Quantum information measures such as the entropy and the mutual information find applications in physics, e.g., as correlation measures. Generalizing such measures based on the Rényi entropies is expected to enhance their scope in applications. We prescribe Rényi generalizations for any quantum information measure which consists of a linear combination of von Neumann entropies with coefficients chosen from the set {-1,0,1}. As examples, we describe Rényi generalizations of the conditional quantum mutual information, some quantum multipartite information measures, and the topological entanglement entropy. Among these, we discuss the various properties of the Rényi conditional quantum mutual information and sketch some potential applications. We conjecture that the proposed Rényi conditional quantum mutual informations are monotone increasing in the Rényi parameter, and we have proofs of this conjecture for some special cases.
The R{é}nyi entropy is one of the important information measures that generalizes Shannon's entropy. The quantum R{é}nyi entropy has a fundamental role in quantum information theory, therefore, bounding this quantity is of vital importance. Another important quantity is R{é}nyi relative entropy on which R{é}nyi generalization of the conditional entropy, and mutual information are defined based. Thus, finding lower bound for R{é}nyi relative entropy is our goal in this paper. We use matrix inequalities to prove new bounds on the entropy of type $β$, R{é}nyi entropy.
We present a family of coherence quantifiers based on the generalized $α-z-$relative R$\acute{e}$nyi entropy. These quantifiers satisfy all the standard criteria for well-defined measures of coherence, and include some existing coherence measures as special cases.
We demonstrate a generalization of quantum discord using a generalized definition of von-Neumann entropy, which is Sharma-Mittal entropy; and the new definition of discord is called Sharma-Mittal quantum discord. Its analytic expressions are worked out for two qubit quantum states as well as Werner, isotropic, and pointer states as special cases. The R{é}nyi, Tsallis, and von-Neumann entropy based quantum discords can be expressed as limiting cases for of Sharma-Mittal quantum discord. We also numerically compare all these discords and entanglement negativity.
University of Birmingham（伯明翰大学）；King’s College London（伦敦国王学院）
A key component of a quantum machine learning model operating on classical inputs is the design of an embedding circuit mapping inputs to a quantum state. This paper studies a transfer learning setting in which classical-to-quantum embedding is carried out by an arbitrary parametric quantum circuit that is pre-trained based on data from a source task. At run time, a binary quantum classifier of the embedding is optimized based on data from the target task of interest. The average excess risk, i.e., the optimality gap, of the resulting classifier depends on how (dis)similar the source and targe…
We propose to quantify the entanglement of pure states of $N \times N$ bipartite quantum system by defining its Husimi distribution with respect to $SU(N)\times SU(N)$ coherent states. The Wehrl entropy is minimal if and only if the pure state analyzed is separable. The excess of the Wehrl entropy is shown to be equal to the subentropy of the mixed state obtained by partial trace of the bipartite pure state. This quantity, as well as the generalized (R{é}nyi) subentropies, are proved to be Schur--convex, so they are entanglement monotones and may be used as alternative measures of entanglement…
Information based uncertainty measures like R{é}nyi entropy (R), Shannon entropy (S) and Onicescu energy (E) (in both position and momentum space) are employed to understand the influence of radial confinement in isotropic harmonic oscillator. The transformation of Hamiltonian in to a dimensionless form gives an idea of the composite effect of oscillation frequency ($ω$) and confinement radius ($r_{c}$). For a given quantum state, accurate results are provided by applying respective \emph{exact} analytical wave function in $r$ space. The $p$-space wave functions are produced from Fourier trans…
Shannon entropy ($S$), R{é}nyi entropy ($R$), Tsallis entropy ($T$), Fisher information ($I$) and Onicescu energy ($E$) have been explored extensively in both \emph{free} H atom (FHA) and \emph{confined} H atom (CHA). For a given quantum state, accurate results are presented by employing respective \emph{exact} analytical wave functions in $r$ space. The $p$-space wave functions are generated from respective Fourier transforms$-$for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. \emph{Exact} mathematical expressions of…
We provide a unifying axiomatics for Renyi's entropy and non-extensive entropy of Tsallis. It is shown that the resulting entropy coincides with Csiszar's measure of directed divergence known from communication theory.
The original definition of quantum discord of bipartite states was defined over projective measurements, in this paper we discuss some generalizations of it. These generalizations are defined over general measurements, rank-one general measurements or Neumark extension measurements. We investigate the nonnegativity, zero-discord sets of all these quantum discords and some properties about them.
After a brief review of classical probability theory (measure theory), we present an observation (due to Sorkin) concerning an aspect of probability in quantum mechanics. Following Sorkin, we introduce a generalized measure theory based on a hierarchy of ``sum-rules.'' The first sum-rule yields classical probability theory, and the second yields a generalized probability theory that includes quantum mechanics as a special case. We present some algebraic relations involving these sum-rules. This may be useful for the study of the higher-order sum-rules and possible generalizations of quantum me…
A generalization of quantum discord to multipartite systems is proposed. A key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems. It is by construction zero only for systems with classically correlated subsystems and is a non-negative quantity, giving a measure of the total non-classical correlations in the multipartite system with respect to a fixed measurement ordering. For the tripartite case, we show that the discord can be decomposed into contributions resulting from changes induced by non-classical correlation breaking measurem…
In this work, we study the phenomena of quantum entanglement by computing de Sitter entanglement entropy from von Neumann measure. For this purpose we consider a bipartite quantum field theoretic setup in presence of axion originating from ${\bf Type~ II~B}$ string theory. We consider the initial vacuum to be CPT invariant non-adiabatic $α$ vacua state under ${\bf SO(1,4)}$ ismometry, which is characterized by a real one-parameter family. To implement this technique we use a ${\bf S^2}$ which divide the de Sitter into two exterior and interior sub-regions. First, we derive the wave function of…
We introduce a new class of quantum and classical correlation measures by generalizing the reflected entropy to multipartite states. We define the new measures for quantum systems in one spatial dimension. For quantum systems having gravity duals, we show that the holographic duals of these new measures are various types of minimal surfaces consist of different entanglement wedge cross sections. One special generalized reflected entropy is $Δ_R$, with the holographic dual proportional to the so called multipartite entanglement wedge cross section $Δ_W$ defined before. We then perform a large $…
Experimental realization of automated error correction is demonstrated through IBM Quantum Experience for Bell and GHZ states using a measurement based approach upon ancilla qubits. The measurement automatically activates error correcting unitary operations to restore the system to its original entangled state. We illustrate the algorithm for the maximally entangled qudit case by applying appropriate Hadamard and Controlled-NOT gates.
Do stochastic Schrödinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system {\em on average} obeys a master equation, the answer is yes. Markovian stochastic Schrödinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic \sch equation introduced by Strunz, Di\' …
We consider the two-state space-inhomogeneous coined quantum walk (QW) in one dimension. For a general setting, we obtain the stationary measure of the QW by solving the eigenvalue problem. As a corollary, stationary measures of the multi-defect model and space-homogeneous QW are derived. The former is a generalization of the previous works on one-defect model and the latter is a generalization of the result given by Konno and Takei (2015).
In this article we discuss the formal structure of a generalized information theory based on the extension of the probability calculus of Kolmogorov to a (possibly) non-commutative setting. By studying this framework, we argue that quantum information can be considered as a particular case of a huge family of non-commutative extensions of its classical counterpart. In any conceivable information theory, the possibility of dealing with different kinds of information measures plays a key role. Here, we generalize a notion of state spectrum, allowing us to introduce a majorization relation and a …
A quantum mechanical generalization of superstatistics is presented here based on the positive operator valued measure transformation property of the system density matrix. This procedure reveals that the origin of the fluctuating factors occurring in the derivation of the superstatistics lies in the choice of the transformation operators governing the dynamics of fluctuations. This generalization addresses situations such as nanosystems based on quantum devices (e.g., superconducting devices, single electron transistors, etc,) operating at low temperatures where cognizance of quantum fluctuat…
The way Quantum Mechanics (QM) is introduced to people used to Classical Mechanics (CM) is by a complete change of the general methodology) despite QM historically stemming from CM as a means to explain experimental results. Therefore, it is desirable to build a bridge from CM to QM. This paper presents a generalization of CM to QM. It starts from the generalization of a point-like object and naturally arrives at the quantum state vector of quantum systems in the complex valued Hilbert space, its time evolution and quantum representation of a measurement apparatus of any size. Each time, when …
