We show that the possible ensembles produced when a separable operation acts on a single pure bipartite entangled state are completely characterized by a majorization condition, a collection of inequalities for Schmidt coefficients, which is identical to that already known for the particular case of local operations and classical communication (LOCC). As a consequence, various known results for LOCC, including some involving monotonicity of entanglement, can be extended to the class of all separable operations.
We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states…
We study the stability of NPT property of an arbitrary pure entangled state under the mixture of arbitrary pure separable states. For bipartite pure states with Schmidt number $n$ $(n>1)$ which is NPT, we show that this state is still NPT when it is mixed with no more than $\frac{n(n-1)}{2}-1$ arbitrary pure separable states. This result is generalized to multipartite case.
We give a conceptually simple necessary condition such that a separable quantum operation can be implemented by local operations on subsystems and classical communication between parties (LOCC), a condition which follows from a novel approach to understanding LOCC. This necessary condition holds for any number of parties and any finite number of rounds of communication and as such, also provides a completely general sufficient condition that a given separable operation cannot be exactly implemented by LOCC. Furthermore, it demonstrates an extremely strong difference between separable operation…
We describe a purely algebraic method for finding the best separable approximation to a mixed state of a composite 2x2 quantum system, consisting of a decomposition of the state into a linear combination of a mixed separable part and a pure entangled one. We prove that, in a generic case, the weight of the pure part in the decomposition equals the concurrence of the state.
For manipulations of multipartite quantum systems, it was well known that all local operations assisted by classical communication (LOCC) constitute a proper subset of the class of separable operations. Recently, Gheorghiu and Griffiths found that LOCC and general separable operations are equally powerful for transformations between bipartite pure states. In this letter we extend this comparison to mixed states and show that in general separable operations are strictly stronger than LOCC when transforming a mixed state to a pure entangled state. A remarkable consequence of our finding is the e…
We present a general description of separable states in Quantum Mechanics. In particular, our result gives an easy proof that inseparabitity (or entanglement) is a pure quantum (noncommutative) notion. This implies that distinction between separability and inseparabitity has sense only for composite systems consisting of pure quantum subsystems. Moreover, we provide the unified characterization of pure-state entanglement and mixed-state entanglement.
We establish relations between Segre variety, conifold, Hopf fibration, and separable sets of pure two-qubit states. Moreover, we investigate the geometry and topology of separable sets of pure multi-qubit states based on a complex multi-projective Segre variety and higher order Hopf fibration.
The optimal (pure state) ensemble length of a separable state, A, is the minimum number of (pure) product states needed in convex combination to construct A. We study the set of all separable states with optimal (pure state) ensemble length equal to k or fewer. Lower bounds on k are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest, the lower bound for non-pure state ensembles is sharp. A consequence of our results is that for all two partic…
Entanglement is the resource to overcome the restriction of operations to Local Operations assisted by Classical Communication (LOCC). The Maximally Entangled Set (MES) of states is the minimal set of n-partite pure states with the property that any truly n-partite entangled pure state can be obtained deterministically via LOCC from some state in this set. Hence, this set contains the most useful states for applications. In this work we characterize the MES for generic three qutrit states. Moreover, we analyze which generic three qutrit states are reachable (and convertible) under LOCC transfo…
We study conditions for the deterministic transformation $\ketψ\longrightarrow\ketϕ$ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a determinis…
We study how multi-partite entanglement evolves under the paradigm of separable operations, which include the local operations and classical communication (LOCC) as a special case. We prove that the average "decay" of entanglement induced by a separable operation is measure independent (among SL-invariant ones) and state independent: the ratio between the average output entanglement and the initial entanglement is solely a function of the separable operation, regardless of the input state and of the SL-invariant entanglement measure being used. We discuss the "disentangling power" of a quantum…
In the first part of this Dissertation, I study the differences between LOCC (local operations and classical communication) and the more general class of separable operations. I show that the two classes coincide for the case of pure bipartite state input, and derive a set of important consequences. Using similar techniques I also generalize the no-cloning theorem when restricted to separable operations and show that cloning becomes much more restrictive, by providing necessary (and sometimes sufficient) conditions. In the second part I investigate graph states and graph codes with carrier qud…
In this paper, we treat an asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state and the completely mixed state by one-way LOCC, two-way LOCC, and separable POVMs. As a result, we derive single-letterized formulas for the Stein's lemma type of optimal error exponents under one-way LOCC, two-way LOCC and separable POVMs, the Chernoff bounds under one-way LOCC POVMs and separable POVMs, and the Hoeffding bounds under one-way LOCC POVMs in the whole region of a parameter and under separable POVMs on a restricted…
We study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of $(D-1)$ pure states, where $D$ is the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of $2\otimes 2$ separable operations not being realizable by local opera…
We introduce an inductive $n$-qubit pure-state estimation method. This is based on projective measurements on states of $2n+1$ separable bases or $2$ entangled bases plus the computational basis. Thus, the total number of measurement bases scales as $O(n)$ and $O(1)$, respectively. Thereby, the proposed method exhibits a very favorable scaling in the number of qubits when compared to other estimation methods. Monte Carlo numerical experiments show that the method can achieve a high estimation fidelity. For instance, an average fidelity of $0.88$ on the Hilbert space of $10$ qubits is achieved …
We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure product states, and can be made separable by mixing them with one or two pure product states.
The class of local operations and classical communication (LOCC) pertains to an important measurement scenario in many quantum communication schemes. While LOCC belongs to the more general class of separable operations (SEP), the exact difference between the two remains a challenging open problem. In this article, we seek to better understand the structure of LOCC and its relationship to SEP by comparing their respective abilities for distilling EPR entanglement from one copy of an $N$-qubit W-class state (i.e. that of the form $\sqrt{x_0}\ket{00...0}+\sqrt{x_1}\ket{10...0}+...+\sqrt{x_n}\ket{…
We consider a class of entangled states of a quantum system (S) and a second system (A) where pure states of the former are correlated with mixed states of the latter, and work out the entanglement measure with reference to the nearest separable state. Such `pure-mixed' entanglement is expected when the system S interacts with a macroscopic measuring apparatus in a quantum measurement, where the quantum correlation is destroyed in the process of environment-induced decoherence whereafter only the classical correlation between S and A remains, the latter being large compared to the former. We p…
We consider an infinite class of unambiguous quantum state discrimination problems on multipartite systems, described by Hilbert space $\cal{H}$, of any number of parties. Restricting consideration to measurements that act only on $\cal{H}$, we find the optimal global measurement for each element of this class, achieving the maximum possible success probability of $1/2$ in all cases. This measurement turns out to be both separable and unique, and by our recently discovered necessary condition for local quantum operations and classical communication (LOCC), it is easily shown to be impossible b…
We show that pure states of multipartite quantum systems are multiseparable (i.e. give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the quantification of multipartite pure-state entanglement are discussed. Further, as an application of the techniques used here, we show that any purification of a bipartite PPT bound entangled state is tri-inseparable, i.e. has none of its three bipartite partial traces separable.
Local operations with classical communication (LOCC) and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement. Separable operations are strictly more powerful than LOCC, but no simple explanation of this phenomenon is known. We show that, in the case of von Neumann measurements, the ability to interpolate measurements is an operational principle that sets apart LOCC and separable operations.
We present a general method for constructing pure-product-state representations for density operators of $N$ quantum bits. If such a representation has nonnegative expansion coefficients, it provides an explicit separable ensemble for the density operator. We derive the condition for separability of a mixture of the Greenberger-Horne-Zeilinger state with the maximally mixed state.
Every entangled state can be perturbed, for instance by decoherence, and stay entangled. For a large class of pure entangled states, we show how large the perturbation can be. Our class includes all pure bipartite and all maximally entangled states. For an entangled state, E, the constucted neighborhood of entangled states is the region outside two parallel hyperplanes, which sandwich the set of all separable states. The states for which these neighborhoods are largest are the maximally entangled ones. As the number of particles, or the dimensions of the Hilbert spaces for two of the particles…
We present a closest separable state to cluster states. We start by considering linear cluster chains and extend our method to cluster states that can be used as a universal resource in quantum computation. We reproduce known results for pure cluster states and show how our method can be used in quantifying entanglement in noisy cluster states. Operational meaning is given to our method that clearly demonstrates how these closest separable states can be constructed from two-qubit clusters in the case of pure states. We also discuss the issue of finding the critical temperature at which the clu…
We introduce algebriac sets in the products of complex projective spaces for multipartite mixed states, which are independent of their eigenvalues and only measure the "position" of their eigenvectors, as their non-local invariants (ie. remaining invariant after local untary transformations). These invariants are naturally arised from the physical consideration of checking multipartite mixed states by measuring them with multipartite separable pure states. The algebraic sets have to be the sum of linear subspaces if the multipartite mixed state is separable, and thus we give a new separability…
Linear maps preserving pure states of a quantum system of any dimension are characterized. This is then used to establish a structure theorem for linear maps that preserve separable pure states in multipartite systems. As an application, a characterization of separable pure state preserving affine maps is obtained.
We classify the completely-positive maps acting on two $d$-dimensional systems which commute with all $U\otimes U$ unitaries, where $U\in SU(d)$. This set of operations map Werner states to Werner states. We find a simple condition for a map being implementable by stochastic local operations and classical communication (SLOCC). We show that all PPT-preserving maps can be implemented by SLOCC. This can be used to prove that the entanglement of Werner states cannot be stochastically increased, even if we allow PPT entanglement for free.
In a recent work [Phys. Rev. Lett. 120, 170502 (2018)], Pallister et al. proposed an optimal strategy to verify non-maximally entangled two-qubit pure states under the constraint that the accessible measurements being locally projective and non-adaptive. Their nice result leads naturally to the question: What is the optimal strategy among general LOCC measurements? In this Letter, we answer this problem completely for two-qubit pure states. To be specific, we give the optimal strategy for each of the following available classes of measurements: (i) local operations and one-way classical commun…
A density operator of a bipartite quantum system is called robustly separable if it has a neighborhood of separable operators. Given a bipartite density matrix, its property to be robustly separable is reduced, using the continuous ensemble method, to a finite number of numerical equations. The solution of this system exists for any robustly separable density operator and provides its representation by a continuous mixture of pure product states.
Quantum separable operations are defined as those that cannot produce entanglement from separable states, and it is known that they strictly surpass local operations and classical communication (LOCC) in a number of tasks, which is sometimes referred to as "quantum nonlocality without entanglement." Here we consider a task with such a gap regarding the trade-off between state discrimination and preservation of entanglement. We show that this task along with the gap has an analogue in a purely classical setup, indicating that the quantum properties are not essential in the existence of a nonzer…
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 t…
We find that when a qubit initialized in a pure state experiences pure dephasing due to interaction with an environment, separable qubit-environment states generated during the evolution also have zero quantum discord with respect to the environment. What follows is that the set of separable states which can be reached during the evolution has zero volume and hence, such effects as sudden death of qubit-environment entanglement are very unlikely. In case of the discord with respect to the qubit, a vast majority of separable states qubit-environment is discordant, but in specific situations zer…
The problem of optimally estimating an unknown unitary quantum operation with the aid of entanglement is addressed. The idea is to prepare an entangled pair, apply the unknown unitary to one of the two parts and then measure the joint output state. This measurement could be an entangled one or it could be separable (e.g., LOCC). A comparison is made between these possibilities and it is shown that by using non-separable measurements one can improve the accuracy of the estimation by a factor of $2(d+1)/d$ where $d$ is the dimension of the Hilbert space on which $U$ acts.
Quantum states transformation under free operations plays a central role in the resource theory of coherence. In this paper, we investigate the transformation from a mixed coherent state into a pure one by using both incoherent operations and stochastic incoherent operations. We show that contrary to the strictly incoherent operations and the stochastic strictly incoherent operations, both the incoherent operations and the stochastic incoherent operations can increase the dimension of the maximal pure coherent subspace of a state. This means that the incoherent operations are generally stronge…
We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a byproduct of the criteria, we prov…
We studied pure state transformations using local operations assisted by finitely many rounds of classical communication ($LOCC_{\mathbb{N}}$) in C. Spee, J.I. de Vicente, D. Sauerwein, B. Kraus, arXiv:1606.04418 (2016). Here, we first of all present the details of some of the proofs and generalize the construction of examples of state transformations via $LOCC_{\mathbb{N}}$ which require a probabilistic step. However, we also present explicit examples of SLOCC classes where any separable transformation can be realized by a protocol in which each step is deterministic (all-det-$LOCC_{\mathbb{N…
We describe a general approach to proving the impossibility of implementing a quantum channel by local operations and classical communication (LOCC), even with an infinite number of rounds, and find that this can often be demonstrated by solving a set of linear equations. The method also allows one to design an LOCC protocol to implement the channel whenever such a protocol exists in any finite number of rounds. Perhaps surprisingly, the computational expense for analyzing LOCC channels is not much greater than that for LOCC measurements. We apply the method to several examples, two of which p…
In a recent paper \cite{mySEPvsLOCC}, we showed how to construct a quantum protocol for implementing a bipartite, separable quantum measurement using only local operations on subsystems and classical communication between parties (LOCC) within any fixed number of rounds of communication, whenever such a protocol exists. Here, we generalize that construction to one that applies for any number of parties. One important observation is that the construction automatically determines the ordering of the parties' measurements, overcoming a significant apparent difficulty in designing protocols for mo…
Existing nonlinear-optic implementations of pure, unfiltered heralded single-photon sources do not offer the scalability required for densely integrated quantum networks. Additionally, lithium niobate has hitherto been unsuitable for such use due to its material dispersion. We engineer the dispersion and the quasi-phasematching conditions of a waveguide in the rapidly emerging thin-film lithium niobate platform to generate spectrally separable photon pairs in the telecommunications band. Such photon pairs can be used as spectrally pure heralded single-photon sources in quantum networks. We est…
We prove that every symmetric separable state admits a convex decomposition into symmetric pure product states. While the result is not new in itself, here we focus on convex geometry. We discuss the decomposition in the context of numerical ranges and ground state problems of infinite bosonic systems.
We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied…
Let S_k be the set of separable states on B(C^m \otimes C^n) admitting a representation as a convex combination of k pure product states, or fewer. If m>1, n> 1, and k \le max(m,n), we show that S_k admits a subset V_k such that V_k is dense and open in S_k, and such that each state in V_k has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains V_k. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and i…
We examine the problem of using local operations and classical communication (LOCC) to distinguish a known pure state from an unknown (possibly mixed) state, bounding the error probability from above and below. We study the asymptotic rate of detecting multiple copies of the pure state and show that, if the overlap of the two states is great enough, then they can be distinguished asymptotically as well with LOCC as with global measurements; otherwise, the maximal Schmidt coefficient of the pure state is sufficient to determine the asymptotic error rate.
We analyze the conditions under which local operations and classical communication enable entanglement transformations within the set of bipartite pure Gaussian states. A set of necessary and sufficient conditions had been found in [Quant. Inf. Comp. 3, 211 (2003)] for the interconversion between such states that is restricted to Gaussian local operations and classical communication. Here, we exploit majorization theory in order to derive more general (sufficient) conditions for the interconversion between bipartite pure Gaussian states that goes beyond Gaussian local operations. While our tec…
Alice and Bob are given an unknown initial state chosen from a set of pure quantum states. Their task is to transform the initial state to a corresponding final pure state using local operations only. We prove necessary and sufficient conditions on the existence of such a transformation. We also provide efficient algorithms that can quickly rule out the possibility of transforming a set of initial states to a set of final states.
Resource theory of quantum coherence originated like entanglement in quantum information theory. However, still now proper classification of quantum states is missing under coherence. In this work, we have provided a classification of states under local incoherent operations. We have succeeded in deriving the necessary and sufficient condition for which two pure multipartite states are equivalent under stochastic local incoherent operations and classical communications (SLICC) and local incoherent operations and classical communications (LICC). In particular, we have succeeded in characterizin…
Motivated by the desire to better understand the class of quantum operations on bipartite systems that preserve positivity of partial transpose (PPT operations) and its relation to the class LOCC (local operations and classical communication), we present some results on deterministic bipartite pure state transformations by PPT operations. Restricting our attention to the case where we start with a rank K maximally entangled state, we give a necessary condition for transforming it into a given pure state, which we show is also sufficient when K is two and the final state has Schmidt rank three.…
Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled…
All causally separable quantum processes are quantum circuits with classical control of causal order
Centre for Quantum Information and Communication (QuIC), École Polytechnique de Bruxelles, C.P. 165, Université libre de Bruxelles（布鲁塞尔自由大学）；Univ. Grenoble Alpes, Inria（格勒诺布尔阿尔卑斯大学）；Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel（格勒诺布尔阿尔卑斯大学）
We present a geometric approach to the characterization of separability and entanglement in pure Gaussian states of an arbitrary number of modes. The analysis is performed adapting to continuous variables a formalism based on single subsystem unitary transformations that has been recently introduced to characterize separability and entanglement in pure states of qubits and qutrits [arXiv:0706.1561]. In analogy with the finite-dimensional case, we demonstrate that the $1 \times M$ bipartite entanglement of a multimode pure Gaussian state can be quantified by the minimum squared Euclidean distan…
In quantum operations, probabilities characterise both the degree of the success of a state transformation and, as density operator eigenvalues, the degree of mixedness of the final state. We give a unified treatment of pure-to-pure state transformations, covering both probabilistic and deterministic cases. We then discuss the role of majorization in describing the dynamics of mixing in quantum operations. The conditions for mixing enhancement for all initial states are derived. We show that mixing is monotonically decreasing for deterministic pure-to-pure transformations, and discuss the rela…
Suppose several parties jointly possess a pure multipartite state, |ψ>. Using local operations on their respective systems and classical communication (i.e. LOCC) it may be possible for the parties to transform deterministically |ψ> into another joint state |ϕ>. In the bipartite case, Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states in the same stochastic LOCC (SLOCC) class. Here we generalize Nielsen m…
