We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM- and POVM-dynamical (quantum) entropies, which are analogues of the classical Kolmogorov-Sinai entropy rate. They quantify the maximal randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy, or equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.
We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM- and POVM-dynamical (quantum) entropies, which are analogues of the classical Kolmogorov-Sinai entropy rate. They quantify the maximal randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy, or equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.
We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.
In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina).
Abstract. In the present paper, we consider the marginal entropy of the a pure state in the tensor product of two Hilbert spaces. The expectation of the marginal entropy of a random pure state with respect to the Haar measure of unitary transformation on the tensor product Hilbert space depends on group integrals of unitary group. For group integrals of ordinary representation matrix elements of U(n), we provide a elementary method to reobtain Weingarton’s result of asymptotical behavior of group integrals. For group integrals of matrix elements of irreducible representations of U(n),we generalize the Weyl-Schur duality theorem and get an algorithm. 1. Mathematical Preliminaries We adopt the following standard conventions of quantum mechanics: The states of an n-level quantum system (and we include the case infinity) are represented by an n-dimensional complex Hilbert space H. In this familiar representation, the states of a composite of two systems A and B say, the first one A n-level, and the second B m-level, is then represented by the tensor product of the respective Hilbert spaces HA and HB. As a result, the composite system (AB) is an (nm)-level system. This also makes sense in the infinite case where we then use standard geometry for Hilbert space, and suitable choices of orthonormal bases (ONBs). For a fixed system with Hilbert space H, the corresponding pure quantum states are vectors in H of norm one, or rather equivalence classes of such vectors: Equivalent vectors v and v ′ in H yield the same rank-one projection operator P, i.e., the projection of H onto Key words and phrases. random pure state, marginal entropy,group integral, Weyl-Schur duality
As quantum analogs of the classical Kolmogorov-Sinai entropy, quantum dynamical entropies have emerged as important tools to characterize complex quantum dynamics. In particular, Alicki-Fannes-Lindblad (AFL) entropy, which quantifies the information production rate of a coherent quantum system subjected to repeated measurement, has received considerable attention as a potential diagnostic for quantum chaos. Despite this interest, the precise behavior of quantum dynamical entropy in the presence of symmetry remains largely unexplored. In this work, we establish rigorous inequalities of the AFL entropy for arbitrary unitary dynamics (single particle and many body) in the presence of various types of symmetry. Our theorems encompass three cases: Abelian symmetry, an anticommuting unitary, and non-Abelian symmetries. In particular, we show that, while the cumulative AFL entropy generally saturates to the dimensional bound at late times for chaotic dynamics, this saturation value is distinctively lower when the measurements respect the symmetries. We motivate our main results with numerical simulations of the perturbed quantum cat maps. Our findings highlight the crucial role of symmetry in quantum dynamics under measurements, and our framework is readily adaptable for investigating symmetry's influence across diverse probes of quantum chaos.
We characterize the asymptotic performance of a class of positive operator valued measurements (POVMs) where the only task is to make measurements on independent and identically distributed quantum states on finite-dimensional systems. The POVMs we utilize here can be efficiently described in terms of a reasonably small set of parameters. Their analysis furthers the development of a quantum method of types. They deliver provably optimal performance in asymmetric hypothesis testing and in the transmission of classical messages over quantum channels. We now relate them to the recently developed α−z divergences Dα,z by giving an operational interpretation for the limiting case limα→1Dα,1−α in terms of probabilities for certain measurement outcomes. This explains one of the more surprising findings of [1] in terms of the theory of group representations. In addition, we provide a Cauchy-Binet type formula for unitary matrices which connects the underlying representation theoretic objects to partial sums of the entries of unitary matrices. At last, we concentrate on the special case of qubits. We are able to give a complete description of the asymptotic detection probabilities for all POVM elements described here. We take the opportunity to define a family of functions on pairs of semi-definite matrices which obeys the quantum generalizations of Rényi's axioms except from the generalized mean value axiom. This family is described by limiting values of α−z divergences for the extremal values of the parameter.
We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary–antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.