It was recently shown that the nonseparable density operators for a bipartite system are trace norm dense if either factor space has infinite dimension. We show here that non-local states -- i.e., states whose correlations cannot be reproduced by any local hidden variable model -- are also dense. Our constructions distinguish between the cases where both factor spaces are infinite-dimensional, where we show that states violating the CHSH inequality are dense, and the case where only one factor space is infinite-dimensional, where we identify open neighborhoods of nonseparable states that do not violate the CHSH inequality but show that states with a subtler form of non-locality (often called "hidden" non-locality) remain dense.
Based on set theoretic ordering properties, a general formulation for constructing a pair of convertibility monotones, which are generalizations of distillable entanglement and entanglement cost, is presented. The new pair of monotones do not always coincide for pure bipartite infinite dimensional states under SLOCC (stochastic local operations and classical communications), demonstrating the existence of SLOCC incomparable pure bipartite states, a new property of entanglement in infinite dimensional systems, with no counterpart in the corresponding finite dimensional systems.
It is shown that the order property of pure bipartite states under SLOCC (stochastic local operations and classical communications) changes radically when dimensionality shifts from finite to infinite. In contrast to finite dimensional systems where there is no pure incomparable state, the existence of infinitely many mutually SLOCC incomparable states is shown for infinite dimensional systems even under the bounded energy and finite information exchange condition. These results show that the effect of the infinite dimensionality of Hilbert space, the ``infinite workspace'' property, remains e…
In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let $M$ be the matrix of the amplitudes of $\ketψ$, we prove $M$ is a compact operator. We also prove $\ketψ$ is separable if and only if $M$ is a bounded linear operator with rank 1, that is the image of $M$ is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting proper…
Given a bipartite quantum system represented by a tensor product of two Hilbert spaces, we give an elementary argument showing that if either component space is infinite-dimensional, then the set of nonseparable density operators is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability, which show that when both component Hilbert spaces are finite-dimensional, there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.
We introduce an algebraic measure of correlations in bipartite quantum systems. The proposed quantity, called maximal mutual correlation, provides the information how much a given state differs from the product state of its marginals. In contrast to the entanglement or quantum discord, maximal mutual correlation can be non - zero for some classically correlated quantum states.
Monogamy of bipartite correlations leads, for arbitrary pure multi-qubit states, to simple conditions able to indicate various types of multipartite entanglement by being capable to exclude the possibility of k-separability.
How can we characterize different types of correlation between quantum systems? Since correlations cannot be generated locally, we take any real function of a multipartite state which cannot increase under local operations to measure a correlation. Correlation measures that can be expressed as an optimization of a linear combination of entropies are particularly useful, since they can often be interpreted operationally. We systematically study such optimized linear entropic functions, and by enforcing monotonicity under local processing we identify four cones of correlation measures for bipart…
The Hilbert space formalism of quantum theory manifests a map between bipartite states and time evolutions, known as Jamiolkowski isomorphism. We extend this map in a physical setting to prove the equality of spatial correlations in bipartite systems and temporal correlations in local systems. We show that these correlations can be observed using weak measurements. This result has several practical and conceptual implications such as manifestation of state independent decoherence, the correspondence between Bell and Leggett-Garg inequalities, multipartite systems, the statistical properties of…
In this work we introduce a nonlinear witness that is a sufficient condition for detecting the vanishment of quantum correlation of bipartite states. Our result directly generalizes the result of [J. Maziero, R. M. Serra, arXiv:1012.3075] to arbitrary dimension based on the Bloch representation of density matrices.
Superquantum ("PR-box") correlations, though designed to respect relativistic causality, violate relativistic causality in the classical limit. Generalizing to all stronger-than-quantum bipartite correlations, I derive Tsirelson's bound from the axioms of nonlocality, relativistic causality and the existence of a classical limit. This derivation of Tsirelson's bound does not assume quantum mechanics yet suggests how Hilbert space is implicit in quantum correlations.
In this paper we discuss the problem of splitting the total correlations for a bipartite quantum state described by the Von Neumann mutual information into classical and quantum parts. We propose a measure of the classical correlations as the difference between the Von Neumann mutual information and the relative entropy of entanglement. We compare this measure with different measures proposed in the literature.
We present a protocol to simulate the quantum correlations of an arbitrary bipartite state, when the parties perform a measurement according to two traceless binary observables. We show that $\log(d)$ bits of classical communication is enough on average, where $d$ is the dimension of both systems. To obtain this result, we use the sampling approach for simulating the quantum correlations. We discuss how to use this method in the case of qudits.
Many three-party correlations, including some that are commonly described as genuinely tripartite nonlocal, can be simulated by a network of underlying subsystems that display only bipartite nonsignaling nonlocal behavior. Quantum mechanics predicts three-party correlations that admit no such simulation, suggesting there are versions of nonlocality in nature transcending the phenomenon of bipartite nonsignaling nonlocality. This paper introduces a rigorous framework for analyzing tripartite correlations that can be simulated by bipartite-only networks. We confirm that expected properties of so…
Based on the relative entropy, we give a unified characterization of quantum correlations for nonlocality, steerability, discord and entanglement for any bipartite quantum states. For two-qubit states we show that the quantities obtained from quantifying nonlocality, steerability, entanglement and discord have strictly monotonic relationship. As for examples, the Bell diagonal states are studied in detail.
A definition of quantum correlation is presented for an arbitrary bipartite quantum state based on the skew information. This definition not only inherits the good properties of skew information such as the contractivity and so on, but also is effective and almost analytically calculated for any bipartite quantum states. We also reveal the relation between our measure and quantum metrology. As applications, we give the exact expressions of quantum correlation for many states, which provides a direct support for our result.
We show that the quantum discord and the measurement induced non-locality (MiN) in a bipartite quantum state is invariant under the action of a local quantum channel if and only if the channel is invertible. In particular, these quantities are invariant under a local unitary channel.
The teleportation channel associated with an arbitrary bipartite state denotes the map that represents the change suffered by a teleported state when the bipartite state is used instead of the ideal maximally entangled state for teleportation. This work presents and proves an explicit expression of the teleportation channel for the teleportation using Weyl's projective unitary representation of the space of 2n-tuples of numbers from Z/dZ for integers d>1, n>0, which has been known for n=1. This formula allows any correlation among the n bipartite mixed states, and an application shows the exis…
Everett's concept of relative state can be viewed as a map that contains information about correlations between measurement outcomes on two quantum systems. We demonstrate how geometric properties of the relative state map can be used to develop operationally well-defined measures of the total correlation in bipartite quantum systems of arbitrary state space dimension. These measures are invariant under local unitary transformations and non-increasing under local operations. We show that some known correlation measures have a natural interpretation in terms of relative states.
For a bipartite quantum system consisting of subsystems A and B it was shown by Zhang et al. (Physics Letters A 376 (2012) 3588-3592) that the amount of classical correlations, which is used to define the quantum discord, is known to be bounded from above by the minimum of the von Neumann entropies of the subsystems A and B. We provide an alternative proof that is shorter and more transparent as it works without defining correlation matrices.
In this work we developed a general approach to the problem of detecting and quantifying different kind of correlations in bipartite quantum systems. Our method is based on the use of distances between quantum states and processes. We rely upon the premise that total correlations can be separated into classical and quantum contributions due to their different nature. In addition, according to recently discussed criteria, we determined the requirements to be satisfied by distances in order to generate correlation measures physically well behaved. The proposed measures allow us to quantify quant…
Necessary and sufficient conditions for the existence of a composite-system statistical operator, and, separately, for the possibility of its being correlated or uncorrelated, are derived in terms of its range dimension and the range dimensions of its reduced statistical operators.
We discuss the problem of separating the total correlations in a given quantum joint probability distribution into nonlocality, contextuality and classical correlations. Bell discord and Mermin discord which qunatify nonlocality and contextuality of quantum correlations are interpreted as distance measures in the nonsignaling polytope. A measure of total correlations is introduced to divide the total amount of correlations into a purely nonclassical part and a classical part. We show that quantum correlations satisfy additivity relations among these three measures.
Quantum discord has been studied extensively as a measure of non-classical correlations which includes entanglement as a subset. Although it is well known that non-zero discord can exist without entanglement, the origin of quantum discord is not well understood as compared to entanglement, which manifests itself more simply as inseparable higher dimensional quantum superposition. In this paper we establish the discordlike correlation of bipartite coherence and then compare it to quantum discord. Consequently, we show that the minimum of the discordlike correlation of coherence coincides with t…
What singles out quantum mechanics as the fundamental theory of Nature? Here we study local measurements in generalised probabilistic theories (GPTs) and investigate how observational limitations affect the production of correlations. We find that if only a subset of typical local measurements can be made then all the bipartite correlations produced in a GPT can be simulated to a high degree of accuracy by quantum mechanics. Our result makes use of a generalisation of Dvoretzky's theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.
In this letter we study thermal quantum correlations as quantum discord and entanglement in bipartite system imposed by external magnetic field with Herring-Flicker coupling ie. $J(R)=1.642 e^{-2 R} R^{5/2}+O(R^{2}e^{-2R})$. The Herring-Flicker coupling strength is the function of $R$, which is the distance between spins and systems carry XXX Heisenberg interaction. By tuning the coupling distance $R$, temperature and magnetic field quantum correlations can be scaled in the bipartite system. We find the long sustainable behaviour of quantum discord in comparison to entanglement over the coupli…
We introduce a class of inequalities based on low order correlations of operators to detect entanglement in bipartite systems. The operators may either be Hermitian or non-Hermitian and are applicable to any physical system or class of states. Testing the criteria on example systems reveals that they outperform other common correlation based criteria, such as those by Duan-Giedke-Cirac-Zoller and Hillery-Zubairy. One unusual feature of the criteria is that it includes a term involving the average of the density matrix squared, related to the purity of the system. We discuss how such a term cou…
The maximum observable correlation between the two components of a bipartite quantum system is a property of the joint density operator, and is achieved by making particular measurements on the respective components. For pure states it corresponds to making measurements diagonal in a corresponding Schmidt basis. More generally, it is shown that the maximum correlation may be characterised in terms of a `correlation basis' for the joint density operator, which defines the corresponding (nondegenerate) optimal measurements. The maximum coincidence rate for spin measurements on two-qubit systems …
Monogamy is a nonclassical property that limits the distribution of quantum correlation among subparts of a multiparty system. We show that monogamy scores for different quantum correlation measures are bounded above by functions of genuine multipartite entanglement for a large majority of pure multiqubit states. The bound is universal for all three-qubit pure states. We derive necessary conditions to characterize the states that violate the bound, which can also be observed by numerical simulation for a small set of states, generated Haar uniformly. The results indicate that genuine multipart…
Simulating quantum nonlocality and steering requires augmenting pre-shared randomness with non-vanishing communication cost. This prompts the question of how one may provide such an operational characterization for the quantumness of correlations due even to unentangled states. Here we show that for a certain class of states, such quantumness can be pointed out by superlocality, the requirement for a larger dimension of the pre-shared randomness to simulate the correlations than that of the quantum state that generates them. This provides an approach to define the nonclassicality of local mult…
We find a single parameter family of genuinely entangled three qubit pure states, called the maximally Bell inequality violating states (MBV), which exhibit maximum Bell inequality violation by the reduced bipartite system for a fixed amount of genuine tripartite entanglement quantified by the so called tangle measure. This in turn implies that there holds a complementary relation between the Bell inequality violation by the reduced bipartite systems and the tangle present in the three qubit states, not necessarily pure. The MBV states also exhibit maximum Bell inequality violation by the redu…
Quantum correlations are of fundamental importance in quantum phenomena and quantum information processing studies. The measure of quantum correlations is one central issue. The recently proposed measure of quantum correlations, the local quantum uncertainty (LQU), satisfies the full physical requirements of a measure of quantum correlations. In this work, by using operator relaxation, a closed form lower bound of the LQU for arbitrary-dimensional bipartite quantum states is derived. We have compared the lower bound and the optimized LQU for several typical quantum states.
Bell nonlocality and Einstein-Podolsky-Rosen (EPR) steering are every important quantum correlations of a composite quantum system. Bell nonlocality of a bipartite state is a quantum correlation demonstrated by some local quantum measurements, while EPR steering is another form of quantum correlations, observed firstly by Schrodinger in the context of famous EPR paradox. In this paper, we give some remarks on Bell nonlocality and EPR steering of bipartite states, including mathematical definitions and characterizations of these two quantum correlations, the convexity and closedness of the set …
We show that some tripartite quantum correlations are inexplicable by any causal theory involving bipartite nonclassical common causes and unlimited shared randomness. This constitutes a device-independent proof that Nature's nonlocality is fundamentally at least tripartite in every conceivable physical theory - no matter how exotic. To formalize this claim we are compelled to substitute Svetlichny's historical definition of genuine tripartite nonlocality with a novel theory-agnostic definition tied to the framework of Local Operations and Shared Randomness (LOSR). A companion article [PRA. 10…
We present a practical scheme for the decomposition of a bipartite mixed state into a sum of direct products of local density matrices, using the technique developed in Li and Qiao (Sci. Rep. 8: 1442, 2018). In the scheme, the correlation matrix which characterizes the bipartite entanglement is first decomposed into two matrices composed of the Bloch vectors of local states. Then we show that the symmetries of Bloch vectors are consistent with that of the correlation matrix, and the magnitudes of the local Bloch vectors are lower bounded by the correlation matrix. Concrete examples for the sep…
One of the last open problems concerning two qubits in a pure state is to find the exact local content of their correlation, in the sense of Elitzur, Popescu and Rohrlich (EPR2) [Phys. Lett. A 162, 25 (1992)]. We propose a new EPR2 decomposition that allows us to prove, for a wide range of states psi(theta)=cos(theta)|00>+sin(theta)|11>, that their local content is pL(theta)=cos(2theta). We also share reflections on how to possibly extend our result to all two-qubit pure states.
Let us consider the set of joint quantum correlations arising from two-outcome local measurements on a bipartite quantum system. We prove that no finite dimension is sufficient to generate all these sets. We approach the problem in two different ways by constructing explicit examples for every dimension d, which demonstrates that there exist bipartite correlations that necessitate d-dimensional local quantum systems in order to generate them. We also show that at least 10 two-outcome measurements must be carried out by the two parties altogether so as to generate bipartite joint correlations n…
We advocate the step change in properties of discrete $d$-level quantum systems, between $d=2$ and $d\geq 3$. Qubit systems, or multipartite systems containing qubit subsystem, are exceptional in their relative simplicity. One faces a step in complexity in valuating measures of quantum correlations for qutrits and then other higher dimensional qudits. There is a growing number of arguments leading to such conclusion: recently found no-go theorem for generalization of the Peres-Horodecki's PPT criterion \cite{sko}, change in geometry of state spaces of qubit and higher degree qudits (the so cal…
We give a set of necessary conditions for locality in bipartite systems, which include and generalize known Bell's inequalities. Each condition corresponds to a specific order of the expansion of random variables defined on graphs, in terms of genuinely n-variable correlation functions. The first non-trivial order leads to known Bell inequalities, while higher orders produce additional, non-equivalent conditions. In particular, in CHSH settings, we obtain at least two additional tight conditions which do not reduce to the CHSH inequality. This shows that tight Bell inequalities are sufficient …
Having the quantum correlations in a general bipartite state in mind, the information accessible by simultaneous measurement on both subsystems is shown never to exceed the information accessible by measurement on one subsystem, which, in turn is proved not to exceed the von Neumann mutual information. A particular pair of (opposite- subsystem) observables are shown to be responsible both for the amount of quasi-classical correlations and for that of the purely quantum entanglement in the pure-state case: the former via simultaneous subsystem measurements, and the latter through the entropy of…
We provide a physically motivated definition for the binding energy (or bond-dissociation) of a bipartite quantum system. We consider coherently applying an external field to cancel out the interaction between the subsystems, to break their bond and separate them as systems from which no work can be extracted coherently by any cyclic evolution. The minimum difference between the average energies of the initial and final states obtained this way is defined as the binding energy of the bipartite system. We show the final optimal state is a passive state. We also discuss how required evolution ca…
A direct analysis of the protocol of randomness amplification using Bell inequality violation is performed in terms of the convex combination of no-signaling boxes required to simulate quantum violation of the inequality. The probability distributions of bits generated by a Santha-Vazirani source are shown to be mixtures of permutations of Bernoulli distributions with parameter defined by the source. An intuitive proof is provided for the range of partial randomness from which perfect randomness can be extracted using quantum correlations violating the chain inequalities. Exact values are deri…
The strong connection between correlations and quantum thermodynamics raises a natural question about the preparation of correlated quantum states from two copies of a thermal qubit. In this work we study the specific forms of allowed and forbidden bipartite correlations. As a consequence, we extend the result to Separable (SEP) but not Absolutely Separable (AbSEP) class of product states. Preparation of a general form of entanglement from arbitrary thermal qubits is studied and as an application we propose a strategy to establish sustained entanglement between two distant parties. The thresho…
Device-independent self-testing allows to uniquely characterize the quantum state shared by untrusted parties (up to local isometries) by simply inspecting their correlations, and requiring only minimal assumptions, namely a no-signaling constraint on the untrusted parties and the validity of quantum mechanics. The device-independent approach exploits the fact that certain non-local correlations can be uniquely achieved by measurements on a particular quantum state. We can think of these correlations as a "classical fingerprint" of the self-tested quantum state. In this work, we answer affirma…
The local and non-local contents of non-local probability distributions are studied using the approach of Elitzur, Popescu and Rohrlich [Phys. Lett. A \textbf{162}, 25 (1992)]. This work focuses on distributions that can be obtained by single-copy von Neumann measurements on bipartite quantum systems. For pure two-qubit states Psi(theta)=cos(theta)|00>+sin(theta)|11>, with cos(theta)>=sin(theta), the local content of the corresponding probability distribution is found to lie between 1-sin(2*theta) and cos(2*theta). For the family Psi(gamma)= (|00>+|11>+gamma*|22>)/sqrt(2+gamma^2) of two-qutrit…
Quantum properties of correlations have a key role in disparate fields of physics, from quantum information processing, to quantum foundations, to strongly correlated systems. We tackle a specific aspect of the fundamental quantum marginal problem: we address the issue of deducing the global properties of correlations of tripartite quantum states based on the knowledge of their bipartite reductions, focusing on relating specific properties of bipartite correlations to global correlation properties. We prove that strictly classical bipartite correlations may still require global entanglement an…
We describe a criterion for the detection of entanglement between two multi-boson systems. The criterion is based on calculating correlations of Gell-Mann matrices with a fixed boson number on each subsystem. This applies naturally to systems such as two entangled spinor Bose-Einstein condensates. We apply our criterion to several experimentally motivated examples, such as an $ S^z S^z $ entangled BECs, ac Stark shift induced two-mode squeezed BECs, and photons under parametric down conversion. We find that entanglement can be detected for all parameter regions for the most general criterion. …
Suppose Alice and Bob share a maximally entangled state of any finite dimension and each perform two-outcome measurements on the respective part of the state. It is known, due to the recent result of Regev and Toner, that if a classical model is augmented with two bits of communication then all the quantum correlations arising from these measurements can be reproduced. Here we show that two bits of communication is in fact necessary for the perfect simulation. In particular, we prove that a pair of maximally entangled four-dimensional quantum systems cannot be simulated by a classical model au…
This thesis explores ways in which quantum channels and correlations (of both classical and quantum types) manifest themselves, and also studies the interplay between these two aspects in various physical settings. Quantum channels represent all possible evolutions of states allowed by quantum mechanics while correlations are intrinsic (nonlocal) properties of composite systems. There are four main problems that are addressed (i) the role played by initial system-bath correlations on the subsystem dynamics of an open quantum system, (ii) computation of correlations for two-qubit X-states, (iii…
Entanglement is at the heart of most quantum information tasks, and therefore considerable effort has been made to find methods of deciding the entanglement content of a given bipartite quantum state. Here, we prove a fundamental limitation to deciding if an unknown state is entangled or not: we show that any quantum measurement which can answer this question necessarily gives enough information to identify the state completely. Therefore, only prior information regarding the state can make entanglement detection less expensive than full state tomography in terms of the demanded quantum resour…
In this paper, two conjectures which were proposed in [Phys. Rev. A \textbf{82}, 052122(2010)] on the correlations in a bipartite state $ρ^{AB}$ are studied. If the mutual information $I\Pa{ρ^{AB}}$ between two quantum systems $A$ and $B$ before any measurement is considered as the total amount of correlations in the state $ρ^{AB}$, then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations $C\Pa{ρ^{AB}}$ in the state $ρ^{AB}$, defined by the maximizing mutual information between two quantum systems $A$ and $B$ after von Neuma…
Spin is a fundamental and distinctive property of the electron, having far-reaching consequences in wide areas of physics. Yet, further to its association with an angular momentum, the physics underpinning its formal treatment remains obscure. In this work we propose to advance in disclosing the meaning behind the formalism, by first recalling some basic facts about the one-particle spin operator. Consistently informed by and in line with the quantum formalism, we then proceed to analyse in detail the spin projection operator correlation function $C_Q(\boldsymbol{a}, \boldsymbol{b}) = \langle(…
Quantumness for a bipartite unsteerable quantum correlation is operationally characterized by the notion of super-unsteerability. Super-unsteerability refers to the requirement of a larger dimension of the random variable that the steering party has to preshare with the party to be steered in the classical simulation protocol to generate an unsteerable correlation than the local Hilbert space dimension of the quantum states (reproducing the given unsteerable correlation) at the steering party's side. In the present study, this concept of super-unsteerability is generalized by defining the noti…
We investigate the correlations of initially separable probability distributions in a globally pure bipartite system with two degrees of freedom for classical and quantum systems. A classical version of the quantum linear mutual information is introduced and the two quantities are compared for a system of oscillators coupled with both linear and non-linear interactions. The classical correlations help to understand how much of the quantum loss of purity are due to intrinsic quantum effects and how much is related to the probabilistic character of the initial states, a characteristic shared by …
The phenomenon of monogamy of Bell inequality violations is interesting both from the fundamental perspective as well as in cryptographic applications such as the extraction of randomness and secret bits. In this article, we derive new and stronger monogamy relations for violations of Bell inequalities in general no-signaling theories. These relations are applicable to the class of binary output correlation inequalities known as XOR games, and to a restricted set of unique games. In many instances of interest, we show that the derived relation provides a significant strengthening over previous…
A bipartite state is classical with respect to party $A$ if and only if party $A$ can perform nondisruptive local state identification (NDLID) by a projective measurement. Motivated by this we introduce a class of quantum correlation measures for an arbitrary bipartite state. The measures utilize the general Schatten $p$-norm to quantify the amount of departure from the necessary and sufficient condition of classicality of correlations provided by the concept of NDLID. We show that for the case of Hilbert-Schmidt norm, i.e. $p=2$, a closed formula is available for an arbitrary bipartite state.…
