Simulating symmetric time evolution with local operations

Toloui, Borzu; Gour, Gilad

doi:10.1088/1367-2630/14/12/123026

Cited by 12 publications

(13 citation statements)

References 30 publications

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“…(Note, however, that the list we provide is not complete; there are many known measures of asymmetry that we do not review here. See, e.g., [4,9,76,77]. )…”

Section: Methods For Deriving Measures Of Coherencementioning

confidence: 99%

How to quantify coherence: Distinguishing speakable and unspeakable notions

2016

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Quantum coherence is a critical resource for many operational tasks. Understanding how to quantify and manipulate it also promises to have applications for a diverse set of problems in theoretical physics. For certain applications, however, one requires coherence between the eigenspaces of specific physical observables, such as energy, angular momentum, or photon number, and it makes a difference which eigenspaces appear in the superposition. For others, there is a preferred set of subspaces relative to which coherence is deemed a resource, but it is irrelevant which of the subspaces appear in the superposition. We term these two types of coherence unspeakable and speakable, respectively. We argue that a useful approach to quantifying and characterizing unspeakable coherence is provided by the resource theory of asymmetry when the symmetry group is a group of translations, and we translate a number of prior results on asymmetry into the language of coherence. We also highlight some of the applications of this approach, for instance, in the context of quantum metrology, quantum speed limits, quantum thermodynamics, and nuclear magnetic resonance (NMR). The question of how best to treat speakable coherence as a resource is also considered. We review a popular approach in terms of operations that preserve the set of incoherent states, propose an alternative approach in terms of operations that are covariant under dephasing, and we outline the challenge of providing a physical justification for either approach. Finally, we note some mathematical connections that hold among the different approaches to quantifying coherence.

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“…(Note, however, that the list we provide is not complete; there are many known measures of asymmetry that we do not review here. See, e.g., [4,9,76,77]. )…”

Section: Methods For Deriving Measures Of Coherencementioning

confidence: 99%

How to quantify coherence: Distinguishing speakable and unspeakable notions

2016

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“…where to get the last equality we have used the fact that the two representations commute, Equation (10). Hence, the characteristic function of the state |eS/e| is equal to the characteristic function of the state G(|eS/e|).…”

Section: Methodsmentioning

confidence: 99%

“…An asymmetry measure quantifies how much the symmetry in question is broken by a given state. More precisely, a function f from states to real numbers is an asymmetry measure [5][6][7][8][9][10][11] if the existence of symmetric dynamics taking r to s implies f(r)Zf(s). A measure for rotational asymmetry, for instance, is a function over states that is non-increasing under rotationally invariant dynamics.…”

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confidence: 99%

Extending Noether’s theorem by quantifying the asymmetry of quantum states

2014

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Noether's theorem is a fundamental result in physics stating that every symmetry of the dynamics implies a conservation law. It is, however, deficient in several respects: for one, it is not applicable to dynamics wherein the system interacts with an environment; furthermore, even in the case where the system is isolated, if the quantum state is mixed then the Noether conservation laws do not capture all of the consequences of the symmetries. Here we address these deficiencies by introducing measures of the extent to which a quantum state breaks a symmetry. Such measures yield novel constraints on state transitions: for nonisolated systems they cannot increase, whereas for isolated systems they are conserved. We demonstrate that the problem of finding non-trivial asymmetry measures can be solved using the tools of quantum information theory. Applications include deriving model-independent bounds on the quantum noise in amplifiers and assessing quantum schemes for achieving high-precision metrology.

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“…Also, note that for closed-system dynamics under Hamiltonian H , any measure of asymmetry (relative to time translation) remains constant, i.e., [20,21,26,29,30,[36][37][38][39][40][41]). In particular, Refs.…”

Section: A Coherence As Asymmetry Relative To Translationsmentioning

confidence: 99%

“…As a matter of fact, it turns out that almost all measures of coherence which have been found recently, have been previously studied in the resource theory of asymmetry. For instance, the function called relative entropy of coherence by Baumgratz et al [25] has been extensively studied as a measure of asymmetry under the names of G asymmetry and relative entropy of asymmetry [29,32,39,41], and it has been generalized to a family of measures of asymmetry, called Holevo asymmetry measures [21,26] (see also [20,26] for measures of asymmetry based on l 1 norm).…”

Section: B Relation Between the Two Approachesmentioning

confidence: 99%

Quantum speed limits, coherence, and asymmetry

2016

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The resource theory of asymmetry is a framework for classifying and quantifying the symmetry-breaking properties of both states and operations relative to a given symmetry. In the special case where the symmetry is the set of translations generated by a fixed observable, asymmetry can be interpreted as coherence relative to the observable eigenbasis, and the resource theory of asymmetry provides a framework to study this notion of coherence. We here show that this notion of coherence naturally arises in the context of quantum speed limits. Indeed, the very concept of speed of evolution, i.e., the inverse of the minimum time it takes the system to evolve to another (partially) distinguishable state, is a measure of asymmetry relative to the time translations generated by the system Hamiltonian. Furthermore, the celebrated Mandelstam-Tamm and Margolus-Levitin speed limits can be interpreted as upper bounds on this measure of asymmetry by functions which are themselves measures of asymmetry in the special case of pure states. Using measures of asymmetry that are not restricted to pure states, such as the Wigner-Yanase skew information, we obtain extensions of the Mandelstam-Tamm bound which are significantly tighter in the case of mixed states. We also clarify some confusions in the literature about coherence and asymmetry, and show that measures of coherence are a proper subset of measures of asymmetry.

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Simulating symmetric time evolution with local operations

Cited by 12 publications

References 30 publications

How to quantify coherence: Distinguishing speakable and unspeakable notions

How to quantify coherence: Distinguishing speakable and unspeakable notions

Extending Noether’s theorem by quantifying the asymmetry of quantum states

Quantum speed limits, coherence, and asymmetry

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