We discuss an alternative to relative entropy as a measure of distance between mixed quantum states. The proposed quantity is an extension to the realm of quantum theory of the Jensen-Shannon divergence (JSD) between probability distributions. The JSD has several interesting properties. It arises in information theory and, unlike the Kullback-Leibler divergence, it is symmetric, always well defined and bounded. We show that the quantum JSD (QJSD) shares with the relative entropy most of the physically relevant properties, in particular those required for a "good" quantum distinguishability measure. We relate it to other known quantum distances and we suggest possible applications in the field of the quantum information theory.
We discuss different statistical distances in probability space, with emphasis on the Jensen-Shannon divergence, vis-a-vis {\it metrics} in Hilbert space and their relationship with Fisher's information measure. This study provides further reconfirmation of Wootters' hypothesis concerning the possibility that statistical fluctuations in the outcomes of measurements be regarded (at least partly) as responsible for the Hilbert-space structure of quantum mechanics.
The notion of distance in Hilbert space is relevant in many scenarios. In particular, distances between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.
A family of quantum measures like the Shannon distinguishability is presented. These measures are defined over the two classes of POVM measurements and related to separate parts in the expression for mutual information. Changes of Ky Fan's norms and the partitioned trace distances under the operation of partial trace are discussed. Upper and lower bounds on the introduced quantities are obtained in terms of partitioned trace distances and Uhlmann's partial fidelities. These inequalities provide a kind of generalization of the well-known bounds on the Shannon distinguishability. The notion of c…
In a recent paper, the generalization of the Jensen Shannon divergence (JSD) in the context of quantum theory has been studied (Phys. Rev. A 72, 052310 (2005)). This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of simulations for mixed states, that its square root verifies the triangle inequality.
The Holevo quantity provides an upper bound for the mutual information between the sender of a classical message encoded in quantum carriers and the receiver. Applying the strong sub-additivity of entropy we prove that the Holevo quantity associated with an initial state and a given quantum operation represented in its Kraus form is not larger than the exchange entropy. This implies upper bounds for the coherent information and for the quantum Jensen--Shannon divergence. Restricting our attention to classical information we bound the transmission distance between any two probability distributi…
Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha is the square of a metric for alpha lies in the interval (0,2], and that the resulting metric space of probability distributions can be i…
In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.
We study metric properties of symmetric divergences on Hermitian positive definite matrices. In particular, we prove that the square root of these divergences is a distance metric. As a corollary we obtain a proof of the metric property for Quantum Jensen-Shannon-(Tsallis) divergences (parameterized by $α\in [0,2]$), which in turn (for $α=1$) yields a proof of the metric property of the Quantum Jensen-Shannon divergence that was conjectured by Lamberti \emph{et al.} a decade ago (\emph{Metric character of the quantum Jensen-Shannon divergence}, Phy.\ Rev.\ A, \textbf{79}, (2008).) A somewhat m…
The notion of distinguishability between quantum states has shown to be fundamental in the frame of quantum information theory. In this paper we present a new distinguishability criterium by using a information theoretic quantity: the Jensen-Shannon divergence (JSD). This quantity has several interesting properties, both from a conceptual and a formal point of view. Previous to define this distinguishability criterium, we review some of the most frequently used distances defined over quantum mechanics' Hilbert space. In this point our main claim is that the JSD can be taken as a unifying dista…
We consider a fixed quantum measurement performed over $n$ identical copies of quantum states. Using a rigorous notion of distinguishability We consider a fixed quantum measurement performed over $n$ identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon's 12th theorem, we show that in the case of a single qubit the number of distinguishable states is $W(α_1,α_2,n)=|α_1-α_2|\sqrt{\frac{2n}{πe}}$, where $(α_1,α_2)$ is the angle interval from which the states are chosen. In the general case of an $N$-dimensional Hilbert space and an area $Ω$ of the dom…
This paper, mostly expository in nature, surveys four measures of distinguishability for quantum-mechanical states. This is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography. Each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions: namely, the probability of an identification error, the Kolmogorov distance, the Bhattacharyya coefficient, and the Shannon distinguishability (as defined through mutual information). These measures have a long history of use in statis…
Shannon information entropies in position and momentum spaces and their sum are calculated as functions of Z (Z=2-54) in atoms. Roothaan-Hartree-Fock electron wave functions are used. The universal property S=a+b lnZ is verified. In addition, we calculate the Kullback-Leibler relative entropy, the Jensen-Shannon divergence, Onicescu's information energy and a complexity measure recently proposed. Shell effects at closed shells atoms are observed. The complexity measure shows local minima at the closed shells atoms indicating that for the above atoms complexity decreases with respect to neighbo…
The uncertainty principle is often interpreted by the tradeoff between the error of a measurement and the consequential disturbance to the followed ones, which originated long ago from Heisenberg himself but now falls into reexamination and even heated debate. Here we show that the tradeoff is switched on or off by the quantum uncertainties of two involved non-commuting observables: if one is more certain than the other, there is no tradeoff; otherwise, they do have tradeoff and the Jensen-Shannon divergence gives it a good characterization.
We prove a concise factor-of-2 estimate for the failure rate of optimally distinguishing an arbitrary ensemble of mixed quantum states, generalizing work of Holevo [Theor. Probab. Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis, MIT, 1979]. A modification to the minimal principle of Cocha and Poor [Proceedings of the 6th International Conference on Quantum Communication, Measurement, and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a suboptimal measurement which has an error rate within a factor of 2 of the optimal by construction. This measurement is quadratically weighted and…
The time evolution of the trace distance between two states of an open quantum system may increase due to initial system-environment correlations, thus exhibiting a breakdown of distance contractivity of the reduced dynamics. We analyze how the time evolution of the distance depends on the chosen distance measure. Here we elucidate the behavior of the trace distance, the Hilbert- Schmidt distance, the Bures distance, the Hellinger distance and the quantum Jensen-Shannon divergence for two system-environment setups, namely a qubit bi-linearly coupled to an infinite and a finite size environment…
We study the time evolution of four distance measures in the presence of initial systemenvironment correlations. It is well-known that the trace distance between two quantum states of an open system may increase due to initial correlations which leads to a breakdown of the contractivity of the reduced dynamics. Here we compare and analyze, for two different models, the time evolution of the trace distance, the Bures metric, the Hellinger distance and the Jensen-Shannon divergence regarding an increase above their initial values, witnessing initial correlations. This work generalizes, deepens a…
We prove the ergodic version of the quantum Stein's lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given.
We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quantified an "amount of information," but, as he pointed out, "no concept of information itself was defined." Logical entropy provides that definition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set--just as the Boole-Laplace logical probability is the normalized quantitative measure of the e…
We analyze a basis-independent definition of quantum coherence. The maximally mixed state is used as the reference state, which allows for a way of defining coherence that is invariant under arbitrary unitary transformations. The basis-independent approach is applied to finding the distri- bution of the coherence within a multipartite system, where the contributions due to correlations between the subsystems and within each subsystem are isolated. The use of the square root of the Jensen-Shannon divergence allows for inequality relations to be derived between these quantities, giving a geometr…
Fermionic (atomic nuclei) and bosonic (correlated atoms in a trap) systems are studied from an information-theoretic point of view. Shannon and Onicescu information measures are calculated for the above systems comparing correlated and uncorrelated cases as functions of the strength of short range correlations. One-body and two-body density and momentum distributions are employed. Thus the effect of short-range correlations on the information content is evaluated. The magnitude of distinguishability of the correlated and uncorrelated densities is also discussed employing suitable measures of d…
We analyze and compare different measures for the degree of non-Markovianity in the dynamics of open quantum systems. These measures are based on the distinguishability of quantum states which is quantified, on the one hand, by the trace distance or, more generally, by the trace norm of the Helstrom matrix, and, on the other hand, by entropic quantifiers: the Jensen-Shannon divergence, the Holevo or the quantum skew divergence. We explicitly construct a qubit dynamics for which the trace norm based non-Markovianity measure is nonzero, while all the entropic measures turn out to be zero. This l…
Physical systems are often neither completely closed nor completely open, but instead they are best described by dynamical systems with partial escape or absorption. In this paper we introduce classical measures that explain the main properties of resonance eigenfunctions of chaotic quantum systems with partial escape. We construct a family of conditionally-invariant measures with varying decay rates by interpolating between the natural measures of the forward and backward dynamics. Numerical simulations in a representative system show that our classical measures correctly describe the main fe…
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 or Bures-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'…
