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[EQUATION] We choose for all logarithms to be base-2, following Tomamichel_12_Framework The von Neumann entropy is [EQUATION] The min entropy is defined as |
[EQUATION] [MATH] denotes the greatest eigenvalue of [MATH] The max entropy is [EQUATION] The Schatten 1-norm is denoted by [MATH] The general Schatten [MATH] -norm of a Hermitian operator |
[MATH] is [EQUATION] for [MATH] Bhatia_97_Matrix [MATH] reflects the discrepancy between [MATH] and the maximally mixed state Tomamichel_12_Framework, , p. 60) The fidelity between normalized states |
[MATH] and [MATH] is [MATH] [MATH] depends on the fidelity through: [MATH] We notate the detector state’s Rényi entropies as [EQUATION] |
following Tomamichel_12_Framework We have now introduced the forward-POVM entropies. The two-detector state [MATH] and the entropy [MATH] are defined analogously. |
[MATH] and [MATH] , like [MATH] quantify rates at which information-processing and thermodynamic tasks can be performed. Applications include quantum key distribution, randomness extraction, erasure, work extraction, and work expenditure (e.g., Renner_05_Security delRio_11_Thermodynamic Tomamichel_12_Framework Berta_13... |
Smoothing introduces an error tolerance [MATH] into the entropies Renner_05_Security Our uncertainty relations for scrambling generalize to smooth entropies. We focus on nonsmooth entropies for simplicity. |
III D Entropic uncertainty relations for QI scrambling We can now reconcile the two notions of quantum operator disagreement, entropic uncertainty relations of pure QI theory and information scrambling of high-energy and condensed-matter theory. |
Theorem 1 The forward and reverse POVMs satisfy entropic uncertainty relations for scrambling, [EQUATION] for [MATH] The bound depends on the OTOC quasiprobability: |
[EQUATION] The real numbers [MATH] and the rest of the [MATH] terms, depend essentially on classical probabilities. Their forms are given below. |
The [MATH] and [MATH] dependences of the [MATH] ’s have been suppressed for conciseness. Inequality ( 25 ) can be smoothed when [MATH] |
The uncertainty relations are proved in App. They follow from three general uncertainty relations: Result 7 in Tomamichel_12_Framework Corollary 2.6 in Krishna_01_Entropic , and Ineq. (13) in Rastegin_08_Uncertainty The OTOC POVMs ( 13 ) and ( 14 are substituted into the general uncertainty relations. The POVMs’ maximu... |
[EQUATION] We substitute in for the [MATH] ’s from Eq. ( 12 ), then multiply out. In each of several terms, two [MATH] ’s contribute [MATH] ’s, while two [MATH] ’s contribute [MATH] ’s. These terms contain quasiprobaiblity values [MATH] We isolate the terms by Taylor-expanding the logarithm in the [MATH] ’s. |
III E Analysis Four points merit analysis: the POVMs’ implications for the butterfly effect, the form of the bound [MATH] simple limits, and conditions that render the bound nontrivial. |
Implications for the butterfly effect: The weak measurements strengthen an analogy between the OTOC and the butterfly effect of classical chaos Shenker_Stanford_14_BHs_and_butterfly Roberts_16_Lieb Aleiner_16_Microscopic Campisi_16_Thermodynamics In the classical butterfly effect, a tiny perturbation snowballs into a d... |
[MATH] should be associated with a weak measurement, Theorem clarifies. The measurement is perturbative in [MATH] Form of the uncertainty bound [MATH] |
for scrambling: The bound ( ) contains three terms dependent on the quasiprobability [MATH] These terms’ proportionality to [MATH] |
accords with intuition: Scrambling is a subtle feature of quantum equilibration, detectable in just many-point correlators. Likewise, the OTOC quasiprobability governs high-order terms in the uncertainty bound. As anticipated in Sec. II the quasiprobability [MATH] is evaluated on the identity operator. The bound highli... |
The quasiprobability-free terms in ( are “background terms”: They contain classical probabilities, accessible without weak measurements. The [MATH] -independent term, |
[EQUATION] dominates [MATH] The Kronecker delta is denoted by [MATH] The two linear terms, [EQUATION] depend on projectors [MATH] |
only through classical probabilities [MATH] This [MATH] equals the conditional probability that, if the system begins maximally mixed over the [MATH] eigenspace of [MATH] if [MATH] is measured, outcome [MATH] will obtain. Such classical dependence characterizes also the [MATH] terms suppressed in Eq. ( ), |
[EQUATION] The dominance of [MATH] , the [MATH] in [MATH] and the min ensure that [MATH] throughout the min’s argument. The first [MATH] has four arguments, |
[MATH] constrained only by the [MATH] In each other [MATH] the first argument must equal the third, even before the minimization is imposed. For example, the second quasiprobability value has the form |
[MATH] The [MATH] eigenvalues equal each other, due to Ineq. ( 27 ). One [MATH] comes from the [MATH] and one, from the [MATH] Nontriviality conditions: |
The Rényi entropies are nonnegative: [MATH] Hence the bound is nontrivial when positive: [MATH] When the coupling is weak, the bound is positive when its first term is positive. The first term simplifies to |
[MATH] The trace is large in the system size, equaling [MATH] in the spin-chain example. One might worry that this trace swells the log, drawing the bound far below zero. |
The probabilities [MATH] can offset the enormity. Let us focus on the spin-chain example and approximate [MATH] Nonnegativity of the log term becomes equivalent to |
[MATH] or [MATH] Strongly measuring a weak-measurement detector must yield one of [MATH] possible outcomes. Weak measurements as in Aharonov_88_How |
satisfy this requirement. Let each detector manifest as a particle, e.g., in a potential that defines a dial. Let [MATH] denote the strongly measured detector observable (e.g., the position [MATH] ). Let [MATH] denote the conjugate observable (e.g., the momentum [MATH] ): |
[MATH] Let be prepared in a Gaussian state that peaks sharply at some [MATH] eigenvalue (e.g., a sharp momentum-space wave packet). The probabilities [MATH] can be small enough that [MATH] We present an example in Sec. IV |
The [MATH] -free log encodes randomness in a measurement of a detector that has never coupled to the system. Hence the log fails to reflect disagreement between [MATH] and [MATH] The disagreement manifests in the [MATH] -dependent terms. |
Simple limits: Three simple limits illuminate the bound’s behavior: early times ( [MATH] ), late times ( [MATH] ), and the weak limit ( [MATH] ). We focus on a chaotic spin chain, for concreteness. Numerical simulations (Sec. IV support these arguments. |
Early times [MATH] ): [MATH] and [MATH] nontrivially transform just far-apart subsystems. Hence [MATH] Also, [MATH] so the projectors nearly commute. Hence |
[MATH] These traces are large, dragging the [MATH] terms in Eq. ( ), and the negative term in ( 30 ), below zero. The [MATH] ’s mitigate the dragging’s magnitude. Still, the bound is expected to be relatively loose before [MATH] |
Late times [MATH] ): [MATH] can fail to commute with [MATH] Traces [MATH] will shrink: Consider a one-qubit system, as a simple illustration. Suppose that [MATH] and that [MATH] Each [MATH] |
translates roughly into a [MATH] The traces’ smallness tightens the uncertainty bound, as expected when the system is scrambled (as explained in the introduction). |
The bound likely does not remain at its maximum possible value at all [MATH] , however. As [MATH] evolves, the bound should fluctuate around a relatively large value. |
Weak limit [MATH] ): The system fails to couple to the detectors. The bound ( ) reduces to [MATH] The probability distribution [MATH] |
has a spread quantified by the Shannon entropy [MATH] The left-hand side of Ineq. ( 24 ) reduces to [MATH] IV Numerical simulations of a spin chain |
We illustrate Theorem with an interacting spin chain. The setup and weak-measurement implementation are described in Sec. IV A The detector probabilities [MATH] the weak-measurement Kraus operators [MATH] the couplings [MATH] and the entropies [MATH] |
are presented in Sec. IV B and calculated in App. We present and analyze results in Sec. IV C IV A Spin-chain setup Consider a one-dimensional (1D) chain of [MATH] qubits. The OTOC operators manifest as single-qubit Pauli operators: |
[MATH] , and [MATH] The operators’ precise forms do not impact our chaotic-system results, however. Model: The chain evolves under the power-law quantum Ising Hamiltonian |
[EQUATION] Swingle_18_Resilience (see Chen_17_SubsystemDiff for a similar model). Each spin [MATH] interacts with each spin that lies within a distance [MATH] The interaction strength declines with distance as a power law controlled by [MATH] We choose [MATH] [MATH] , and [MATH] as in Swingle_18_Resilience Planck’s con... |
flips from site to site. The transverse-field Ising model with a longitudinal field reproduces our results’ qualitative features. But the power-law quantum Ising model mimics all-to-all interactions, such as in the SYK model Sachdev_93_Gapless Kitaev_15_Simple Polchinski_16_Spectrum Maldacena_16_Remarks Around [MATH] ,... |
Weak-measurement implementation: Section III E guides our implementation, which parallels Aharonov_88_How We illustrate with the forward-protocol weak measurement, temporarily reinstating operators’ hats. |
The detector consists of a particle that scatters off the system. The detector could manifest as a photon, as in circuit QED deLange_14_Reversing |
and in purely photonic experiments Lundeen_11_Direct Let [MATH] denote the longitudinal direction, which points from the detector’s initial position to the system. |
Let [MATH] denote a transversal direction; and [MATH] , the [MATH] component of the detector’s initial state. [MATH] consists of a Gaussian, |
[EQUATION] centered on the transverse-momentum eigenvalue [MATH] [MATH] denotes the Gaussian’s standard deviation. The displaced detector position [MATH] |
couples to the system’s [MATH] [The displacement prevents the minimization in ( from choosing the detector-measurement outcome [MATH] This choice would set [MATH] to [MATH] eliminating the weak measurement.] The interaction unitary has the form |
[EQUATION] The interaction strength [MATH] governs the outcome-dependent coupling [MATH] Numerical experiments show that [MATH] and [MATH] |
keep [MATH] perturbatively small while strengthening the bound. The detector’s [MATH] is measured strongly. Let [MATH] denote the measurement’s precision. Positions [MATH] and [MATH] can be distinguished if they lie a distance [MATH] apart. Hence the classical register has a discrete spectrum [MATH] We simulated a regi... |
IV B Analytical ingredients in spin-chain uncertainty relation Analytical results are presented here: the detector probability [MATH] the weak-measurement Kraus operators |
[MATH] the coupling strengths [MATH] and the entropies [MATH] We derive these results and check their practicality in App. Consider preparing the detector in [MATH] then measuring [MATH] The measurement has a probability |
[MATH] of yielding a position within [MATH] of [MATH] By Eq. ( 32 ), [EQUATION] The weak-measurement Kraus operators have the form |
[EQUATION] The outcome-dependent coupling is [EQUATION] The Rényi- [MATH] entropy limits, as [MATH] , to [EQUATION] The other entropies have analogous forms. |
Entropies [MATH] Let us remove operators’ hats. We illustrate the entropies’ analytical forms with [EQUATION] The measurement operators have the form |
[EQUATION] by Eq. ( 13 ). We substitute in from Eq. ( 12 ), multiply out, and substitute into Eq. ( 40 ): [EQUATION] The other entropies have analogous forms. |
IV C Spin-chain results Figures illustrate the entropic uncertainty relations for information scrambling [Ineqs. ( 24 and ( 25 )] in the characteristic parameter regime detailed in Sec. IV B Time is measured in units of the inverse coupling, [MATH] The scrambling time [MATH] as reflected by (i) the quasiprobability’s s... |
Figure shows the greatest time-dependent contributions to the bound [MATH] [Eq. ( )]. Choosing [MATH] tightens the bound (see App. ), so we focused on [MATH] and [MATH] The bound grows at [MATH] , confirming expectations: At the scrambling time, the OTOC drops. A decayed OTOC reflects noncommutation of [MATH] and [MATH... |
Figure shows the quasiprobability’s contribution to the uncertainty bound ( ). Figure shows the LHS of Ineq. ( 24 [MATH] ), the LHS of Ineq. ( 25 at [MATH] |
[MATH] ), and the shared RHS [MATH] . Figure is more zoomed-out than Fig. hence the tightening is too small to detect. This reduced visibility is expected: Scrambling is a subtle, high-order stage of quantum equilibration. It manifests in the [MATH] terms of [MATH] , just as |
[MATH] can be inferred from high-order terms in weak-measurement experiments NYH_17_Jarzynski NYH_18_Quasi The LHSs lie [MATH] bits above the bound. The gap stems from the |
[MATH] in Eq. ( ). This gap bodes ill for the large-system limit, [MATH] of interest in holography. But the gap scales only linearly, not exponentially, with [MATH] Furthermore, small gaps would follow from many of today’s experiments (e.g., Li_16_Measuring ). Additionally, Sec. VI presents weak-measurement entropic un... |
Figure illustrates how tight the bound can grow in an exceptional parameter regime. The top curves represent [MATH] and [MATH] These curves dip at [MATH] because (i) [MATH] is a [MATH] eigenstate and (ii) the POVMs’ [MATH] measurements are fine-grained—are replaced with measurements of |
[MATH] The POVM outcomes become highly predictable around [MATH] , so the bound grows tight to within 0.53 bits. Our numerics emphasize the scrambling Hamiltonian [MATH] which is nonintegrable. Integrable Hamiltonians’ OTOCs revive and decay repeatedly, as information recollects from across the system and spreads again... |
Extension to higher-point OTOCs Higher-point OTOCs reflect later, subtler stages of QI scrambling and many-body equilibration. [MATH] has been generalized to the [MATH] -fold OTOC |
Roberts_16_Chaos Haehl_17_Classification Haehl_17_Thermal NYH_18_Quasi Dressel_18_Strengthening Haehl_18_Fine [EQUATION] We follow the notation in NYH_18_Quasi This [MATH] -point correlator is labeled by |
[MATH] The conventional OTOC corresponds to [MATH] If [MATH] the correlator encodes [MATH] time reversals, as concretized in Schwinger-Keldysh path integrals Haehl_17_Classification |
and in the weak-measurement scheme NYH_17_Jarzynski NYH_18_Quasi Higher-point OTOCs [MATH] equilibrate at later times [MATH] Haehl_18_Fine |
and can be inferred from sequences of [MATH] weak measurements. [MATH] equals a coarse-graining of a quasiprobability distribution |
[MATH] NYH_18_Quasi [MATH] governs terms [MATH] in an entropic uncertainty relation for scrambling. Denote the eigenvalues of [MATH] |
by [MATH] Denote the eigensubspace projectors by [MATH] The forward POVM consists of a weak measurement of [MATH] followed by a weak measurement of [MATH] and so on, until a weak measurement of [MATH] followed by a strong measurement of [MATH] The reverse POVM consists of a strong measurement of [MATH] followed by a we... |
The weak measurement of an observable [MATH] is represented by a Kraus operator [MATH] The [MATH] denotes the weak measurement’s outcome, |
[MATH] denotes the detector probability, and [MATH] denotes the outcome-dependent weak-coupling strength. The von Neumann uncertainty relation has the form |
[EQUATION] The term [EQUATION] contains the quasiprobability behind the [MATH] -fold OTOC. Hence our entropic uncertainty relations extend to arbitrary-point OTOCs. |
VI Entropic uncertainty relations for weak values beyond scrambling Weak values , like OTOCs, involve time reversals and measurement sequences Aharonov_88_How Dressel_14_Understanding Consider preparing a quantum system in a state [MATH] at a time [MATH] evolving the system for a time [MATH] under a unitary [MATH] meas... |
[MATH] and obtaining the outcome [MATH] Let [MATH] denote a nondegenerate observable that fails to commute with [MATH] Which value can most reasonably be attributed, retrodictively, to the [MATH] at a time [MATH] given that [MATH] was prepared and that the measurement yielded [MATH] The weak value |
[EQUATION] is the expectation value conditioned on the preselection and postselection. [MATH] and [MATH] denote time-evolved states. |
Consider eigendecomposing [MATH] then factoring out the sum and eigenvalues. Multiplying the numerator and denominator by [MATH] yields |
[EQUATION] wherein [MATH] denotes a conditioned probability. The numerator is a Kirkwood-Dirac quasiprobability Kirkwood_33_Quantum Dirac_45_On an extension of which is the OTOC quasiprobability NYH_18_Quasi The Kirkwood-Dirac quasiprobability governs the conditional quasiprobability |
[MATH] that, if [MATH] is prepared and the [MATH] measurement yields [MATH] [MATH] is the value most reasonably attributable to [MATH] retrodictively. |
[MATH] generalizes to arbitrary initial states [MATH] and to degenerate observables [MATH] and [MATH] [EQUATION] The time-evolved state |
[MATH] and the conditional probability [MATH] One can infer [MATH] experimentally by preparing [MATH] , evolving the system for a time [MATH] measuring [MATH] weakly, evolving the system for a time [MATH] and measuring [MATH] strongly. One performs this protocol in many trials. |
[MATH] is inferred from the measurement statistics. [MATH] can range outside the spectrum of [MATH] as advertised in the foundational paper Aharonov_88_How Hence the physical significances of [MATH] have galvanized debate (e.g., Ferrie_14_How Vaidman_14_Comment Cohen_14_Comment Aharonov_14 Sokolovski_14_Comment Brodutc... |
and disturbances by measurements Dressel_12_Significance Kirkwood-Dirac quasiprobabilities have been interpreted in terms of operator decompositions Lundeen_11_Direct Lundeen_12_Procedure |
and Bayesian retrodiction Aharonov_88_How Johansen_04_Nonclassical Hall_01_Exact Hall_04_Prior Dressel_15_Weak We introduce another physical significance: Weak values govern first-order-in- [MATH] terms in entropic uncertainty bounds for POVMs that involve weak measurements. Kirkwood-Dirac quasiprobabilities play an an... |
VI A Entropic uncertainty relations for weak values and Kirkwood-Dirac quasiprobabilities Consider a quantum system associated with a Hilbert space [MATH] Let [MATH] denote any state of the system. Let [MATH] |
[MATH] and [MATH] be eigenvalue decompositions of observables. (The index [MATH] signifies “initial” and should not be confused with [MATH] .) |
The uncertainty relation for [MATH] features a POVM that we label I. One measures [MATH] weakly, then [MATH] strongly: [MATH] The weak-measurement Kraus operator |
[MATH] The [MATH] signifies terms of second order in the Hamiltonian’s coupling parameter (e.g., the [MATH] in the spin-chain example of Sec. IV ). We define as POVM II a strong measurement of [MATH] |
[MATH] Define the entropies [MATH] , and [MATH] via analogy with the QI-scrambling entropies (Sec. III C ). One can infer the weak value |
[EQUATION] by preparing the state [MATH] measuring [MATH] weakly, and postselecting a strong [MATH] measurement on [MATH] Theorem 2 |
POVMs I and II obey entropic uncertainty relations dependent on the weak value [MATH] [EQUATION] The bound has the form [EQUATION] |
The Rényi orders [MATH] and [MATH] satisfy [MATH] , and [MATH] denotes an arbitrary state. The proof is analogous to the proof of Theorem The forward and reverse POVMs are replaced with POVMs I and II. One can prove analogous uncertainty relations in which Kirkwood-Dirac quasiprobabilities replace [MATH] The weak measu... |
VI B Qubit example Let us illustrate the uncertainty relation ( 52 for [MATH] The system, denoted by a subscript [MATH] consists of a qubit. So does the detector, denoted by [MATH] Let [MATH] [MATH] , and |
[MATH] The weak measurement manifests as follows: The detector begins in the state [MATH] [MATH] -controlled [MATH] couples the system to the detector weakly, and the detector’s [MATH] is measured strongly. The weak values [MATH] |
are imaginary and so nonclassical Dressel_12_Significance [MATH] has only real eigenvalues [MATH] but the conditioned average [MATH] is imaginary. |
We illustrate the uncertainty relation’s LHS with [MATH] The inequality is calculated in App. [MATH] If [MATH] , as in Sec. IV the relation approximates to [MATH] The bound is satisfied and is tight at order [MATH] |
VII Discussion We have reconciled two measures of disagreement between quantum operators: entropic uncertainty relations and out-of-time-ordered correlators (OTOCs). The reconciliation unites several subfields of physics: (i) quasiprobabilities and weak measurements tie (ii) quantum information theory to (iii) condense... |
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