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We give systematic ways of defining monotone quantum relative entropies and (multi-variate) quantum Rényi divergences starting from a set of monotone quantum relative entropies. Interestingly, despite its central importance in information theory, only two additive and monotone quantum extensions of the classical relative entropy have been known so far, the Umegaki and the Belavkin-Staszewski relative entropies, which are the minimal and the maximal ones, respectively, with these properties. Using the Kubo-Ando weighted geometric means, we give a general procedure to construct monotone and additive quantum relative entropies from a given one with the same properties; in particular, when starting from the Umegaki relative entropy, this gives a new one-parameter family of monotone (even under positive trace-preserving (PTP) maps) and additive quantum relative entropies interpolating between the Umegaki and the Belavkin-Staszewski ones on full-rank states. In a different direction, we use a generalization of a classical variational formula to define multi-variate quantum Rényi quantities corresponding to any finite set of quantum relative entropies (D q x ) x ∈ X and real weights (P (x)) x ∈ X summing to 1, as Q P b , q ((ϱ x) x ∈ X) : = sup τ ≥ 0 ⁡ { Tr τ − ∑ x P (x) D q x (τ ‖ ϱ x) }. We analyze in detail the properties of the resulting quantity inherited from the generating set of quantum relative entropies; in particular, we show that monotone quantum relative entropies define monotone Rényi quantities whenever P is a probability measure. With the proper normalization, the negative logarithm of the above quantity gives a quantum extension of the classical Rényi α -divergence in the 2-variable case (X = { 0 , 1 } , P (0) = α). We show that if both D q 0 and D q 1 are lower semi-continuous, monotone, and additive quantum relative entropies, and at least one of them is strictly larger than the Umegaki relative entropy then the resulting barycentric Rényi divergences are strictly between the log-Euclidean and the maximal Rényi divergences, and hence they are different from any previously studied quantum Rényi divergence. [ABSTRACT FROM AUTHOR]

We consider a recently introduced framework for the description of memory effects based on quantum state distinguishability quantifiers, in which entropic quantifiers can be included. After briefly presenting the approach, we validate it considering the performance of different quantifiers in the characterization of the reduced dynamics of a two-level system undergoing decoherence. We investigate the different behaviors of these quantifiers in the dependence on physical features of the model, such as environmental temperature and coupling strength. It appears that the performance of the different quantifiers conveys the same physical information, though with different sensitivities, thus supporting robustness of the approach. [ABSTRACT FROM AUTHOR]

We present a criterion to establish the local continuity of the relative entropy of resource, i.e. the relative entropy distance to the set of free states, in any quantum resource theory. Several basic corollaries of this criterion are presented. Applications to the relative entropy of entanglement in multipartite quantum systems are considered. It is shown, in particular, that local continuity of any relative entropy of multipartite entanglement follows from local continuity of the quantum mutual information. [ABSTRACT FROM AUTHOR]

A revised version of the compactness criterion for families of quantum operations in the strong convergence topology obtained in [1] is presented, along with a more detailed proof and the examples showing the necessity of this revision. Several criteria for the existence of limit points of a sequence of quantum operations w.r.t. the strong convergence are obtained and discussed. Applications in different areas of quantum information theory are described. [ABSTRACT FROM AUTHOR]

In this letter, we obtain the precise range of the values of the parameter α such that Petz–Rényi α -relative entropy D α (ρ | | σ) of two faithful displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states ρ and σ with inverse temperature parameters r 1 , r 2 , … , r n and s 1 , s 2 , … , s n , respectively, 0 < r j , s j < ∞ , for all j, we have D α (ρ | | σ) < ∞ ⇔ α < min s j s j - r j : j ∈ { 1 , … , n } such that r j < s j , where we adopt the convention that the minimum of an empty set is equal to infinity. This result is particularly useful in the light of operational interpretations of the Petz–Rényi α -relative entropy in the regime α > 1 . Along the way, we also prove a special case of a conjecture of Seshadreesan et al. (J Math Phys 59(7):072204, 2018. ). [ABSTRACT FROM AUTHOR]

A quantum channel is sufficient with respect to a set of input states if it can be reversed on this set. In the approximate version, the input states can be recovered within an error bounded by the decrease of the relative entropy under the channel. Using a new integral representation of the relative entropy in Frenkel (Integral formula for quantum relative entropy implies data processing inequality, Quantum 7, 1102 (2023)), we present an easy proof of a characterization of sufficient quantum channels and recoverability by preservation of optimal success probabilities in hypothesis testing problems, equivalently, by preservation of L 1 -distance. [ABSTRACT FROM AUTHOR]

Observational entropy provides a general notion of quantum entropy that appropriately interpolates between Boltzmann's and Gibbs' entropies, and has recently been argued to provide a useful measure of out-of-equilibrium thermodynamic entropy. Here we study the mathematical properties of observational entropy from an information-theoretic viewpoint, making use of recently strengthened forms of the monotonicity property of quantum relative entropy. We present new bounds on observational entropy applying in general, as well as bounds and identities related to sequential and post-processed measurements. A central role in this work is played by what we call the 'coarse-grained' state, which emerges from the measurement's statistics by Bayesian retrodiction, without presuming any knowledge about the 'true' underlying state being measured. The degree of distinguishability between such a coarse-grained state and the true (but generally unobservable) one is shown to provide upper and lower bounds on the difference between observational and von Neumann entropies. [ABSTRACT FROM AUTHOR]

Quantum relative entropy has been studied extensively, and many forms have been derived due to different parameters. Maximum relative entropy and minimum relative entropy are obtained by taking specific conditions for parameters. Our goal in this paper is to propose a new bipartite entanglement monotone based on minimum relative entropy of any bipartite quantum entanglement state. We also demonstrate that entanglement monotone satisfies some basic properties as an entanglement measure. [ABSTRACT FROM AUTHOR]

One of the beautiful and very simple inequalities for a convex function is the Hermite–Hadamard inequality. The concept of log‐convexity is a strong variant of convexity. In this paper, by the Hermite–Hadamard inequality, we introduce two‐parametric Tsallis quantum relative entropy, two‐parametric Tsallis–Lin quantum relative entropy, and two‐parametric quantum Jensen–Shannon divergence in quantum information theory. Then some properties of quantum Tsallis–Jensen–Shannon divergence for two density matrices are investigated by this inequality. [ABSTRACT FROM AUTHOR]

The idea of this paper is to investigate and realizing the quantum relative entropy normally used in quantum information using physical systems for two quantum systems. We have described these respectively by Hamiltonian and Lindblad operators (Hk, Lk), k = 1,2,..., and we calculate the rate at which the relative entropy distance between the states evolving according to their two systems evolve. A consequence of the Baker-Campbell-Haussdorff formula in lie algebra theory, the computation uses the differentiation formula for the exponential map. The asymptotic formula for this relative entropy rate is obtained in terms of the scattering matrix corresponding to the two Hamiltonian. [ABSTRACT FROM AUTHOR]

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen–Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description. [ABSTRACT FROM AUTHOR]

Observational entropy provides a general notion of quantum entropy that appropriately interpolates between Boltzmann’s and Gibbs’ entropies, and has recently been argued to provide a useful measure of out-of-equilibrium thermodynamic entropy. Here we study the mathematical properties of observational entropy from an information-theoretic viewpoint, making use of recently strengthened forms of the monotonicity property of quantum relative entropy. We present new bounds on observational entropy applying in general, as well as bounds and identities related to sequential and post-processed measurements. A central role in this work is played by what we call the ‘coarse-grained’ state, which emerges from the measurement’s statistics by Bayesian retrodiction, without presuming any knowledge about the ‘true’ underlying state being measured. The degree of distinguishability between such a coarse-grained state and the true (but generally unobservable) one is shown to provide upper and lower bounds on the difference between observational and von Neumann entropies.

We prove the Gibbs variational formula in terms of quantum relative entropy density that characterizes translation invariant thermal equilibrium states in quantum lattice systems. It is a natural quantum extension of the similar statement established by Föllmer for classical systems. [ABSTRACT FROM AUTHOR]

Consider a function F(X,Y ) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute F(X,Y ) = XpYq. Our first main result gives conditions on F such that Tr[X log(F(Z,Y ))] ≤Tr[X(p log X + q log Y )] for all X,Y,Z such that TrZ = TrX. (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X,Y,Z instead of just X,Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy D(X∥Y ) = Tr[X(log X − log Y ]), and two others, the Donald relative entropy DD(X∥Y ), and the Belavkin–Stasewski relative entropy DBS(X∥Y ). They are known to satisfy DD(X∥Y ) ≤ D(X∥Y ) ≤ DBS(X∥Y ). We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on Tr[X log(F(Z,Y ))] in a number of cases in which F(Z,Y ) is homogeneous of degree 1 in Z and − 1 in Y. We also investigate the Legendre transforms in X of DD(X∥Y ) and DBS(X∥Y ), and show how our results for these lead to new refinements of the Golden–Thompson inequality.

Consider a function F(X,Y) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute F(X, Y) = XpYq. Our first main result gives conditions on F such that Tr[X log (F (Z,Y))] < Tr[X(p log X + q log Y)] for all X, Y, Z such that TrZ = TrX. (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X, Y, Z instead of just X, Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy D(X||Y) = Tr[X(logX -- logY]), and two others, the Donald relative entropy Dd (X||Y), and the Belavkin-Stasewski relative entropy DB s (X||Y). They are known to satisfy Dd(X||Y) < D(X||Y) < DB S(X|| Y). We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on Tr[X log(F(Z, Y))] in a number of cases in which F (Z, Y) is homogeneous of degree 1 in Z and --1 in Y. We also investigate the Legendre transforms in X of Dd(X||Y) and DBs(X|| Y), and show how our results for these lead to new refinements of the Golden-Thompson inequality. [ABSTRACT FROM AUTHOR]

We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained. [ABSTRACT FROM AUTHOR]

Abstract Given a function f : (0 , ∞) → R and a positive semidefinite n × n matrix P , one may define a trace functional on positive definite n × n matrices as A ↦ Tr (P f (A)). For differentiable functions f , the function A ↦ Tr (P f (A)) is differentiable at all positive definite matrices A. Under certain continuity conditions on f , this function may be extended to certain non-positive-definite matrices A , and the directional derivatives of Tr (P f (A)) may be computed there. This note presents conditions for these directional derivatives to exist and computes them. These conditions hold for the function f (x) = log ⁡ (x) and for the functions f p (x) = x p for all p > − 1. The derivatives of the corresponding trace functionals are computed here, and an alternative derivation of the directional derivatives using integral representations is provided. [ABSTRACT FROM AUTHOR]

The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations. [ABSTRACT FROM AUTHOR]

Thermodynamics establishes a relation between the work that can be obtained in a transformation of a physical system and its relative entropy with respect to the equilibrium state. It also describes how the bits of an informational reservoir can be traded for work using Heat Engines. Therefore, an indirect relation between the relative entropy and the informational bits is implied. From a different perspective, we define procedures to store information about the state of a physical system into a sequence of tagging qubits. Our labeling operations provide reversible ways of trading the relative entropy gained from the observation of a physical system for adequately initialized qubits, which are used to hold that information. After taking into account all the qubits involved, we reproduce the relations mentioned above between relative entropies of physical systems and the bits of information reservoirs. Some of them hold only under a restricted class of coding bases. The reason for it is that quantum states do not necessarily commute. However, we prove that it is always possible to find a basis (equivalent to the total angular momentum one) for which Thermodynamics and our labeling system yield the same relation.

We prove a number of quantitative stability bounds for the cases of equality in Petz’s monotonicity theorem for quasi-relative entropies $S_{f}(\rho ||\sigma)$ defined in terms of an operator monotone decreasing functions $f$ and, in particular, the Rényi relative entropies. Included in our results are bounds in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras, where the results are already non-trivial and of interest in quantum information theory. [ABSTRACT FROM AUTHOR]

The so-called q -z- Rényi Relative Entropies provide a huge two parameter family of relative entropies which include almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant ( 0 , 2 ) symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined differential forms on a suitable unfolding space without the need to introduce a specific set of coordinates. To make the reader acquainted with the intrinsic formalism introduced, we first perform the computation for the qubit case, and then, we extend the computation of the metric-like tensors to a generic n -level system. By suitably varying the parameters q and z , we are able to recover well-known examples of quantum metric tensors that, in our treatment, appear written in terms of globally defined geometrical objects that do not depend on the coordinates system used. In particular, we obtain a coordinate-free expression for the von Neumann–Umegaki metric, for the Bures metric and for the Wigner–Yanase metric in the arbitrary n -level case. [ABSTRACT FROM AUTHOR]

The property of superadditivity of the quantum relative entropy states that, in a bipartite system $ {\mathcal H}_{AB}= {\mathcal H}_{A} \otimes {\mathcal H} _{B}$ , for every density operator $\rho _{AB}$ , one has $ D(\rho _{AB}||\sigma _{A}\otimes \sigma _{B}) \ge D(\rho _{A}||\sigma _{A})+ D(\rho _{B}||\sigma _{B})$. In this paper, we provide an extension of this inequality for arbitrary density operators $\sigma _{AB}$. More specifically, we prove that $ \alpha (\sigma _{AB})\cdot D(\rho _{AB}||\sigma _{AB}) \ge D(\rho _{A}||\sigma _{A})+ D(\rho _{B}||\sigma _{B})$ holds for all bipartite states $\rho _{AB}$ and $\sigma _{AB}$ , where $\alpha (\sigma _{AB})= 1+2 \|\sigma _{A}^{-1/2} \otimes \sigma _{B}^{-1/2} \, \sigma _{AB} \, \sigma _{A}^{-1/2} \otimes \sigma _{B}^{-1/2} - {{\mathsf{1}}}_{AB}\|_\infty $. [ABSTRACT FROM AUTHOR]

We find that the standard relative entropy and the Umegaki entropy are designed for the purpose of inferentially updating probabilities and density matrices, respectively. From the same set of inferentially guided design criteria, both of the previously stated entropies are derived in parallel. This formulates a quantum maximum entropy method for the purpose of inferring density matrices in the absence of complete information. [ABSTRACT FROM AUTHOR]

A recent canonical divergence, which is introduced on a smooth manifold M endowed with a general dualistic structure ( g , ∇ , ∇ * ) , is considered for flat α -connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical α -divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum α -connections on the manifold of all positive definite Hermitian operators. In this case as well, we obtain that the recent canonical divergence is the quantum α -divergence.

Building upon work by Matsumoto, we show that the quantum relative entropy with full-rank second argument is determined by four simple axioms: (i) Continuity in the first argument; (ii) the validity of the data-processing inequality; (iii) additivity under tensor products; and (iv) super-additivity. This observation has immediate implications for quantum thermodynamics, which we discuss. Specifically, we demonstrate that, under reasonable restrictions, the free energy is singled out as a measure of athermality. In particular, we consider an extended class of Gibbs-preserving maps as free operations in a resource-theoretic framework, in which a catalyst is allowed to build up correlations with the system at hand. The free energy is the only extensive and continuous function that is monotonic under such free operations. [ABSTRACT FROM AUTHOR]

We find that the standard relative entropy and the Umegaki entropy are designed for the purpose of inferentially updating probabilities and density matrices, respectively. From the same set of inferentially guided design criteria, both of the previously stated entropies are derived in parallel. This formulates a quantum maximum entropy method for the purpose of inferring density matrices in the absence of complete information.

The dynamical properties of quantum coherence in the system of two-coupled-cavities, each of which resonantly interacts with a two-level atom, is investigated via the relative entropy measure. We focus on the coherences for the atom-atom, atom-cavity and cavity-cavity subsystems and find that the dynamical behaviors of these coherences depend largely on the cavity-cavity coupling, which may indicate the Mott insulator-superfluid transition in the thermodynamic limit. We also study the influences of the initial cavity-cavity correlation on the coherences and show that the initial correlation of the cavity-cavity subsystem can enhance the revival ability for the atom-atom and cavity-cavity coherences while reduce that for the atom-cavity coherence. Besides, we demonstrate the qualitative difference of dynamics between coherence and entanglement. Finally, the influences of dissipations including cavity losses and atomic decays on the coherence are explored. [ABSTRACT FROM AUTHOR]

We consider the relative operator entropy constructed by the spectral geometric mean and see its properties analogous to those of the Tsallis relative operator entropy by the usual geometric mean. Furthermore, we define the quantum relative entropy constructed by the spectral geometric mean and derive its subadditivity under tensor product. [ABSTRACT FROM AUTHOR]

The fact that the quantum relative entropy is non-increasing with respect to quantum physical evolutions lies at the core of many optimality theorems in quantum information theory and has applications in other areas of physics. In this work, we establish improvements of this entropy inequality in the form of physically meaningful remainder terms. One of the main results can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum physical evolution is relatively small, then it is possible to perform a recovery operation, such that one can perfectly recover one state while approximately recovering the other. This can be interpreted as quantifying how well one can reverse a quantum physical evolution. Our proof method is elementary, relying on the method of complex interpolation, basic linear algebra and the recently introduced Rényi generalization of a relative entropy difference. The theorem has a number of applications in quantum information theory, which have to do with providing physically meaningful improvements to many known entropy inequalities. [ABSTRACT FROM AUTHOR]

We present a novel spontaneous collapse model where size is no longer the property of a physical system which determines its rate of collapse. Instead, we argue that the rate of spontaneous localization should depend on a system's quantum Integrated Information (QII), a novel physical property which describes a system's capacity to act like a quantum observer. We introduce quantum Integrated Information, present our QII collapse model and briefly explain how it may be experimentally tested against quantum theory. [ABSTRACT FROM AUTHOR]

The entropic uncertainty relation with quantum side information (EUR-QSI) from (Berta et al 2010 Nat. Phys. http://dx.doi.org/10.1038/nphys1734 6 http://dx.doi.org/10.1038/nphys1734 ) is a unifying principle relating two distinctive features of quantum mechanics: quantum uncertainty due to measurement incompatibility, and entanglement. In these relations, quantum uncertainty takes the form of preparation uncertainty where one of two incompatible measurements is applied. In particular, the ‘uncertainty witness’ lower bound in the EUR-QSI is not a function of a post-measurement state. An insightful proof of the EUR-QSI from (Coles et al 2012 Phys. Rev. Lett. http://dx.doi.org/10.1103/PhysRevLett.108.210405 108 http://dx.doi.org/10.1103/PhysRevLett.108.210405 ) makes use of a fundamental mathematical consequence of the postulates of quantum mechanics known as the non-increase of quantum relative entropy under quantum channels. Here, we exploit this perspective to establish a tightening of the EUR-QSI which adds a new state-dependent term in the lower bound, related to how well one can reverse the action of a quantum measurement. As such, this new term is a direct function of the post-measurement state and can be thought of as quantifying how much disturbance a given measurement causes. Our result thus quantitatively unifies this feature of quantum mechanics with the others mentioned above. We have experimentally tested our theoretical predictions on the IBM quantum experience and find reasonable agreement between our predictions and experimental outcomes.