cee3cf331f747a0ae5d951d060 | subsection | 27 | 50 | Measurements | Measurements are modeled in two ways:as Positive Operator Valued Measures (POVMs),
measurements with post-selection.In both cases a measurement is treated as a special case of a quantum channel
(operation). | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.00... |
c07c297f0f839d4216926fa1b11873bc5e9fcc34 | subsection | 28 | 50 | Positive Operator Valued Measure measurement | A POVM measurement is
defined as follows. Let \mu :\Gamma \rightarrow \mathrm {P}(\mathcal {X}) be a mapping from
a finite alphabet of measurement outcomes to the set of linear positive
operators. If \sum _{\xi \in \Gamma } {\mu (\xi )=\mathbb {I}_{\mathcal {X}}} then
\mu is a POVM measurement. The set of positive semi... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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-0.0... |
cbbd67a98130f3551a46576cfc916cbe5b7992cf | subsection | 29 | 50 | Measurement with post-selection | When a quantum system after being
measured is not destroyed one can be interested in its state after the
measurement. This state depends on the measurement outcome. In this case the
measurement process is defined in the following way.Let \mu :\Gamma \rightarrow \mathrm {L}(\mathcal {X}, \mathcal {Y}) be a mapping from ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0... |
9295a79fdbde2817709f04d4d39dd10780e1c41b | subsection | 30 | 50 | Measurement with post-selection | The outcome \xi will be obtained with
probability (\rho _\xi ).
julia> PM = PostSelectionMeasurement(E1)
PostSelectionMeasurementArrayComplexFloat64,2
dimensions: (3, 3)
ComplexFloat64
[0.0+0.0im 0.0+0.0im 0.0+0.0im;
0.0+0.0im 1.0+0.0im 0.0+0.0im;
0.0+0.0im 0.0+0.0im 1.0+0.0im]julia> iseffect(PM)
truejulia> PM(ρ)
3×3 ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.... |
eb9d9878767bbe365f999c0e9a4ededb0c240471 | subsection | 31 | 50 | Random quantum objects | In this section we present the implementation of the sub-package
RandomMatrices. The justification for including these functionalities
in our package is twofold. First, the application of random matrix theory (RMT)
in quantum information is a blooming field of research with a plethora of
interesting
results , , , , , ,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1063/1.4936880",
"end": 328,
"openalex_id": "https://openalex.org/W1914973616",
"raw": "Collins B, Nechita I. Random matrix techniques in quantum information theory. Journal of Mathematical Physics. 2016;57(1):015215.",
"source_ref... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.009699156507849693,
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0.0... |
80808ab8e07656d35624beb6fa8e6fa29e73d04b | subsection | 32 | 50 | Ginibre matrices | In this section we introduce the Ginibre random matrices
ensemble . This ensemble is at the core of a vast
majority of algorithms for generating random matrices presented in later
subsections. Let (G_{ij})_{1 \le i \le m, 1 \le j \le n} be a m\times n
table of independent identically distributed (i.i.d.) random variabl... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1063/1.1704292",
"end": 67,
"openalex_id": "https://openalex.org/W1978864992",
"raw": "Ginibre J. Statistical ensembles of complex, quaternion, and real matrices. Journal of Mathematical Physics. 1965;6(3):440–449.",
"source_ref_id... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.05749345198273659,
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-0.021407460793852806,
0.031889... |
a5349a55b7b11e66d43cfdd6b1bd7aed43f3d4ec | subsection | 33 | 50 | Wishart matrices | Wishart matrices form an ensemble of random positive semidefinite matrices. They
are parametrized by two factors. First is the Dyson index \beta which is equal
to one for real matrices, two for complex matrices and four for symplectic
matrices. The second parameter, K, is responsible for the rank of the
matrices. They ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.04906122758984566,
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0.06730663776397705,
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0.01861153542995453,
-0.0026887566782534122,
0... |
ed23e0297c9d537f3262c10e6fa187ba9a6e9986 | subsection | 34 | 50 | Circular ensembles | Circular ensembles are measures on the space of unitary matrices. There are
three main circular ensembles. Each of this ensembles has an associated Dyson
index \beta Circular orthogonal ensemble (COE), \beta =1.
Circular unitary ensemble (CUE), \beta =2.
Circular symplectic ensemble (CSE), \beta =4.They can be chara... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1070/sm1967v001n04abeh001994",
"end": 212,
"openalex_id": "https://openalex.org/W2060581589",
"raw": "Mehta ML. Random matrices. vol. 142. Elsevier; 2004.",
"source_ref_id": "6fe3922ccf8c602d8f0da3e0e9f30f421e56e7cc",
"start"... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.06452754884958267,
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0.056408192962408066,
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0.009057971648871899,
0.003666684962809086,
0.01... |
6d27d365f938d65fbac80a4a96975df03a26b9d0 | subsection | 35 | 50 | Circular ensembles | For detailed analysis of this
algorithm see . This procedure can be generalized in
order to obtain a random isometry. The only required changed is the dimension of
G. We simply start with G \in \mathrm {L}(\mathcal {X}, \mathcal {Y}), where \dim (\mathcal {X})\ge \dim (\mathcal {Y}).Furthermore, we may introduce two ad... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 45,
"openalex_id": "https://openalex.org/W2964303560",
"raw": "Mezzadri F. How to generate random matrices from the classical compact groups. Notices of the American Mathematical Society. 2007;54(5):592 – 604.",
"source_ref_... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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e2e30a3beb3006dfbe9ef5afd21a6e11fc33b8c5 | subsection | 36 | 50 | Random quantum states | In this section we discuss the properties and methods of generating random
quantum states. We will treat quantum channels as a special case of quantum
states. | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.0076510487124323845,
0.01... |
957cc05d1c0b874c6f4dfa0f801c33d0a8c63186 | subsection | 37 | 50 | Pure states | Pure states are elements of the unit sphere in \mathcal {X}. Thus it is straightforward
to generate them randomly. We start with a vector of \dim (\mathcal {X}) independent
complex numbers sampled from the standard normal distribution. What remains is
to normalize the length of this vector to unity.This is implemented ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.026715559884905815,
0.011473501101136208,... |
791b880c30c197bf685c7c802f5d3ba1be67dff0 | subsection | 38 | 50 | Mixed states | Random mixed states can be generated in one of two equivalent ways. The first
one comes from the partial trace of random pure states. Suppose we have a pure
state {\psi } \in \mathcal {X}\otimes \mathcal {Y}. Then we can obtain a random mixed as\rho = _\mathcal {Y}{\psi }{\psi }.Note that in the case \dim (\mathcal {X}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1172,
"openalex_id": "",
"raw": "Wootters WK. Random quantum states. Foundations of Physics. 1990;20(11):1365–1378.",
"source_ref_id": "3d958d4e3c0d0c325755d715ba22c622c6df172a",
"start": 1092
},
{
"arxiv... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.0064061484299600124,
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... |
efff1d8f6b86bb1c79755bb1a3e58d9b4b1a129d | subsection | 39 | 50 | Random quantum channels | Quantum channels are a special subclass of quantum states with constraints
imposed on their partial trace as well as trace. Formally, we start with
a Ginibre matrix G \in \mathrm {L}(\mathcal {X}\otimes \mathcal {Y}, \mathcal {Z}). We obtain a random
Choi-Jamiołkowski matrix J_\Phi corresponding to a channel \Phi asJ_\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.physleta.2008.11.043",
"end": 657,
"openalex_id": "https://openalex.org/W2057238140",
"raw": "Bruzda W, Cappellini V, Sommers HJ, Życzkowski K. Random quantum operations. Physics Letters A. 2009;373(3):320–324.",
"source_ref... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0... |
1b57ed53ffdd814b6e6fd37a8ab3008b41247304 | subsection | 40 | 50 | Example | As an example we provide the teleportation protocol in the
presence of noise. Imagine we have an entangled pair of particles in the state{\psi } = \frac{1}{\sqrt{2}} \left( {00} + {11} \right).One of the particles stays with Alice and another is sent through a noisy
channel to Bob. As a noise model we chose the amplitu... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
-0.029949728399515152,
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-0.... |
5e0eb42b319e28175f4a210c0660944fc3e9771d | subsection | 41 | 50 | Benchmarks | In the benchmarks we compare our library to the state-of-the-art Python
library, QuTiP , . We perform the following tests:sampling a random unitary matrix,
sampling a random pure state,
sampling a random mixed state,
sampling a random channel,
calculating the trace distance of a random mixed state from the maximall... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.cpc.2012.02.021",
"end": 90,
"openalex_id": "https://openalex.org/W2118492081",
"raw": "Johansson J, Nation P, Nori F. QuTiP: An open-source Python framework for the dynamics of open quantum systems. Computer Physics Communication... | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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-0.029207414016127586,... |
fa485565cf1b860dde1a9312d2e5011016777f78 | subsection | 42 | 50 | Sampling a random unitary matrix | The Julia code for this test is
using QuantumInformation
function randomunitary(steps::Int, d::Int)
dist = CUE(d)
for i=1:steps U = rand(dist) end
end
The Python implementation reads
import qutip as q
def randomunitary(steps, d):
for in range(steps):
q.randunitaryhaar(d)
The benchmark results are presented in Fig. ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.022974343970417976,
-0.017604511231184006,
-0... |
958ade8ef727fb3336e0fd504240df821957b2f1 | subsection | 43 | 50 | Sampling a random pure state | The Julia code for this test is
using QuantumInformation
function randompurestate(steps::Int, d::Int)
dist = HaarKet(d)
for i=1:steps ψ = rand(dist) end
end
The Python implementation reads
import qutip as q
def randompurestate(steps, d):
for in range(steps):
q.randkethaar(d)
The benchmark results are presented in F... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.010253860615193844,
-0.0033740848302841187,
-0.... |
1b08627efe122a220feafb48b70dca10687ba9e4 | subsection | 44 | 50 | Sampling a random mixed state | The Julia code for this test is
using QuantumInformation
function randommixedstate(steps::Int, d::Int)
dist = HilbertSchmidtStates(d)
for i=1:steps ρ =rand(dist) end
end
The Python implementation reads
import qutip as q
def randommixedstate(steps, d):
for in range(steps):
q.randdmhs(d)
The benchmark results are pre... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.00... |
f1cd35d2592779ae0fb934be947b2a7744f64191 | subsection | 45 | 50 | Sampling a random channel | The Julia code for this test is
using QuantumInformation
function randomchannel(steps::Int, d::Int)
dist = ChoiJamiolkowskiMatrices(round(Int, sqrt(d)))
for i=1:steps Φ = convert(SuperOperatorMatrixComplexF64, rand(dist)) end
end
The Python implementation reads
import qutip as q
def randomchannel(steps, d):
for in r... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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28a066e499bf48ca82c0322120f3bb30efac4d9c | subsection | 46 | 50 | Calculating the trace distance form the maximally mixed state | The Julia code for this test is
using QuantumInformation
function tracedistancemaxmixed(steps::Int, d::Int)
dist = HilbertSchmidtStates(d)
ρ = 𝕀(d)/d
for i=1:steps tracedistance(rand(dist), ρ) end
end
The Python implementation reads
import qutip as q
def tracedistancemaxmixed(steps, d):
rho = q.Qobj(np.eye(d) / d)
... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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42e25d4594bde12fa468a94229dbb29afe8e58d0 | subsection | 47 | 50 | Calculating the trace distance between two random mixed states | The Julia code for this test is
using QuantumInformation
function tracedistancerandom(steps::Int, d::Int)
dist = HilbertSchmidtStates(d)
for i=1:steps tracedistance(rand(dist), rand(dist)) end
end
The Python implementation reads
import qutip as q
def tracedistancerandom(steps, d):
for in range(steps):
q.metrics.trac... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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0.... |
ecd008152221d4777220b9699b3b68f9d609d59a | subsection | 48 | 50 | Calculating the entropy of the stationary state of a random
channel | The Julia code for this test is
using QuantumInformation
function randomunitary(steps::Int, d::Int)
dist = CUE(d)
for i=1:steps U = rand(dist) end
end
The Python implementation reads
import qutip as q
def randomunitary(steps, d):
for in range(steps):
q.randunitaryhaar(d)
The benchmark results are presented in Fig. ... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
"quant-ph"
] | 2,018 | en | Physics | [
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56fd76dfeeebf10ef9f90c822a8c30e52f84dc52 | subsection | 49 | 50 | Conclusions and future work | Numerical investigations are important
part of research in many fields of science, especially in quantum information.
The Julia language is a modern programming language, which provides
strong support for linear algebra and posses an extensive type system. One of
the important feature of Julia is high performance appro... | {
"cite_spans": []
} | 10.1371/journal.pone.0209358 | 1806.11464 | QuantumInformation.jl---a Julia package for numerical computation in
quantum information theory | [
"Piotr Gawron",
"Dariusz Kurzyk",
"Łukasz Pawela"
] | [
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da13278cbcbbf624f678a5d43b8af5c31282f694 | abstract | 0 | 19 | Abstract | Detecting anomalous behavior in network traffic is a major challenge due to
the volume and velocity of network traffic. For example, a 10 Gigabit Ethernet
connection can generate over 50 MB/s of packet headers. For global network
providers, this challenge can be amplified by many orders of magnitude.
Development of nov... | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
"Chansup Byun",
"Matthew Hubbell",
"Micheal Houle",
"Micheal Jones",
"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
"Albert ... | [
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d37cf762aa7ba64cbf7c4c928aa999bdf2f513e8 | subsection | 1 | 19 | Introduction | The rapid rise of sophisticated cyber threats is well documented and a growing threat to our information systems , . Understanding internet phenomenology is challenging due to the variety of new threats and volume of new data being generated. To demonstrate the variety challenges, there are approximately 250,000 new ma... | {
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b90f3ead05f8cc32bcb7eeb4cf05fe815dd4cc10 | subsection | 2 | 19 | Introduction | Using MIT SuperCloud and D4M together allows cyber network analysts to overcome the scalability wall shown in Figure REF .
[Figure: D4M and MIT SuperCloud allows high performance without significantly compromising coding effort]The rest of the article is organized as follows: Section describes the tools used to develo... | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
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"William Arcand",
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"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
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1005e0b13189ac3299b117c24fa17d6c2080e2a6 | subsection | 3 | 19 | Tools | The scalable architecture described in this article leverages prior
work on D4M, Associative Arrays and the MIT SuperCloud computing platform. | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
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"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
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d5b6f85c815e44bc897ad2a18b01274c7e8efb00 | subsection | 4 | 19 | D4M | The Dynamic Distributed Dimensional Data Model (D4M) is a software
library developed at MIT Lincoln Laboratory that is used in a number
of applications for processing large amounts of data. D4M is made up
of three components:Support for a mathematical data object called associative arrays;
A schema that is used to rep... | {
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83c7b1ecb0cfcbe1ad03f53498afcb545696f773 | subsection | 5 | 19 | Associative Arrays | Associative Arrays generalize matrices to better match the intuitions
of spreadsheets, databases, and tables, all while supporting the power
of linear algebra. The indices of an associative array can range over
arbitrary (though usually totally ordered and finite) sets, while the
entries of an associative array may lie... | {
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93d8e24617ae074c122807cc5f2ed5e70a3928ac | subsection | 6 | 19 | MIT SuperCloud | The MIT SuperCloud is a high performance computing environment
developed at the Massachusetts Institute of Technology. Unlike
traditional supercomputing systems that are tuned for large-scale batch
processing, the MIT SuperCloud is designed for data scientists
interested in iterative analysis of machine learning and A... | {
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cb1f8d48975371d2834ffe868aa4cd83a3db5728 | subsection | 7 | 19 | Developing Scalable Pipeline and Analytics | Developing network analytics involves developing a pipeline that can
scale with the massive amount of data collected by packet capture
devices. This section details the network data used and computational pipeline developed. | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
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e49ac4296d1350922999e820d407a6e48b0c5371 | subsection | 8 | 19 | IP Network Traces | IP network packets form the basic unit in which information is
transmitted across the internet. An individual network packet
consists of a header and payload. The header consists of typical
information that one needs to correctly route a particular
packet such as source IP, destination IP, etc. Headers are typically
40... | {
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"Andrew Prout",
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fc291cc0832c11693f59f30179c122c20241b12c | subsection | 9 | 19 | Computing Pipeline | Figure REF describes the pipeline used to extract, store and
process packet capture data described in
Section REF .
[Figure: Analytics pipeline used for processing network packet capture data.].Each step of this pipeline is described below:Uncompress: Data from packet capture appliances is
often written in a binary com... | {
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569e2e51fab7b2087e75b866c9f020d9d0130c90 | subsection | 10 | 19 | Experimental Results | In order to test the pipeline of Figure REF ,
we use data products made available by the MAWI working group. The dataset we use called a “Day in the
Life” (DITL) internet traces, consists of 4 days (96 hours) of 1 Gigabit packet capture
headers collected on two days in 2015 and two days in 2017.In total, the raw data i... | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
"Chansup Byun",
"Matthew Hubbell",
"Micheal Houle",
"Micheal Jones",
"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
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0c63e9e68c4aa7f20920a6974f02d1c3d81289f0 | subsection | 11 | 19 | Step1: Uncompress Raw Data | In this step, we take 385 compressed input files (corresponding to the
number of computing nodes used in the experiment) and convert them to
385 uncompressed output files. Each input and output file corresponds
with roughly 15 minutes of network flows. Each 2GB file expands
to 6GB after uncompressing which translates t... | {
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} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
"Chansup Byun",
"Matthew Hubbell",
"Micheal Houle",
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"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
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15b85521de6c3b7186d332d6428a29a2e414b5bf | subsection | 12 | 19 | Step 2: Split Uncompressed Files | Once the uncompressed output files are generated from the previous step,
we split these files into smaller chunks in order to optimize later steps in
the pipeline. We first use tcpdump to convert the 385 binary
.pcap files into ASCII versions, then split these files into
into approximately 500,000 smaller output .pcap ... | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
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"Matthew Hubbell",
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"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
"Albert ... | [
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3a9add699fe4105f268b95450feda7e23727037f | subsection | 13 | 19 | Step 3: Parse Split Files | With the approximately 500,000 split and uncompressed .pcap files, we convert
these files into a human readable format using a tool such as
tshark . Using tshark, we convert these
.pcap into a tab separated value (TSV) format while also
filtering the headers for the fields shown in
Section REF . Each output TSV file at... | {
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} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
"Chansup Byun",
"Matthew Hubbell",
"Micheal Houle",
"Micheal Jones",
"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
"Albert ... | [
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5af3b0216c932642387a06ac96bbfa7220876344 | subsection | 14 | 19 | Step 4: Dense Array Construction (Sort) | In order to convert the 500,000 files in the previous step to a format
amenable for further processing, we use D4M to convert these TSV files
into associative array format (which also sorts the data
during construction). Each of the 5 MB input files expands to roughly
50 MB (total of 20 TB) during this step. At this po... | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
"Chansup Byun",
"Matthew Hubbell",
"Micheal Houle",
"Micheal Jones",
"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
"Albert ... | [
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24632118ea2b690ec893a2b9f069a5865b3559f0 | subsection | 15 | 19 | Step 5: Graph Construction (Sparse) | With the dense associative arrays from the previous step stored on
disk, we can now, in parallel, generate the sparse version of the
network graph. This sparse representation directly corresponds to the
incidence matrix of the graph. Each of the 50 MB input associative
array is converted to a sparse representation usin... | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
"Chansup Byun",
"Matthew Hubbell",
"Micheal Houle",
"Micheal Jones",
"Peter Michaleas",
"Julie Mullen",
"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
"Albert ... | [
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4071bfbd2b83b13f0e79996e59062e71ed91190d | subsection | 16 | 19 | Step 6: Ingest | With the sparse data products of the previous step, it is easy to use
D4M to directly insert this data into Apache Accumulo. Our prior work
has demonstrated that Accumulo is capable of extremely high ingest
rates suitable for applications such as internet traffic analysis. In
our experiment, we create various Accumulo ... | {
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"Vijay Gadepally",
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8659ade9c07c049d795989185293265fc7f4da2f | subsection | 17 | 19 | Performance Analysis | To assess the performance of each step, the D4M code included
timers. For each of the first five steps of the pipeline, the time
measured includes the time for reading the file from disk, performing
the operation and writing the file back to disk. For the insertion
step, we measure the time taken to load the file and i... | {
"cite_spans": []
} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
"Vijay Gadepally",
"Jeremy Kepner",
"Lauren Milechin",
"William Arcand",
"David Bestor",
"Bill Bergeron",
"Chansup Byun",
"Matthew Hubbell",
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"Andrew Prout",
"Antonio Rosa",
"Charles Yee",
"Siddharth Samsi",
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4b0abf45c56e31e0e4f52b95df7d9f5ed1f68e70 | subsection | 18 | 19 | Conclusions | Network and cyber security of the future will largely rely on massive quantities of data. Internet network analysis will continue to be challenged by the fast pace of analytic changes coupled with massive quantities of data. In order to address these challenges, it is important that researchers leverage high level prog... | {
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} | 1808.08353 | Hyperscaling Internet Graph Analysis with D4M on the MIT SuperCloud | [
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94a8305ddfdc60423e2516ae1ac5d1a8e9304b1c | abstract | 0 | 77 | Abstract | A step forward is made in a long standing Lov\'{a}sz's problem regarding
hamiltonicity of vertex-transitive graphs by showing that every connected
vertex-transitive graph of order a product of two primes, other than the
Petersen graph, contains a Hamilton cycle. Essential tools used in the proof
range from classical re... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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30ad1ec8310877208cd06612c1a4094d0110460a | subsection | 1 | 77 | Introduction | The following question asked by Lovász in 1970
tying together traversability and symmetry,
two seemingly unrelated graph-theoretic concepts,
remains unresolved after all these years.Problem 1.1
Does every finite connected vertex-transitive graph have a Hamilton path?No connected vertex-transitive graph without a Hami... | {
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b92474489e7f56ed5e12a5752342dba431f69c74 | subsection | 2 | 77 | Introduction | Despite the fact that, somewhat paradoxically, it is precisely the above problem that is responsible for much of the work directed towards obtaining such structural results by
opening up new research directions.
Such is the case for example with the so-called polycirculant
conjecture which states that every vertex-tran... | {
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primes | [
"Shaofei Du",
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824d249451bdafcdc5010e85b463f931c41019f3 | subsection | 3 | 77 | Introduction | Here we will use an approach which combines a variety
of graph-theoretic and number-theoretic tools.
For example, we will use the fact that
the polycirculant conjecture has been settled for
certain vertex-transitive graphs. In particular, a vertex-transitive graph
of order pq, p>q, contains a fixed-point-free automorph... | {
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... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
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270fdde4b620e95f19ca35ce10bb22f07d9b4a72 | subsection | 4 | 77 | Basic definitions and notation | Throughout this paper graphs are finite, simple and undirected,
and groups are finite, unless specified otherwise.
Furthermore, a multigraph is a generalization
of a graph in which we allow multiedges and loops.
Given a graph X we let V(X) and E(X) be the
vertex set and the edge set of X, respectively.
For adjacent ver... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
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a4c2de3ce9d8f55e81bf5a769418715a244b50c7 | subsection | 5 | 77 | Generalized orbital graphs | In this subsection we recall the orbital graph construction which
is used throughout the rest of the paper.
Orbital graphs can be constructed for any group
action but in view of the fact that
any transitive action of a group G is isomorphic to
the action of G on the coset space of
a subgroup of G, we give this construc... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
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891a4da10d2a619c34f4dfdfbd5da68a6236aed2 | subsection | 6 | 77 | Generalized orbital graphs | In the action of G on G/H\times G/H there
are two non-diagonal orbitals, both self-paired.
The corresponding
nontrivial suborbits of H are, respectively, of length 3 and 6,
giving the Petersen graph and its complement. | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
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f847a432b4a37c0e0fd7658cac30f41d019584f1 | subsection | 7 | 77 | Semiregular automorphisms and quotient (multi)graphs | Let m\ge 1 and n\ge 2 be integers. An automorphism \rho
of a graph X is called (m,n)-semiregular
(in short, semiregular)
if as a permutation on V(X) it has a cycle decomposition consisting
of m cycles of length n.
If m=1 then X is called a circulant; it is
in fact a Cayley graph of a cyclic group of order n.
Let \math... | {
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0.0033750564325600863,
0.... | |
252fd3ab2a38c1d02164c2ba79681a9b243f1529 | subsection | 8 | 77 | Semiregular automorphisms and quotient (multi)graphs | The symbol n/R, where R \subseteq \mathbb {Z}_n\setminus \lbrace 0\rbrace ,
inside the circle corresponding to
the orbit S_i indicates that for each j\in \mathbb {Z}_n,
the vertex v_i^j is adjacent to all the vertices v_i^{j+r}, where r \in R.
When X\langle S_i \rangle is an independent set of vertices
we simply write ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4153/cjm-1982-020-8",
"end": 1217,
"openalex_id": "https://openalex.org/W2333628352",
"raw": "B. Alspach and T. D. Parsons, A construction for vertex-transitive graphs, Canad. J. Math. 34 (1982), 307–318.",
"source_ref_id": "2b23bb... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.047415487468242645,
0.003619473660364747,
-0.04119106009602547,
0.040763892233371735,
-0.0025496503803879023,
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0.019802220165729523,
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0.06450214982032776,
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-0.029062582179903984,
0.014775386080145836,
0... | |
780bcf0b5e84aafa625a1dd3b92828a584f6709b | subsection | 9 | 77 | Semiregular automorphisms and quotient (multi)graphs | The second class consists of the so-called Fermat graphs ,
mentioned in the introduction as the class of
vertex-transitive graphs of order pq admitting an
imprimitive subgroup of automorphisms with blocks of size q
and no imprimitive subgroup of automorphisms with blocks of size p,
where p is the largest of the two pri... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 416,
"openalex_id": "",
"raw": "D. Marušič and R. Scapellato, Characterizing vertex-transitive pq-graphs with an imprimitive subgroup of automorphisms, J. Graph Theory 16 (1992), 375–387.",
"source_ref_id": "df5f9ec28857ecc5... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.0458541065454483,
-0.002328171394765377,
-0.016657579690217972,
0.017740627750754356,
-0.00852709449827671,
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0.0004990981542505324,
0.006891082040965557,
0.0... | |
111834aedbfe1a2ec7fc9d68ee6357cf61f015cd | subsection | 10 | 77 | Some group-theoretic terminology | For group-theoretic terms not defined here
we refer the reader to .
The following classical groups appear in the description of
primitive group actions, of degree
a product of two distinct primes,
which do not have imprimitive subgroups
(see Table REF ):P\Omega ^\epsilon (2d,2): the projective orthogonal group of a vec... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 67,
"openalex_id": "https://openalex.org/W2798943694",
"raw": "H. Wielandt, Permutation groups, Academic Press, New York, 1966.",
"source_ref_id": "8a2cf968027c31fecc14e57ca6fb71bcd0903446",
"start": 0
}
]
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.023462051525712013,
-0.0008661928586661816,
-0.00002353785021114163,
-0.013668399304151535,
0.016368519514799118,
-0.04121876880526543,
0.016246479004621506,
-0.004847247619181871,
0.006044758018106222,
0.02038056030869484,
-0.004225610289722681,
-0.04610033705830574,
-0.03176072239875793... | |
8439f13d5f391c6d6c45861e2f9b8b8ed1f2f728 | subsection | 11 | 77 | Useful number theory facts | For a prime power r a finite field \hbox{\rm GF}(r) of order r
will be denoted by F_r,
with the subscript r being omitted whenever the order of the field
is clear from the context.
The set of nonzero
quadratic residues modulo r, that is, elements of F^*=F\setminus \lbrace 0\rbrace
that are congruent to a perfect squar... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 777,
"openalex_id": "https://openalex.org/W1524896264",
"raw": "J. H. Silverman, A friendly introduction to number theory, 4th Edition, Pearson, 2012.",
"source_ref_id": "b4822283ae13d8dcee46f1c5cd1464f5378cf4b4",
"sta... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.01909085549414158,
0.014505082741379738,
-0.04532361403107643,
-0.02516452595591545,
0.008057531900703907,
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0.06623046845197678,
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0.015504644252359867,
0.04556778073310852,
-0.028765998780727386,
-0.007462373003363609,
-0.046544451266527176,
0.0... | |
3db3b372ac4414d1cc412f9749f1d72d723f48c6 | subsection | 12 | 77 | Useful number theory facts | Then |(A\setminus B) \cup (B \setminus A)|\ge 2, that is,
|A\cup B|\ge |A|+2.Proof.
First observe that there must exist three consecutive elements
s-1, s, s+1 of the field F
such that s-1 and s are squares but s+1 is not.
Therefore s\in S^*\cap N^*-1 but s\notin S^*\cap N^*+1,
and so s\in B\setminus A. But then, since ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.01724346913397312,
0.0161600299179554,
-0.04745768755674362,
-0.007439107168465853,
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0.024064557626843452,
0.0027467471081763506,
-0.00007808677764842287,
-0.0362570621073246,
-0.011376111768186092,
-0.030458373948931694,
... | |
e40f0e90bf7f3e2a03e433286b59efcf7e9be772 | subsection | 13 | 77 | Theorems about existence of Hamilton cycles | In this subsection we list three results about existence of Hamilton
cycles in particular graphs that will prove useful in the subsequent sections.
A recent result about the existence of
Hamilton paths in those generalized Petersen graphs which do
not have a Hamilton cycle
is also given as it will be needed in Subsecti... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 490,
"openalex_id": "",
"raw": "V. Chvátal, On Hamilton's Ideals, J. Combin. Theory 12 (1972), 163–168.",
"source_ref_id": "7283946c8c906b77e6c82b1c348e5733990acc7c",
"start": 328
},
{
"arxiv_id": "",
... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.045162711292505264,
-0.01089397817850113,
-0.01588323712348938,
0.009612333960831165,
0.01009295042604208,
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0.0008139014826156199,
0.011305934749543667,
0.004916784819215536,
0.006461624056100845,
-0.008521409705281258,
0.01681395433843136,
-0.005828430876135826,
0.... | |
e03ab28d536edbcc4f37780da41226e493861063 | subsection | 14 | 77 | Theorems about existence of Hamilton cycles | If n \equiv {2\pmod {6}} and \lbrace x, y\rbrace \ne \lbrace v_i, v_{i+4+6t} \rbrace ,
for any i and t, then there is a Hamilton path joining x and y in
GP(n, 2).
If n \equiv {4\pmod {6}} and \lbrace x, y\rbrace is none of the pairs
\lbrace u_i, u_{i+2}\rbrace , \lbrace u_i, v_{i\pm 1}\rbrace , \lbrace u_i, v_{i+2+6t}... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.02996106445789337,
-0.0027268535923212767,
0.00852761510759592,
0.02550656907260418,
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-0.01583481952548027,
-0.005812202580273151,
-0.030388208106160164,
... | |
00f90dd0a520d26fb250d9ba1f8ffb4d4187b40f | subsection | 15 | 77 | Polynomials of degree | In early eighties, motivated by a question posed by Alspach, Heinrich and Rosenfeld
in the context of decompositions of complete symmetric digraphs,
Madden and Vélez investigated polynomials that represent quadratic residues at primitive roots. They proved that,
with finally many exceptions,
for any finite field F of ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf02761904",
"end": 246,
"openalex_id": "https://openalex.org/W1987000845",
"raw": "B. Alspach, K. Heinrich and M. Rosenfeld, Edge partitions of the complete symmetric directed graph and related designs, Israel J. Math. 40 (1981), 1... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.012391476891934872,
-0.012093898840248585,
-0.06647447496652603,
-0.030993953347206116,
0.019670706242322922,
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0.0762411579489708,
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0.005291557405143976,
-0.005108432378619909,
-0.016710182651877403,
-0.018648257479071617,
-0.028704887256026268,
... | |
581608fe493a611ca4a30abb48d6c812864e4f99 | subsection | 16 | 77 | Polynomials of degree | Thend(k(m)+1,m)-c_r(k(m)+1,m)>1.Proof.
Since 2\le k(m)\le \frac{m}{2}-1, we have m\ge 6.
Let \Omega be the increasing sequence of all prime numbers.
For a given prime q, let
\mathcal {I}_q=\lbrace w_1=2, w_2, w_3, \ldots , w_{k(m)}=q, w_{k(m)+1},\ldots , w_m\rbrace
be a subsequence of \Omega not missing any prime in... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.02078373171389103,
0.04107915237545967,
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0.015549649484455585,
0.011749980971217155,
0.00580250658094883,
-0.008995601907372475,
-0.027375929057598114,
-0.0320454016327858,
-0.03827136382460594,
0.02... | |
9cbc91aca5c76cbfc598329849a68e7094421e94 | subsection | 17 | 77 | Polynomials of degree | Note that this is true if w_m\ge 13, which is the case since m\ge 7.
Next, note that for either m being even and l<\frac{m}{2}-2 or m being odd,
(REF ) holds.
So we may assume that m is even and that
l=m/2-1\ge 2.Now we prove that (REF ) holds under this assumption
for any even integer m\ge 8 by induction. Suppose fir... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2140/pjm.1982.98.123",
"end": 1336,
"openalex_id": "https://openalex.org/W1966377157",
"raw": "D. J. Madden and W. Y. Vélez, Polynomials that represent quadratic residues at primitive roots, Pacific J. Math. 98 (1982), 123–137.",
"... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.017486942932009697,
0.0338447131216526,
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0.012665064074099064,
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0.013664535246789455,
-0.02627619169652462,
0.0... | |
a46ac4665c7c12db8be3e2ddee351c3c1aab0f3b | subsection | 18 | 77 | Polynomials of degree | Then there exist s and t such thats and t are coprime,
a prime q divides p-1 if and only if q divides st, and
2\phi (t)/t>1+(4s-2)\sqrt{p}/(p-1)+(4s+2)/(p-1).Proof.
Since m\le 2k(m)+1 the four cases (i) - (iv) of Proposition REF
need to be considered. In each case, as in ,
we will prescribe a choice for s (which the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2140/pjm.1982.98.123",
"end": 508,
"openalex_id": "https://openalex.org/W1966377157",
"raw": "D. J. Madden and W. Y. Vélez, Polynomials that represent quadratic residues at primitive roots, Pacific J. Math. 98 (1982), 123–137.",
"s... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.009933114983141422,
0.0020655845291912556,
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0.0013017187593504786,
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0.015410823747515678,
0.009559288620948792,
0.029951928183436394,
-0.03503292426466942,
0.0020751208066940308,
-0.04592730849981308,
0.... | |
0e63b205012f9d6a8a50bc28b66168c75bb33e77 | subsection | 19 | 77 | Polynomials of degree | Since q_i\ge 5 for i\in \lbrace 3,4,\ldots , m-1\rbrace one can see that
either m=3 or m=4. In other words, either t=q_3 or t=q_3q_4, and thus
we can improve the value for \alpha with2\frac{\phi (t)}{t}-1\ge 2(1-\frac{1}{5})(1-\frac{1}{131}) -1 \ge 0.58778.In this case p satisfies (REF )
with \alpha =0.58778 if and onl... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.013184191659092903,
0.001206452725455165,
-0.02288922108709812,
-0.03044266439974308,
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0.036561716347932816,
0.011024975217878819,
0.002731447108089924,
0.036561716347932816,
-0.036561716347932816,
0.010582449845969677,
-0.05056992173194885,
0... | |
b3200666e316e26b239c712d57f7fc7e08c59bf9 | subsection | 20 | 77 | Polynomials of degree | \rule {2.5mm}{3mm}In order to proceed with the proof of
Theorem REF we now need to
identify all those sequences \lbrace 2=q_1,q_2,\ldots ,q_m\rbrace
with q_m<131 for which one cannot choose s=q_1q_2\cdots q_n
and t=q_{n+1}q_{n+2}\cdots q_m so as to satisfy (REF ).
Since Lemma REF holds for each q_m
we can assume that ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.030693411827087402,
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0.034280162304639816,
-0.013133605942130089,
0.006688904948532581,
-0.030647624284029007,
... |
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