Axiomatic Characterization of the Quantum Relative Entropy and Free Energy

Wilming, Henrik; Gallego, Rodrigo; Eisert, Jens

doi:10.3390/e19060241

Cited by 55 publications

(54 citation statements)

References 30 publications

(41 reference statements)

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“…In the first regime, the von Neumann entropy (vNE) or quantities directly related to it prevail, such as the standard quantum relative entropy or mutual information, while in the second regime quantities such as quantum Rényi divergences [8][9][10][11] and smoothed versions of the above [12,13] become important. This common wisdom is, however, recently being challenged [14][15][16][17][18][19], as it has been shown that the vNE determines possible single-shot state transitions in quantum mechanicsunder unitary evolutions -provided that three assumptions hold [18]: i) one can prepare a suitable catalyst, i.e. an auxiliary system that does not change its state during the process but might become correlated with the system on which the transition is performed; ii) one has access to an environment, or source of randomness, that is modelled as a large system in the maximally mixed state; iii) one has full control over system, catalyst and the environment, in the sense that one can implement any unitary on the joint system.…”

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confidence: 99%

Von Neumann Entropy from Unitarity

et al. 2019

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The von Neumann entropy is a key quantity in quantum information theory and, roughly speaking, quantifies the amount of quantum information contained in a state when many identical and independent (i.i.d.) copies of the state are available, in a regime that is often referred to as being asymptotic. In this work, we provide a new operational characterization of the von Neumann entropy which neither requires an i.i.d. limit nor any explicit randomness. We do so by showing that the von Neumann entropy fully characterizes single-shot state transitions in unitary quantum mechanics, as long as one has access to a catalyst -an ancillary system that can be re-used after the transition -and an environment which has the effect of dephasing in a preferred basis. Building upon these insights, we formulate and provide evidence for the catalytic entropy conjecture, which states that the above result holds true even in the absence of decoherence. If true, this would prove an intimate connection between single-shot state transitions in unitary quantum mechanics and the von Neumann entropy. Our results add significant support to recent insights that, contrary to common wisdom, the standard von Neumann entropy also characterizes single-shot situations and opens up the possibility for operational single-shot interpretations of other standard entropic quantities. We discuss implications of these insights to readings of the third law of quantum thermodynamics and hint at potentially profound implications to holography.

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“…Notice from the previous definition that the main difference between the relative entropy and D + A,B lies in the fact that the latter lacks the property of monotonicity. Indeed, as mentioned above, since D + A,B verifies the properties of continuity, additivity and superadditivity, we know that it cannot verify the property of monotonicity (i.e., data processing for every quantum channel), as it would imply that it is a multiple of the relative entropy [51]. This motivates the appearance of the property of "semi-monotonicity".…”

Section: Conditional Relative Entropymentioning

confidence: 97%

Quantum conditional relative entropy and quasi-factorization of the relative entropy

Capel¹,

Lucia²,

Pérez-Garcı́a³

2018

J. Phys. A: Math. Theor.

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The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-σ-algebras.In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasifactorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.arXiv:1804.09525v2 [quant-ph]

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“…While Definition 2 does not require the catalyst to be uncorrelated with S at the end of the protocol, and in this sense goes beyond the conventional notion of catalysis discussed in the resource-theoretic literature on quantum thermodynamics [6,7], the more general notion of catalysis that we employ here is receiving increasing interest in quantum thermodynamics, where it was shown to single out the quantum relative entropy, free energy and von Neumann entropy [15,8,16], to be useful in the context of algorithmic cooling [16,17] and to show the energetic instability of passive states [18]. Finally, let us briefly comment on the interpretation of the random variable W as work in the setting of β-catalytic channels and the role of the Hamiltonian of the catalyst.…”

Section: Violations Of Nmw and Je Via β-Catalytic Channelsmentioning

confidence: 99%

By-passing fluctuation theorems

Boës

Gallego

et al. 2020

Quantum

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Fluctuation theorems impose constraints on possible work extraction probabilities in thermodynamical processes. These constraints are stronger than the usual second law, which is concerned only with average values. Here, we show that such constraints, expressed in the form of the Jarzysnki equality, can be bypassed if one allows for the use of catalystsadditional degrees of freedom that may become correlated with the system from which work is extracted, but whose reduced state remains unchanged so that they can be re-used. This violation can be achieved both for small systems but also for macroscopic many-body systems, and leads to positive work extraction per particle with finite probability from macroscopic states in equilibrium. In addition to studying such violations for a single system, we also discuss the scenario in which many parties use the same catalyst to induce local transitions. We show that there exist catalytic processes that lead to highly correlated work distributions, expected to have implications for stochastic and quantum thermodynamics.Definition 1 (No macroscopic work). Given a sequence of N -particle systems initially at thermal equilibrium with inverse temperature β and channels C (implicitly depending on N ), we say that the processes represented by C fulfill the no macroscopic work (NMW) condition if the probability of an event extracting work per particle larger or equal than is arbitrarily small as N → ∞,As is clear from the above, channels that satisfy the JE, such as unitary channels, also satisfy NMW and Av-SL. We now turn to investigate violations of JE and NMW for non-unitary channels. Violations of NMW and JEThe first main result of this work is to introduce a physically motivated family of channels C that violates both Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0.

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Axiomatic Characterization of the Quantum Relative Entropy and Free Energy

Cited by 55 publications

References 30 publications

Von Neumann Entropy from Unitarity

Von Neumann Entropy from Unitarity

Quantum conditional relative entropy and quasi-factorization of the relative entropy

By-passing fluctuation theorems

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