Continuity bounds on the quantum relative entropy

The quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory. We have derived many such inequalities in previous work. The present paper is a follow-up on this, and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned. The result obtained here is more general than the corresponding one from our previous work. As a corollary, we obtain a sharp upper bound on the regularised relative entropy introduced by Lendi, Farhadmotamed, and van Wonderen.

The relative entropy of entanglement E_R is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, E_R is everywhere lower semi-continuous. We use this to derive a dual variational expression for E_R in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multipartite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity—more generally, all relative entropy distances from the sets of states with non-negative lambda -quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason, the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for E_R and closely related quantities in the presence of an energy constraint.

The problem of quantum cloning is closely connected to quantum cryptography. In quantum cryptography, an eavesdropper on the quantum channel can not copy perfectly the sent quantum states, however in many cases the cloning machine is known to be the most powerful eavesdropping strategy against quantum cryptographic protocols. The geometric interpretation of quantum states investigates distances between two different quantum states. In our method we use quantum relative entropy as an informational distance between quantum states. We show a geometrical approach to analyze the security of quantum cryptography, based on quantum relative entropy and Delaunay triangulation on the Bloch sphere. In our security analysis, we use an approximation algorithm from classical computational geometry to determine the smallest enclosing ball of balls using core-sets.

We consider a two-parameter family of Rényi relative entropies Dα,z(ρ ∥ σ) that are quantum generalisations of the classical Rényi divergence Dα(p ∥ q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi’s axioms for a divergence. We consider the range of the parameters α, z for which the data-processing inequality holds. We also investigate a variety of limiting cases for the two parameters, obtaining explicit formulas for each one of them.

The quantum relative entropy between two states satisfies a monotonicity\nproperty meaning that applying the same quantum channel to both states can\nnever increase their relative entropy. It is known that this inequality is only\ntight when there is a "recovery map" that exactly reverses the effects of the\nquantum channel on both states. In this paper we strengthen this inequality by\nshowing that the difference of relative entropies is bounded below by the\nmeasured relative entropy between the first state and a recovered state from\nits processed version. The recovery map is a convex combination of rotated Petz\nrecovery maps and perfectly reverses the quantum channel on the second state.\nAs a special case we reproduce recent lower bounds on the conditional mutual\ninformation such as the one proved in [Fawzi and Renner, Commun. Math. Phys.,\n2015]. Our proof only relies on elementary properties of pinching maps and the\noperator logarithm.\n

We derive a strengthened monotonicity inequality for quantum relative entropy by employing properties of $\alpha$-R\'{e}nyi relative entropy. We develop a unifying treatment towards the improvement of some quantum entropy inequalities. In particular, an emphasis is put on a lower bound of quantum conditional mutual information (QCMI) as it gives a Pinsker-like lower bound for the QCMI. We also give some improved entropy inequalities based on R\'{e}nyi relative entropy. The inequalities obtained, thus, extends some well-known ones. We also obtain a condition under which a tripartite operator becomes a Markov state. As a by-product we provide some trace inequalities of operators, which are of independent interest.

The quantum relative entropy is known to play a key role in determining the asymptotic convertibility of quantum states in general resource-theoretic settings, often constituting the unique monotone that is relevant in the asymptotic regime. We show that this is no longer the case when one allows stochastic protocols that may only succeed with some probability, in which case the quantum relative entropy is insufficient to characterize the rates of asymptotic state transformations, and a new entropic quantity based on a regularization of the Hilbert projective metric comes into play. Such a scenario is motivated by a setting where the cost associated with transformations of quantum states, typically taken to be the number of copies of a given state, is instead identified with the size of the quantum memory needed to realize the protocol. Our approach allows for constructing transformation protocols that achieve strictly higher rates than those imposed by the relative entropy. Focusing on the task of resource distillation, we give broadly applicable strong converse bounds on the asymptotic rates of probabilistic distillation protocols, and show them to be tight in relevant settings such as entanglement distillation with nonentangling operations. This generalizes and extends previously known limitations that are only applicable to deterministic protocols. Our methods are based on recent results for probabilistic one-shot transformations as well as a new asymptotic equipartition property for the projective relative entropy.

Quantum state estimation (or state tomography) is an indispensable task in quantum information processing. Because full state tomography that determines all elements of the density matrix is computationally demanding, one usually takes the strategy of assuming a certain model of quantum states and identifying the model parameters. However, it is difficult to make a valid assumption given little prior knowledge on a quantum state of interest, and thus we need a reasonable model selection method for quantum state estimation. Actually, in the classical statistical estimation theory, several types of information criteria have been established and widely used in practice for appropriately choosing a classical statistical model. In this study, we propose quantum information criteria for evaluating the quality of the estimated quantum state in terms of the quantum relative entropy, which is a natural quantum analogue of the classical information criterion defined in terms of Kullback–Leibler divergence. In particular, we derive two quantum information criteria depending on the type of an estimator for the quantum relative entropy; one uses the log-likelihood and the other uses the classical shadow. The general role of information criteria is to predict the performance of an estimated model for unseen data, although it is a function of only sampled data; this generalization capability of the proposed quantum information criteria is evaluated in numerical simulations.

It is shown by way of examples that the quantum correspondent to Fisher information in statistics named quantum information metric is an extensive quantity in spite of the nonextensivity of the quantum q-deformed relative entropy. A general proof is presented, and some related topics on the quantum relative entropies vs the metrics are discussed.

In the dynamics of open quantum systems, the backflow of information to the reduced system under study has been suggested as the actual physical mechanism inducing memory and thus leading to non-Markovian quantum dynamics. To this aim, the trace-distance revivals between distinct evolved system states have been shown to be subordinated to the establishment of system-environment correlations or changes in the environmental state. We show that this interpretation can be substantiated also for a class of entropic quantifiers. We exploit a suitably regularized version of Umegaki's quantum relative entropy, known as telescopic relative entropy, that is tightly connected to the quantum Jensen-Shannon divergence. In particular, we derive general upper bounds on the telescopic relative entropy revivals conditioned and determined by the formation of correlations and changes in the environment. We illustrate our findings by means of examples, considering in particular the Jaynes-Cummings model and a phase covariant dynamics.

We prove that the quantum relative entropy is a rate function in large deviation principle. Next, we define information criteria for quantum states and estimate the accuracy of the use of them. Most of the results in this paper are essentially based on Hiai-Ohya-Tsukada theorem.

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

This short note is concerned with maps on the cone of positive definite matrices which preserve certain generalized distance measures. We mean sorts of distances which are connected with quantum divergences, relative entropies, or related to geometrical structures on the positive definite cone. The novelty here is that we do not assume that our maps, the so-called generalized isometries, are surjective. As we will see, in some cases, those transformations turn to be automatically surjective, while in other cases, we can determine the precise structure of all not necessarily surjective such maps. In spite of the presented results, a number of related questions are left open and proposed to be the objects of further research.

Schumacher and Westmoreland [Phys. Rev. A 64, 42304 (2001)] have established a quantum analog of a well-known classical information theory result on a role of relative entropy as a measure of nonoptimality in (classical) data compression. In this paper, we provide an alternative simple and constructive proof of this result by constructing quantum compression codes (schemes) from classical data compression codes. Moreover, as the quantum data compression or coding task can be effectively reduced to a (quasi)classical one, we show that relevant results from classical information theory and data compression become applicable and therefore can be extended to the quantum domain.

We present a systematic procedure to derive a quantum master equationfor thermal relaxation in real scalar field theory, expanding on the method proposed in Koide and Nicacio [Phys. Lett. A 494, 129277 (2024)0375-960110.1016/j.physleta.2023.129277]. We begin by introducing a generalized model for a classical scalar field interacting with a Brownian thermostat, consistent with stochastic thermodynamics. Applying canonical quantization to this model, we derive the corresponding quantum master equation, which is applicable to any form of the scalar field Hamiltonian. While its evolution is generally non-CPTP (completely positive and trace preserving), it can be adjusted to describe a CPTP evolution, such as those found in the Gorini-Kossakowski-Sudarshan-Lindblad equationby appropriately tuning the parameters of the model. In this framework, we define heat, work, and entropy in a way that satisfies the first and second laws of quantum thermodynamics. This suggests that the quantum-classical correspondence extends beyond closed systems governed by unitary time evolution to open systems as well. We further investigate the relation between the second law in quantum thermodynamics and relative entropy, providing insights into the study of quantum fluctuations through information-theoretical techniques in quantum field theory.