From Quasi-Entropy to Various Quantum Information Quantities

Monotone metrics on matrix spaces

EPR steering is an asymmetric quantum correlation, as such, it is an important resource for quantum information protocols, like those used in quantum key distribution that are of tremendous relevance nowadays. The security of the aforementioned protocols can be shown by using entropic uncertainty measures, which are based on quantum information quantities, among which the entropies are found. In this work, we propose an entropic measure of steering, starting from the detection of steering given in a previous entropic uncertainty relation-based criterion, as well as using a generalised entropy, the Tsallis entropies. We define the entropic measure of steering in terms of an indicator, a quantity depending on the entropies of the measurements of the system, which assigns a value to the steering content of quantum states, while also comprising the asymmetry property of steering. We investigate the properties of the proposed quantity as a measure of steering and its asymmetry and show how it can be used for specific examples. For instance, a class of noisy two-qubit states that includes the Werner states is used to demonstrate the potential of such a proposal. Furthermore, a comparison is also made with the detection of steering in these states using existing steering criteria and the amount of steering a geometric measure for the Werner state gives.

The two principal/immediate influences—which we seek to interrelate here—upon the undertaking of this study are papers of Życzkowski and Słomczyński [J. Phys. A 34, 6689 (2001)] and of Petz and Sudár [J. Math. Phys. 37, 2262 (1996)]. In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudár work (completing a program of Chentsov), the quantum analog of the (classically unique) Fisher information (monotone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudár? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys’ prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions—recently studied by Srednicki [Phys. Rev. A 71, 052107 (2005)]—formed by normalizing the volume elements of the various information metrics, is also discussed in our context.

In quantum estimation theory, the Holevo bound is known as a lower bound of weighed traces of covariances of unbiased estimators. The Holevo bound is defined by a solution of a minimization problem, and in general, explicit solution is not known. When the dimension of Hilbert space is two and the number of parameters is two, a explicit form of the Holevo bound was given by Suzuki. In this paper, we focus on a logarithmic derivative lies between the symmetric logarithmic derivative (SLD) and the right logarithmic derivative (RLD) parameterized by $\beta\in[0,1]$ to obtain lower bounds of weighted trace of covariance of unbiased estimator. We introduce the maximum logarithmic derivative bound as the maximum of bounds with respect to $\beta$. We show that all monotone metrics induce lower bounds, and the maximum logarithmic derivative bound is the largest bound among them. We show that the maximum logarithmic derivative bound has explicit solution when the $d$ dimensional model has $d+1$ dimensional $\mathcal{D}$ invariant extension of the SLD tangent space. Furthermore, when $d=2$, we show that the maximization problem to define the maximum logarithmic derivative bound is the Lagrangian duality of the minimization problem to define Holevo bound, and is the same as the Holevo bound. This explicit solution is a generalization of the solution for a two dimensional Hilbert space given by Suzuki. We give also examples of families of quantum states to which our theory can be applied not only for two dimensional Hilbert spaces.

Human perception like thinking seems to be essentially synthetic. We never perceive an object by scanning it point by point. We instead form right away a Gestalt, i.e. a global idea of it. Rational activity, as well, seems to be essentially based on gestaltic patterns. Quantum computation has suggested a new holistic semantics, which makes essential use of the characteristic “holistic” features of the quantum‐theoretic formalism. In this framework, meanings are represented by quantum information quantities: systems of qubits or, more generally, mixtures of systems of qubits. The global meaning of a compound expression (which may correspond to an entangled state) determines the meanings of its parts, but generally not the other way around.

Topological semimetals are a class of many-body systems exhibiting novel macroscopic quantum phenomena at the interplay between high energy and condensed matter physics. They display a topological quantum phase transition (TQPT) which evades the standard Landau paradigm. In the case of Weyl semimetals, the anomalous Hall effect is a good non-local order parameter for the TQPT, as it is proportional to the separation between the Weyl nodes in momentum space. On the contrary, for nodal line semimetals (NLSM), the quest for an order parameter is still open. By taking advantage of a recently proposed holographic model for strongly-coupled NLSM, we explicitly show that entanglement entropy (EE) provides an optimal probe for nodal topology. We propose a generalized c-function, constructed from the EE, as an order parameter for the TQPT. Moreover, we find that the derivative of the renormalized EE with respect to the external coupling driving the TQPT diverges at the critical point, signaling the rise of non-local quantum correlations. Finally, we show that these quantum information quantities are able to characterize not only the critical point but also features of the quantum critical region at finite temperature.

We try to find a continuous functionu defined on a real right half-line with the range (0, ∞) such thatu −1 is operator monotone. We then look for another functionv such thatv(u −1) is operator monotone, namely,u(A)≦u(B) impliesv(A)≦v(B) for self-adjoint operatorsA andB.

A capital letter means a bounded liner operator on a complex Hilbert space H , and B(H) denotes the algebra of all bounded linear operators on H equipped with operator norm ‖ · ‖ . An operator T is said to be positive (denoted by T 0) if (Tx,x) 0 for all x∈H and an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. Chaotic order is defined by logA logB for strictly positive operators A and B . A continuous real-valued function f defined on an interval J is called operator monotone if A B implies that f (A) f (B) for all self-adjoint operators A , B with spectra in J . The well known celebrated Lower-Heinz inequality asserts that if A B 0, then Aα Bα for any α ∈ [0,1] , which means that t → tα is operator monotone. Another well known example of operator monotone function is t → log t on (0,∞) . Recently, Furuta [5] showed that if A > B , then f (Aα ) > f (Bα) for all α ∈ (0,1] and if logA > logB , then there exists β ∈ (0,1] such that f (Aα) > f (Bα) for all α ∈ (0,β ] , where f (t) is a non-constant operator monotone function on [0,∞) . Comprehensive survey on related order preserving operator inequalities were given in [2,4]. In this paper, we will obtain an improvement which leads to the result of Moslehian and Najafi [6].

A short proof is given based on C*-algebra theory for the well-known theorem that if S and Tare bounded selfadjoint operators on a Hubert space such that 0^5^T then Sai|7a for each0^a 0; and S+e/and T+el are both invertible. Since (S+eI)x converges to Sx in norm when e-0 for each a>0, and since the positive operators in B(%>) form a norm closed set, it suffices to prove the theorem assuming that S and T are invertible. (The case a=0 can be verified directly, since S° is the range projection of 5.) Let E denote the set of exponents a in [0, 1] for which the function t—t is operator monotone. Trivially 0e£ and 1 eE. Since the function a-S is continuous from [0, 1] to F(§) in the norm topology we see that Fis a closed set. The proof will be complete when we show that Fis convex. Take a and s in E. Then S°^TX; hence r^^F72^/. It follows that \\S/2T-'*\\£l. Similarly \\Ss'2T-s/2\\^\. With P(A) the spectral radius of an operator A we have p(AB) = p(BA). Therefore /j*-U+s)/lMa+s)/2j--<+s1li) _ D('r'.a-s)li'j—iix+s'iHc{a+s)lij-l.!i+s)H-Y-t.!ii-s)li _ „(T—f/!c(»+j!)/2T-a/2 <; II j—0/2c(«t+/»/2<T--«/2t| <; \\T-pl2Ssl2 ||S°,/2T_,I/21| 1. Received by the editors April 27, 1972. AMS 1970 subject classifications. Primary 47B15; Secondary 46L05.

Loewner's theorem in several variables

An operator monotone function [formula omitted] and strictly chaotic order

Coherence is a basic feature of quantum systems and a common necessary condition for quantum correlations. It is also an important physical resource in quantum information processing. In this paper, using relative entropy, we consider a more general definition of the cohering power of quantum operations. First, we calculate the cohering power of unitary quantum operations and show that the amount of distributed coherence caused by non-unitary quantum operations cannot exceed the quantum-incoherent relative entropy between system of interest and its environment. We then find that the difference between the distributed coherence and the cohering power is larger than the quantum-incoherent relative entropy. As an application, we consider the distributed coherence caused by purification.

In communication processes, a channel has an activity to communicate information of input system to output system. The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983, and he extended it to general quantum systems by means of the relative entropy of Araki and Uhlmann. The capacity shows the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of Ohya mutual entropy with respect to a certain subset of the initial state space. One of the most important theorems in quantum communication theory is coding theorem. The quantum coding theorems are discussed by using the mutual entropy — type measures introduced in the studies of quantum information.In this paper, we compare with these mutual entropy‐type measures in order to obtain most suitable one for discussing the quantum coding theorem. We concretely calculate the quantum capacity for quantum channels.

The main aim of this survey article is to present recent developments of matrix versions of the arithmetic–geometric mean inequality. Among others, we show improvements and generalizations of the arithmetic–geometric mean inequality for unitarily invariant norms via the Hadamard product, and for singular values via the operator monotone functions.

Let $A, B$ be positive operators and let f be any operator monotone function. We obtain inequalities for $|||f(A)X - Xf(B)|||$ in terms of $|||f\left(|AX - XB| \right) |||$ for every unitarily invariant norm. The case $X = I$ was considered by T. Ando [Math. Z., 197 (1988), pp. 403--409], and some of our results reduce to his results in this special case. Some related inequalities are obtained.