Relative Entropy Of Entanglement Research Articles

New Additivity Properties of the Relative Entropy of Entanglement and Its Generalizations

New Additivity Properties of the Relative Entropy of Entanglement and Its Generalizations

Continuity of the relative entropy of resource

We present a criterion to establish the local continuity of the relative entropy of resource, i.e. the relative entropy distance to the set of free states, in any quantum resource theory. Several basic corollaries of this criterion are presented. Applications to the relative entropy of entanglement in multipartite quantum systems are considered. It is shown, in particular, that local continuity of any relative entropy of multipartite entanglement follows from local continuity of the quantum mutual information.

Resource-efficient quantum principal component analysis

Principal component analysis (PCA) is an important dimensionality reduction method in machine learning and data analysis. Recently, the quantum version of PCA has been established to diagonalize quantum states. Although these quantum algorithms promise quantum advantages, they require substantial resources beyond the reach of state-of-the-art quantum technologies. This work aims to reduce resource requirements and improve the efficiency of quantum PCA. Assuming that the quantum state is accessed through a purified quantum query model and a sampling model, we propose quantum algorithms that use minimal resource requirements for ancillary qubits to reveal properties of eigenvectors and eigenvalues of a state. In particular, we show that estimating eigenvalue λ with error ε and success probability larger than λ(1−η) requests a query complexity O~(ϵ−1) and a sample complexity O~(ϵ−2η−1) , respectively. To our knowledge, our result is the first quantum speedup that achieves asymptotic linear scaling in 1/ϵ for quantum PCA. As applications, we discuss estimating the minimum relative entropy of entanglement of bipartite pure-states and performing quantum state discrimination tasks. We show that quantum speedups are maintained when the pure state has a low Schmidt number and states of discrimination have a low rank. This study opens up a new quantum PCA method for high-dimensional quantum data analysis and discusses its application in quantum information processing tasks.

Entanglement preservation in tripartite quantum systems under dephasing dynamics

Protecting entanglement from decoherence is a critical aspect of quantum information processsing. For many-body quantum systems evolving under decoherence, estimating multipartite entanglement is challenging. This challenge can be met up by considering a distance-based measure such as relative entropy of entanglement which decisively measures entanglement in both pure as well as mixed states. In this work, we investigate the tripartite entanglement dynamics of pure and mixed states in the presence of a structured dephasing environment at finite temperature. We show that the robustness of the quantum system to decoherence is dependent on the distribution of entanglement and its relation to different configurations of the bath. If the bath is structured individually such that each qubit has its own environment, the system has different dynamics compared to when the bath is common to all the three qubits. From the results we conjecture that there is a connection between the distribution of entanglement among the qubits and the distribution of bath degrees of freedom, and the interplay of these two distributions determines the decay rate of the entanglement dynamics. The sustainability of tripartite entanglement is shown to be enhanced significantly in the presence of reservoir memory.

Detecting entanglement by pure bosonic extension

In the realm of quantum information theory, the detection and quantification of quantum entanglement stand as paramount tasks. The relative entropy of entanglement (REE) serves as a prominent measure of entanglement, with extensive applications spanning numerous related fields. The positive partial transpose (PPT) criterion, while providing an efficient method for the computation of REE, unfortunately, falls short when dealing with bound entanglement. In this study, we propose a method termed “pure bosonic extension” to enhance the practicability of k-bosonic extensions, which approximates the set of separable states from the “outside”, through a hierarchical structure. It enables efficient characterization of the set of k-bosonic extendible states, facilitating the derivation of accurate lower bounds for REE. Compared to the semi-definite programming (SDP) approach, such as the symmetric/bosonic extension function in QETLAB, our algorithm supports much larger dimensions and higher values of extension k. Published by the American Physical Society 2024

Entanglement Monogamy via Multivariate Trace Inequalities

Entropy is a fundamental concept in quantum information theory that allows to quantify entanglement and investigate its properties, for example its monogamy over multipartite systems. Here, we derive variational formulas for relative entropies based on restricted measurements of multipartite quantum systems. By combining these with multivariate matrix trace inequalities, we recover and sometimes strengthen various existing entanglement monogamy inequalities. In particular, we give direct, matrix-analysis-based proofs for the faithfulness of squashed entanglement by relating it to the relative entropy of entanglement measured with one-way local operations and classical communication, as well as for the faithfulness of conditional entanglement of mutual information by relating it to the separably measured relative entropy of entanglement. We discuss variations of these results using the relative entropy to states with positive partial transpose, and multipartite setups. Our results simplify and generalize previous derivations in the literature that employed operational arguments about the asymptotic achievability of information-theoretic tasks.

Algorithm for evaluating distance-based entanglement measures

Abstract Quantifying entanglement in quantum systems is an important yet challenging task due to its NPhard nature. In this work, we propose an efficient algorithm for evaluating distance-based entanglement measures. Our approach builds on Gilbert’s algorithm for convex optimization, providing a reliable upper bound on the entanglement of a given arbitrary state. We demonstrate the effectiveness of our algorithm by applying it to various examples, such as calculating the squared Bures metric of entanglement as well as the relative entropy of entanglement for GHZ states, W states, Horodecki states, and chessboard states. These results demonstrate that our algorithm is a versatile and accurate tool that can quickly provide reliable upper bounds for entanglement measures.

Attainability and Lower Semi-continuity of the Relative Entropy of Entanglement and Variations on the Theme

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.

Upper Bounds on Device-Independent Quantum Key Distribution Rates in Static and Dynamic Scenarios

In this work, we develop upper bounds for key rates for device-independent quantum key distribution (DI-QKD) protocols and devices. We study the reduced cc-squashed entanglement and show that it is a convex functional. As a result, we show that the convex hull of the currently known bounds is a tighter upper bound on the device-independent key rates of standard CHSH-based protocol. We further provide tighter bounds for DI-QKD key rates achievable by any protocol applied to the CHSH-based device. This bound is based on reduced relative entropy of entanglement optimized over decompositions into local and non-local parts. In the dynamical scenario of quantum channels, we obtain upper bounds for device-independent private capacity for the CHSH based protocols. We show that the device-independent private capacity for the CHSH based protocols on depolarizing and erasure channels is limited by the secret key capacity of dephasing channels.

Capacity of entanglement for a nonlocal Hamiltonian

The notion of capacity of entanglement is the quantum information theoretic counterpart of the heat capacity which is defined as the second cumulant of the entanglement spectrum. Given any bipartite pure state, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the reduced state of any of the subsystems. Here, we study the dynamics of this quantity under a nonlocal Hamiltonian. Specifically, we address the following question: Given an arbitrary nonlocal Hamiltonian, what is the capacity of entanglement that the system can possess? As a useful application, we show that the quantum speed limit for creating the entanglement is not only governed by the fluctuation in the nonlocal Hamiltonian, but also depends inversely on the time average of the square root of the capacity of entanglement. Furthermore, we discuss this quantity for a general self-inverse Hamiltonian and provide a bound on the rate of capacity of entanglement. Towards the end, we generalize the capacity of entanglement for bipartite mixed states based on the relative entropy of entanglement and show that the above definition reduces to the capacity of entanglement for pure bipartite states. Our results can have several applications in diverse areas of physics.

Heat current and entropy production rate in local non-Markovian quantum dynamics of global Markovian evolution

We examine the elements of the balance equation of entropy in open quantum evolutions and their response as we go from a Markovian to a non-Markovian situation. In particular, we look at the heat current and entropy production rate in the non-Markovian reduced evolution, as well as a Markovian limit of the same, experienced by one of two interacting systems immersed in a Markovian bath. The analysis naturally leads us to define a heat current deficit and an entropy production rate deficit, which are differences between the global and local versions of the corresponding quantities. The investigation leads, in certain cases, to a complementarity of the time-integrated heat current deficit and the relative entropy of entanglement between the two systems.

Quantum transitions, ergodicity, and quantum scars in the coupled top model.

We consider an interacting collective spin model known as coupled top (CT), exhibiting a rich variety of phenomena related to quantum transitions, ergodicity, and formation of quantum scars, discussed in our previous work [Mondal, Sinha, and Sinha, Phys. Rev. E 102, 020101(R) (2020)2470-004510.1103/PhysRevE.102.020101]. In this work, we present a detailed analysis of the different type of transitions in the CT model, and find their connection with the underlying collective spin dynamics. Apart from the quantum scarring phenomena, we also identify another source of deviation from ergodicity due to the presence of nonergodic multifractal states. The degree of ergodicity of the eigenstates across the energy band is quantified from the relative entanglement entropy as well as multifractal dimensions, which can be probed from nonequilibrium dynamics. Finally, we discuss the detection of nonergodic behavior and different types of quantum scars using "out-of-time-order correlators," which has relevance in the recent experiments.

Characterizing entanglement using quantum discord over state extensions

We propose a framework to characterize entanglement with quantum discord, both asymmetric and symmetric, over state extensions. In particular, we show that the minimal Bures distance of discord over state extensions is equivalent to the Bures distance of entanglement. This equivalence places quantum discord in a more primitive position than entanglement conceptually in the sense that entanglement can be interpreted as an irreducible part of discord over all state extensions. Based on this equivalence, we also offer an operational meaning of the Bures distance of entanglement by connecting it to quantum state discrimination. Moreover, for the relative entropy part, we prove that the entanglement measure introduced by Devi and Rajagopal [A. R. U. Devi and A. K. Rajagopal, Phys. Rev. Lett. 100, 140502 (2008)] is actually equivalent to the relative entropy of entanglement. We also provide several quantifications of entanglement based on discord measures.

Geometry of Krylov complexity

We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics, and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues toward computation of Lanczos coefficients, and it sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples, among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy, and capacity of entanglement.

Quantifying multipartite quantum entanglement in a semi-device-independent manner

We propose two semi-device-independent approaches that are able to quantify unknown multipartite quantum entanglement experimentally, where the only information that has to be known beforehand is quantum dimension, and the concept that plays a key role is nondegenerate Bell inequalities. Specifically, using the nondegeneracy of multipartite Bell inequalities, we obtain useful information on the purity of target quantum state. Combined with an estimate of the maximal overlap between the target state and pure product states, and a continuous property of the geometric measure of entanglement, we shall prove the information on purity allows us to give a lower bound for this entanglement measure. In addition, we show that a different combination of the above results also converts to a lower bound for the relative entropy of entanglement. As a demonstration, we apply our approach on 5-partite qubit systems with the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and show that useful lower bounds for the geometric measure of entanglement can be obtained if the Bell expression value is larger than 3.60, and those for the relative entropy of entanglement can be given if the Bell expression value is larger than 3.80, where the Tsirelson bound is 4.

All entangled states are quantum coherent with locally distinguishable bases

We find that a bipartite quantum state is entangled if and only if it is quantum coherent with respect to complete bases of states in the corresponding system that are distinguishable under local quantum operations and classical communication. The corresponding minimal quantum coherence is the entanglement of formation. Connections to the relative entropy of entanglement and quantum coherence, and generalizations to the multiparty case are also considered.

Quantifying resources for the Page-Wootters mechanism: Shared asymmetry as relative entropy of entanglement

Recently, some attention has been given to the so-called Page-Wootters mechanism of quantum clocks. Among the various proposals to explore the mechanism using more modern techniques, some have chosen to use a quantum information perspective, defining and using informational measures to quantify how well a quantum system can stand as a reference frame for other quantum system. In this work, we explore the proposal based on a resource theory of asymmetry, known as mutual or shared asymmetry, which actually is equivalent to the approach from coherence theory in the case of interest here: quantum reference frames described by the $\text{U}(1)$ compact group. We extend some previous results in the literature about shared asymmetry and the Page-Wootters mechanism to more general cases, culminating in the enunciation of a theorem relating shared asymmetry of a bipartite state ${\ensuremath{\rho}}_{SR}$ with the relative entropy of entanglement of internal states ${\ensuremath{\rho}}_{M}$ on the charge sectors of the Hilbert space ${\mathcal{H}}_{S}\ensuremath{\bigotimes}{\mathcal{H}}_{R}$. Using this result, we reinterpret the relation between the Page-Wootters mechanism and entanglement, and we also open some paths to further studies.

The verification of a requirement of entanglement measures

The quantification of quantum entanglement is a central issue in quantum information theory. Recently, Gao \emph{et al}. ( \href{http://dx.doi.org/10.1103/PhysRevLett.112.180501}{Phys. Rev. Lett. \textbf{112}, 180501 (2014)}) pointed out that the maximum of entanglement measure of the permutational invariant part of $\rho$ ought to be a lower bound on entanglement measure of the original state $\rho$, and proposed that this argument can be used as an additional requirement for (multipartite) entanglement measures. Whether any individual proposed entanglement measure satisfies the requirement still has to prove. In this work, we show that most known entanglement measures of bipartite quantum systems satisfy the new criterion, include all convex-roof entanglement measures, the relative entropy of entanglement, the negativity, the logarithmic negativity and the logarithmic convex-roof extended negativity. Our results give a refinement in quantifying entanglement and provide new insights into a better understanding of entanglement properties of quantum systems.

Closest separable state when measured by a quasi-relative entropy

It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.

Genuine secret-sharing states

Quantum secret sharing allows each player to have classical information for secret sharing in quantum mechanical ways. In this work, we construct a class of quantum states on which players can quantumly perform secret sharing secure against dishonest players as well as eavesdropper. We here call them the genuine secret-sharing states. In addition, we show that if $N$ players share an $N$-party genuine secret-sharing state, then arbitrary $M$ players out of the total players can share an $M$-party genuine secret-sharing state by means of local operations and classical communication on the state. We also define the distillable rate with respect to the genuine secret-sharing state, and explain the connection between the distillable rate and the relative entropy of entanglement.