A continuous-variable Bell inequality, valid for an arbitrary number of observers measuring observables with an arbitrary number of outcomes, was recently introduced in [Cavalcanti \emph{et al.}, Phys. Rev. Lett. {\bf 99}, 210405 (2007)]. We prove that any $n$-mode quantum state violating this inequality with quadrature measurements necessarily has a negative partial transposition. Our results thus establish the first link between nonlocality and bound entanglement for continuous-variable systems.
The partial transposition of a two-qubit state has at most one negative eigenvalue and all the eigenvalues lie in [-1/2,1]. In this Brief Report, we extend this result by Sanpera et al. [A. Sanpera, R. Tarrach and G. Vidal, Phys. Rev. A 58, 826 (1998)] to arbitrary bipartite states. We show that partial transposition of an $m\otimes n$ state can not have more than (m-1)(n-1) number of negative eigenvalues. Low-dimensional states have been studied to show the tightness of this result and explicit examples have been provided for $mn\le 9$. It is also shown that all the eigenvalues of partial tra…
Bound entanglement with a nonpositive partial transposition (NPT) does not exist. For any NPT entangled state a distillation procedure can be based on a certain number of copies. This number is the minimal Schmidt rank of a pure state needed to witness the NPT entanglement under study.
It is well known that, under some assumptions, the limit distribution of random block matrices and their partial transposition converges to the distributions of random variables in some noncommutative probability space. Using free probability theory, we obtain the relation between the free cumulants of the corresponding random variables. As an application, we are able to derive a new family of co-completely positive and k-positive maps by using the Wishart ensemble.
Many of the properties of the partial transposition are not clear so far. Here the number of the negative eigenvalues of K(T)(the partial transposition of K) is considered carefully when K is a two-partite state. There are strong evidences to show that the number of negative eigenvalues of K(T) is N(N-1)/2 at most when K is a state in Hilbert space N*N. For the special case, 2*2 system(two qubits), we use this result to give a partial proof of the conjecture sqrt(K(T))(T)>=0. We find that this conjecture is strongly connected with the entanglement of the state corresponding to the negative eig…
We study the distillability of a certain class of bipartite density operators which can be obtained via depolarization starting from an arbitrary one. Our results suggest that non-positivity of the partial transpose of a density operator is not a sufficient condition for distillability, when the dimension of both subsystems is higher than two.
We present a generalized partial transposition separability criterion for the density matrix of a multipartite quantum system. This criterion comprises as special cases the famous Peres-Horodecki criterion and the recent realignment criterion in [O. Rudolph, quant-ph/0202121] and [K. Chen, L.A. Wu, quant-ph/0205017]. It involves only straightforward matrix manipulations and is easy to apply. A quantitative measure of entanglement based on this criterion is also obtained.
We construct a new class of PPT states for bipartite "d x d" systems. This class is invariant under the maximal commutative subgroup of U(d) and contains as special cases almost all known examples of PPT states. Theses states may be used to test the atomic property of positive maps which are crucial in studying quantum entanglement.
We study robustness of bipartite entangled states that are positive under partial transposition (PPT). It is shown that almost all PPT entangled states are unconditionally robust, in the sense, both inseparability and positivity are preserved under sufficiently small perturbations in its immediate neighborhood. Such unconditionally robust PPT entangled states lie inside an open PPT entangled ball. We construct examples of such balls whose radii are shown to be finite and can be explicitly calculated. This provides a lower bound on the volume of all PPT entangled states. Multipartite generaliza…
It is challenging task to detect and measure genuine multipartite entanglement. We investigate the problem by considering the average based positive partial transposition(PPT) criterion and the realignment criterion. Sufficient conditions for detecting genuine tripartite entanglement are presented. We also derive lower bounds for the genuine tripartite entanglement concurrence with respect to the conditions. While the PPT criterion and the realignment criterion are powerful for detecting bipartite entanglement and for providing lower bounds of bipartite concurrences, our results give an effect…
The discovery of entangled quantum states from which one cannot distill pure entanglement constitutes a fundamental recent advance in the field of quantum information. Such bipartite bound-entangled (BE) quantum states \emph{could} fall into two distinct categories: (1) Inseparable states with positive partial transposition (PPT), and (2) States with negative partial transposition (NPT). While the existence of PPT BE states has been confirmed, \emph{only one} class of \emph{conjectured} NPT BE states has been discovered so far. We provide explicit constructions of a variety of multi-copy undis…
It is shown that any separable state on Hilbert space ${\cal H}={\cal H}_1\otimes{\cal H}_2$, can be written as a convex combination of N pure product states with $N\leq (dim{\cal H})^2$. Then a new separability criterion for mixed states in terms of range of density matrix is obtained. It is used in construction of inseparable mixed states with positive partial transposition in the case of $3\times 3$ and $2\times 4$ systems. The states represent an entanglement which is hidden in a more subtle way than it has been known so far.
We prove an analytic positive lower bound for the geometric distance between entangled positive partial transpose (PPT) states of a broad class and any private state that delivers one secure key bit. Our proof holds for any Hilbert space of finite dimension. Although our result is proven for a specific class of PPT states, we show that our bound nonetheless holds for all known entangled PPT states with non-zero distillable key rates whether or not they are in our special class.
We show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of a "momentum-like" variable of one of the particles in the Wigner function for the state. This generalizes a result obtained for continuous-variable systems to the discrete-variable system case. We show that, in principle, quantum mechanics allows measuring the expectation value of an observable in a partially transposed state, in spite of the fact that the latter may not be a physical state. We illustrate this result with the example of an "…
We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical-half-numerical method that allows to construct multipartite PPT entangled symmetric states (PPTESS) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing to search for extremal elements in the convex set of bipartite PPT states [J. M. Leinaas, J. Myrheim, and E. Ovrum, Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we s…
We construct a family of bipartite states of arbitrary dimension whose eigenvalues of the partially transposed matrix can be inferred directly from the block structure of the global density matrix. We identify from this several subfamilies in which the PPT criterion is both necessary and sufficient. A sufficient criterion of separability is obtained, which is fundamental for the discussion. We show how several examples of states known to be classifiable by the PPT criterion indeed belong to this general set. Possible uses of these states in numerical analysis of entanglement and in the search …
In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds for any two matrices $A$ and $B$ for which the product $AB$ is defined. Here $T$ denote the usual transposition. In this work, we explore the possibility of deriving the matrix equality $(AB)^Γ=A^ΓB^Γ$ for any $4 \times 4$ matrices $A$ and $B$, where $Γ$ denote the partial transposition. We found that, in general, $(AB)^Γ\neq A^ΓB^Γ$ holds for $4 \times 4$ matrices $A$ and $B$ but there exist particular set of $4 \times 4$ matrices for which $(AB)^Γ= A^ΓB^Γ$ holds. We have exploited this matrix equality to investigate th…
From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with positive partial transposition and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)] With the present contribution we continu…
We study bipartite entanglement in systems of N identical bosons distributed in M different modes. For such systems, a definition of separability not related to any a priori Hilbert space tensor product structure is needed and can be given in terms of commuting subalgebras of observables. Using this generalized notion of separability, we classify the states for which partial transposition turns out to be a necessary and sufficient condition for entanglement detection.
Usual separability criteria applicable to distinguishable particles are not applicable to identical particles. Here we show that Partial transposition and symmetrization (or anti symmetrization) of density matrix of bipartite boson systems (fermion systems) can be used to check whether a state is separable or not.By using determinant based separability criteria it has been found that bipartite states of identical particles are always separable. It is found that partial transposition and symmetrization is equivalent to realignment criteria proposed by Wu.
We explore the link between two concepts: the level of violation of a Bell inequality by a quantum state and discrimination between two states by means of restricted classes of operations, such as local operations and classical communication (LOCC) and separable ones. For any bipartite Bell inequality, we show that its value on a given quantum state cannot exceed the classical bound by more than the maximal quantum violation shrunk by a factor related to distinguishability of this state from the separable set by means of some restricted class of operations. We then consider the general scenari…
Tight Bounds for Purity and Product Testing from Partial Transposition
Central High School（中央高中）；Department of Mathematics, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校数学系）；IQUIST, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校 IQUIST）；Department of Physics, University of Rhode Island（罗德岛大学物理系）
We construct a large class of quantum "d x d" states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure - that is way we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well known examples of PPT states from the literature and gives rise to a huge…
Effects of the detection loophole on rival entanglement attestation techniques
Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute（哈里什-昌德拉研究所，霍米·巴巴国立学院成员机构）；Departamento de Física Teórica, Universidad Complutense（康普卢滕塞大学理论物理系）
Let W be a Wishart random matrix of size d^2 times d^2, considered as a block matrix with d times d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d tends to infinity. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "W…
Quantized detector networks (QDN) deals with quantum information exchange between observers and their apparatus rather than with systems under observation. Partial observations in QDN involve subsets of the elementary signal detectors which constitute an apparatus. We use them to prove that QDN is consistent with Einstein locality and violations of Bell-type inequalities.
Genuine high-dimensional entanglement, i.e. the property of having a high Schmidt number, constitutes a resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT), on the other hand, are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question, whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e. linear, scaling in local dimension should be possible, albeit without providing an insight into the pos…
The recent proposed realignment separability criterion for mixed is analyzed. We identify the essential part of this criterion is a swap operator followed by a partial transposition. Then we analyze the separability criterion of permutation indices of the density matrix for multipartite state which is a generalization of the realignment criterion. We give a method to solve the equivalence problem of the separability criterion of permutation indices of the density matrix. For tripartite state, we show the non-trivial separability criteria are either partial transposition criterion or realignmen…
We first show that partial transposition for pure and mixed two-particle states in a discrete $N$-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a re…
We investigate the genuine entanglement in tripartite systems based on partial transposition and the norm of correlation tensors of the density matrices. We first derive an analytical sufficient criterion to detect genuine entanglement of tripartite qubit quantum states combining with the partial transposition of the density matrices. Then we use the norm of correlation tensors to study genuine entanglement for tripartite qudit quantum states and obtain a genuine entanglement criterion by constructing certain matrices. With detailed examples our results are seen to be able to detect more genui…
The entanglement produced by a bilinear Hamiltonian in continuous variables has been thoroughly studied and widely used. In contrast, the physics of entanglement resulting from nonlinear interaction described by partially degenerate high-order Hamiltonians remains unclear. Here, we derive a hierarchy of sufficient and necessary conditions for the positive-partial-transposition separability of bipartite nonlinear quantum states. The proposed criteria detect the nonpositive-partial-transposition inseparability of higher-order moments of states, which provides a systematic framework for the chara…
The verification of quantum entanglement under the influence of realistic noise and decoherence is crucial for the development of quantum technologies. Unfortunately, a full entanglement characterization is generally not possible with most entanglement criteria such as entanglement witnesses or the partial transposition criterion. In particular, so called bound entanglement cannot be certified via the partial transposition criterion. Here we present the full entanglement verification of dephased qubit and qutrit Werner states via entanglement quasiprobabilities. Remarkably, we are able to reve…
We analyze a class of entangled states for bipartite $d \otimes d$ systems, with $d$ non-prime. The entanglement of such states is revealed by the construction of canonically associated entanglement witnesses. The structure of the states is very simple and similar to the one of isotropic states: they are a mixture of a separable and a pure entangled state whose supports are orthogonal. Despite such simple structure, in an opportune interval of the mixing parameter their entanglement is not revealed by partial transposition nor by the realignment criterion, i.e. by any permutational criterion i…
We construct a large class of bipartite M x N quantum states which defines a proper subset of states with positive partial transposes (PPT). Any state from this class is PPT but the positivity of its partial transposition is recognized with respect to canonical factorization of the original density operator. We propose to call elements from this class states with strong positive partial transposes (SPPT). We conjecture that all SPPT states are separable.
Partial transposition of state operator is a well known tool to detect quantum correlations between two parts of a composite system. In this letter, the global partial transpose (GPT) is linked to conceptually multipartite underlying structures in a state - the negativity fonts. If K-way negativity fonts with non zero determinants exist, then selective partial transposition of a pure state, involving K of the N qubits (K leq N) yields an operator with negative eigevalues, identifying K-body correlations in the state. Expansion of GPT interms of K-way partially transposed (KPT) operators reveal…
We analyze a class of positive partial transpose states (PPT) such that the positivity of its partial transposition is recognized with respect to canonical factorization of the original density operator (Cholesky block decomposition). We call such PPT states strong PPT states (SPPT). This property, contrary to PPT, is basis dependent. It is shown that there exists a proper subset of SPPT states which are separable and provide a separable decomposition for any of these states.
We investigate non-locality distillation protocols for isotropic correlations. These correlations are the hardest instances which respect to distillability and only partial results are known about their behaviour under non-locality distillation protocols. We completely resolve this issue by proving that non-locality distillation is impossible for all non-local isotropic correlations.
Non-local correlations are one of the most fascinating consequences of quantum physics from the point of view of information: Such correlations, although not allowing for signaling, are unexplainable by pre-shared information. The correlations have applications in cryptography, communication complexity, and sit at the very heart of many attempts of understanding quantum theory -- and its limits -- better in terms of classical information. In these contexts, the question is crucial whether such correlations can be distilled, i.e., whether weak correlations can be used for generating (a smaller …
It has been recently shown, that some of the tripartite boxes admitting bilocal decomposition, lead to non-locality under wiring operation applied to two of the subsystems [R. Gallego et al. Physical Review Letters 109, 070401 (2012)]. In the following, we study this phenomenon quantitatively. Basing on the known classes of boxes closed under wirings, we introduced multipartite monotones which are counterparts of bipartite ones - the non-locality cost and robustness of non-locality. We then provide analytical lower bounds on both the monotones in terms of the Maximal Non-locality which can be …
By constructing the quantum state in high-dimensional probability tensor, we find the quantum magic square(QMS) may stand as an ideal means of characterizing the non-local phenomena, i.e. the separability, entanglement, two/one-way steering, and Bell non-locality, etc. In this scheme, different types of non-locality exhibit distinctive inner structures of the probability tensor, which are observable in form of the partial sum of the tensor components. In application, we prove the Bell and GHZ theorems, and demonstrate that the uncertainty relation may rate the non-locality, from Bell locality …
We present separability criteria for both bipartite and multipartite quantum states. These criteria include the criteria based on the correlation matrix and its generalized form as special cases. We show by detailed examples that our criteria are more powerful than the positive partial transposition criterion, the realignment criterion and the criteria based on the correlation matrices.
We investigate the canonical forms of positive partial transposition (PPT) density matrices in ${\cal C}^2 \otimes {\cal C}^M \otimes {\cal C}^N$ composite quantum systems with rank $N$. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.
Bipartite states with vanishing quantum discord are necessarily separable and hence positive partial transpose (PPT). We show that 2 x N states satisfy additional property: the positivity of their partial transposition is recognized with respect to the canonical factorization of the original density operator. We call such states SPPT (for strong PPT). Therefore, we provide a natural witness for a quantum discord: if a 2 x N state is not SPPT it must contain nonclassical correlations measured by quantum discord. It is an analog of the celebrated Peres-Horodecki criterion: if a state is not PPT …
One of the formulations of Heisenberg uncertainty principle, concerning so-called measurement uncertainty, states that the measurement of one observable modifies the statistics of the other. Here, we derive such a measurement uncertainty principle from two comprehensible assumptions: impossibility of instantaneous messaging at a distance (no-signaling), and violation of Bell inequalities (non-locality). The uncertainty is established for a pair of observables of one of two spatially separated systems that exhibit non-local correlations. To this end, we introduce a gentle form of measurement wh…
We provide necessary and sufficient conditions for the partial transposition of bipartite harmonic quantum states to be nonnegative. The conditions are formulated as an infinite series of inequalities for the moments of the state under study. The violation of any inequality of this series is a sufficient condition for entanglement. Previously known entanglement conditions are shown to be special cases of our approach.
Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the inertias of EWs, i.e., the triplet of the numbers of negative, zero, and positive eigenvalues respectively. We focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit-qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose …
Necessary and sufficient observable conditions for the nonnegativity of all partial transpositions of multi-mode quantum states are derived. The result is a hierarchy of inequalities for minors in terms of moments of the given state. Violations of any inequality is a sufficient condition for entanglement. Full entanglement can be certified for a manifold of multi-mode quantum states. A \textit{Mathematica} package is given for a systematic test of the hierarchy of conditions.
Recently, a new and powerful separability criterion was introduced in [O. Rudolph, quant-ph/0202121] and [Chen {\it et al.}, quant-ph/0205017]. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely triparite entanglement. We also show that for multipartite sys…
We provide a simple construction of bipartite entangled states that are positive under partial transposition, and hence undistillable. The construction makes use of the properties of the projectors onto the symmetric and antisymmetric subspaces of the Hilbert space of two identical systems. The resulting states can be considered as generalizations of the celebrated Werner states.
We analyze the separability properties of density operators supported on $\C^2\otimes \C^N$ whose partial transposes are positive operators. We show that if the rank of $ρ$ equals N then it is separable, and that bound entangled states have rank larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed 3N. If it exceeds this number we show that one can subtract product vectors until decreasing it to 3N, while keeping the positivity of $ρ$ and its partial transpose. This automatically gi…
When conducting a Bell test, it is normal to assume that the preparation of the quantum state is independent of the measurements performed on it. Remarkably, the violation of local realism by entangled quantum systems can be certified even if this assumption is partially relaxed. Here, we allow such measurement dependence to correlate multiple runs of the experiment, going beyond previous studies that considered independent and identically distributed (i.i.d.) runs. To do so, we study the polytope that defines block-i.i.d. measurement-dependent local models. We prove that non-i.i.d. models are…
