Partial Transpose Research Articles

Low-rank extremal positive-partial-transpose states and unextendible product bases

It is known how to construct, in a bipartite quantum system, a unique low rank entangled mixed state with positive partial transpose (a PPT state) from an unextendible product basis (a UPB), defined as an unextendible set of orthogonal product vectors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by non-singular product transformations, unitary or non-unitary. The characteristic property of a state $\rho$ in such a family is that its kernel $\Ker\rho$ has a generalized UPB, a basis of product vectors, not necessarily orthogonal, with no product vector in $\Im\rho$, the orthogonal complement of $\Ker\rho$. The generalized UPB in $\Ker\rho$ has the special property that it can be transformed to orthogonal form by a product transformation. In the case of a system of dimension $3\times 3$, we give a complete parametrization of orthogonal UPBs. This is then a parametrization of families of rank 4 entangled (and extremal) PPT states, and we present strong numerical evidence that it is a complete classification of such states. We speculate that the lowest rank entangled and extremal PPT states also in higher dimensions are related to generalized, non-orthogonal UPBs in similar ways.

Numerical studies of entangled positive-partial-transpose states in composite quantum systems

We report here on the results of numerical searches for PPT states in a series of bipartite quantum systems of low dimensions. PPT states are represented by density matrices that remain positive semidefinite under partial transposition with respect to one of the subsystems, and our searches are for such states with specified ranks for the density matrix and its partial transpose. For a series of different ranks extremal PPT states and nonextremal entangled PPT states have been found. The results are listed in tables and charted in diagrams. Comparison of the results for systems of different dimensions reveals several regularities. We discuss lower and upper bounds on the ranks of extremal PPT states.

Entangled states with strong positive partial transpose

Chruscinski, Jurkowski, and Kossakowski [Phys. Rev. A 77, 022113 (2008)] studied quantum states with strong positive partial transpose (SPPT) and conjectured that all SPPT states are separable. We construct a two-parameter class of 3 x 3 entangled SPPT states, so the conjecture does not hold true for general SPPT states.

Witnessing quantum discord in2×Nsystems

Bipartite states with vanishing quantum discord are necessarily separable and hence positive partial transpose (PPT). We show that $2\ifmmode\times\else\texttimes\fi{}N$ 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 strong PPT (SPPT). Therefore, we provide a natural witness for a quantum discord: if a $2\ifmmode\times\else\texttimes\fi{}N$ 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 it must be entangled.

Difficulties in analytic computation for relative entropy of entanglement

It is known that relative entropy of entanglement for an entangled state $\ensuremath{\rho}$ is defined via its closest separable (or positive partial transpose) state $\ensuremath{\sigma}$. Recently, it has been shown how to find $\ensuremath{\rho}$ provided that $\ensuremath{\sigma}$ is given in a two-qubit system. In this article we study the reverse process, that is, how to find $\ensuremath{\sigma}$ provided that $\ensuremath{\rho}$ is given. It is shown that if $\ensuremath{\rho}$ is of a Bell-diagonal, generalized Vedral-Plenio, or generalized Horodecki state, one can find $\ensuremath{\sigma}$ from a geometrical point of view. This is possible due to the following two facts: (i) the Bloch vectors of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ are identical to each other; (ii) the correlation vector of $\ensuremath{\sigma}$ can be computed from a crossing point between a minimal geometrical object, in which all separable states reside in the presence of Bloch vectors, and a straight line, which connects the point corresponding to the correlation vector of $\ensuremath{\rho}$ and the nearest vertex of the maximal tetrahedron, where all two-qubit states reside. It is shown, however, that these properties are not maintained for the arbitrary two-qubit states.

PPT classifiable families of bipartite states

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 of PPT bound entangled states are briefly discussed.

Direct derivation of the Peres–Horodecki criterion for the two-qubit states from the Hill–Wootters formula for the entanglement of formation

In this paper, we show a direct method of deriving the Peres–Horodecki criterion for the two-qubit states from the Hill–Wootters formula for the entanglement of formation. Although the Peres–Horodecki criterion and the Hill–Wootters formula are established results in the field of quantum information theory, they are proved independently and connections between them are not discussed precisely. In this paper, we clarify these connections. First, we replace the original Peres–Horodecki criterion with an equivalent statement found by Augusiak et al. [Augusiak, R.; Demianowicz, M.; Horodecki, M.; Horodecki, R. Rev. Mod. Phys. 2009, 81, 865–942]. Second, we obtain an analytical form of the concurrence of an arbitrary two-qubit state ρ, using Ferrari's method to solve a quartic equation for eigenvalues . Finally, with the above preparations, we accomplish the direct derivation of the Peres–Horodecki criterion from the Hill–Wootters formula.

Advances in delimiting the Hilbert–Schmidt separability probability of real two-qubit systems

We seek to derive the probability—expressed in terms of the Hilbert–Schmidt (Euclidean or flat) metric—that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres–Horodecki test on the partial transposes (PTs) of the associated 4 × 4 density matrices (ρ). But the full implementation of the test—requiring that the determinant of the PT be nonnegative for separability to hold—appears to be, at least presently, computationally intractable. So, we have previously implemented—using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)—the weaker implied test of nonnegativity of the six 2 × 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of . Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 × 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of . Then, we conclude that a still further improved upper bound of can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 × 3 principal minors. Numerical simulations—as opposed to exact symbolic calculations—indicate, on the other hand, that the true probability is certainly less than . Our analyses lead us to suggest a possible form for the true DESF, yielding a separability probability of , while the absolute separability probability of provides the best exact lower bound established so far. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESFs that comes close to reproducing our numerical estimate of the true separability function.

Approximating the set of separable states using the positive partial transpose test

The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, is efficient. In this paper, we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states. More precisely, we prove that, as the dimension of the space tends to infinity, the maximum trace distance of a positive partial transpose state from separable states tends to 1. Using similar techniques, we show that the same result holds for other well-known separability criteria such as reduction criterion, majorization criterion, and symmetric extension criterion. We also bring in evidence that the sets of positive partial transpose states and separable states have totally different shapes.

Greenberger-Horne-Zeilinger andWentanglement witnesses for the noninteracting Fermi gas

The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three classes of parametrized entanglement witnesses (EWs) are introduced with the aim of detecting genuine tripartite entanglement in the three-body reduced density matrix and discriminating between the presence of the two types of genuine tripartite entanglement, $W\B$ and $\mathrm{GHZ}\W$ (the convex set of $B$ states is comprised of mixed states of product and biseparable states; that of $W$ states is comprised of mixed states of $B$ states and $W$-type pure entangled states; and the GHZ (Greenberger-Horne-Zeilinger) set contains generic mixtures of any kind for a tripartite system). By choosing appropriate EW operators, the problem of finding GHZ and $W$ EWs is reduced to linear programming. Specifically, we devise $W$ EWs based on a spin-chain model with periodic boundary conditions, and we construct a class of parametrized GHZ EWs by linearly combining projection operators corresponding to all the different state-vector types arising for a three-fermion system. A third class of EWs is provided by a GHZ stabilizer operator capable of distinguishing $W\B$ from $\mathrm{GHZ}\B$ entanglement, which is not possible with $W$ EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of $W$ type, and hence states containing $\mathrm{GHZ}\W$ genuine tripartite entanglement do not arise. Some genuine tripartite entangled states that have a positive partial transpose (PPT) with respect to some bipartition are detected. Finally, it is demonstrated that a NIFG does not exhibit ``pure'' $W\B$ genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.

Entropic characterization of separability in Gaussian states

We explore separability of bipartite divisions of mixed Gaussian states based on the positivity of the Abe-Rajagopal (AR) $q$-conditional entropy. The AR $q$-conditional entropic characterization provide more stringent restrictions on separability (in the limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$) than that obtained from the corresponding von Neumann conditional entropy ($q=1$ case)---similar to the situation in finite dimensional states. Effectiveness of this approach, in relation to the results obtained by partial transpose criterion, is explicitly analyzed in three illustrative examples of two-mode Gaussian states of physical significance.

A Few Steps More Towards NPT Bound Entanglement

We consider the problem of existence of bound entangled states with non-positive partial transpose (NPT). As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover it would rule out a simple mathematical description of the set of distillable states. Distillability is equivalent to so called n-copy distillability for some n. We consider a particular state, known to be 1-copy nondistillable, which is supposed to be bound entangled. We study the problem of its two-copy distillability, which boils down to show that maximal overlap of some projector Q with Schmidt rank two states does not exceed 1/2. Such property we call the the half-property. We first show that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. We then attack the problem in twofold way: a) prove the half-property for some classes of Schmidt rank two states b) bound the required overlap from above for all Schmidt rank two states. We have succeeded to prove the half-property for wide classes of states, and to bound the overlap from above by c<3/4. Moreover, we translate the problem into the following matrix analysis problem: bound the sum of the squares of the two largest singular values of matrix A \otimes I + I \otimes B with A,B traceless 4x4 matrices, and Tr A^\dagger A + Tr B^\dagger B = 1/4.

Teleportation via a mixture of a two qubit subsystem of a N-qubit W and GHZ state

In this work we study a state which is a random mixture of a two qubit subsystem of a N-qubit W state and GHZ state. We analyze several possibilities like separability criterion (Peres-Horodecki criterion [M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996); A. Peres, Phys. Rev. Lett. 77, 1413 (1996)]), non violation of Bell’s inequality [J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 80 (1969)] (M(ρ)<1) and teleportation fidelity [N. Gisin, Phys. Lett. A 210, 157 (1996); R. Horodecki, P. Horodecki, M. Horodecki, Phys. Lett. A 200, 340 (1995); S. Massar, S. Popescu, Phys. Rev. Lett. 74, 1259 (1995); S. Popescu, Phys. Rev. Lett. 72, 797 (1994); C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993)] $(F_{max}>\frac{2}{3})$ for this state. We also obtain a relationship between N (number of qubits) and p (the classical probability of random mixture) for each of these possibilities. Finally we present a detailed analysis of all these possibilities for N=3,4,5 qubit systems. We also report that for N=3 and $p\in(0.75,1]$ , this entangled state can be used as a teleportation channel without violating Bell’s inequality.

Entanglement of a coarse grained quantum field in the expanding universe

We investigate the entanglement of a quantum field in the expanding universe. By introducing a bipartite system using a coarse-grained scalar field, we apply the separability criterion based on the partial transpose operation and numerically calculate the bipartite entanglement between separate spatial regions. We find that the initial entangled state becomes separable or disentangled after the spatial separation of two points exceed the Hubble horizon. This provides the necessary conditions for the appearance of classicality of the quantum fluctuation. We also investigate the condition of classicality that the quantum field can be treated as the classical stochastic variables.

Separability criteria and entanglement witnesses for symmetric quantum states

We study the separability of symmetric bipartite quantum states and show that a single correlation measurement is sufficient to detect the entanglement of any bipartite symmetric state with a non-positive partial transpose. We also discuss entanglement conditions and entanglement witnesses for states with a positive partial transpose.

Searching for extremal PPT entangled states

We study extremality in various sets of states that have positive partial transposes. One of the tools we use for this purpose is the recently formulated criterion allowing to judge if a given state is extremal in the set of PPT states. First we investigate qubit–ququart states and show that the only candidates for extremal PPT entangled states (PPTES) have ranks of the state and its partial transposition (5, 5) or (5, 6) (equivalently (6, 5)). Then, examples of extremal states of (5, 5) type and the so-called edge states of type (5, 6) are provided. We also make an attempt to explore the set of PPT states with ranks (5, 6). Finally, we discuss what are the possible configurations of ranks of density matrices and their respective partial transposition in general three-qubit and four-qubit symmetric states for which there may exist extremal entangled PPT states. For instance in the first case we show that the only possibilities are (4, 4, 4) and (4, 4, 5).

Power of symmetric extensions for entanglement detection

In this paper, we present new progress on the study of the symmetric extension criterion for separability. First, we show that a perturbation of order O(1/N) is sufficient and, in general, necessary to destroy the entanglement of any state admitting an N Bose symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from N Bose symmetric extensions with Positive Partial Transpose (PPT) decreases at least as fast as O(1/N^2). From these results, we derive upper bounds on the time and space complexity of the weak membership problem of separability when attacked via algorithms that search for PPT symmetric extensions. Finally, we show how to estimate the error we incur when we approximate the set of separable states by the set of (PPT) N -extendable quantum states in order to compute the maximum average fidelity in pure state estimation problems, the maximal output purity of quantum channels, and the geometric measure of entanglement.

Dynamical maps and density matrices

The relations between dynamical maps and quantum states of bipartite systems are analyzed from the perspective of quantum conditional probability. In particular, we explore new interesting relations between completely positive maps, which correspond to quantum channels, and states of bipartite systems which correspond to correlations between the initial and final states. The new connection emerges in a natural way from the generalisation of the classical concept of conditional probability.We develop applications of these relations which prove to be very useful in both directions, either for the classification of positive maps which are not completely positive, the classification of non-decomposable dynamical maps or for the classification of positive partial transpose and entangled states.

Necessary and Sufficient Condition for Greenberger-Horne-Zeilinger Diagonal States to Be Full N-partite Entangled

We show that any N-qubit state which is diagonal in the Greenberger-Horne-Zeilinger basis is full N-qubit entangled state if and only if no partial transpose of the multiqubit state is positive with respect to any partition.

Distinguishability of Quantum States Under Restricted Families of Measurements with an Application to Quantum Data Hiding

We consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements. Let the bias of a POVM be defined as the total variational distance between the outcome distributions for the two states to be distinguished. The performance of a set of measurements can then be defined as the ratio of the bias of this POVM and the largest bias achievable by any measurements. We first provide lower bounds on the performance of various POVMs acting on a single system such as the isotropic POVM, and spherical 2 and 4-designs, and show how these bounds can lead to certainty relations. Furthermore, we prove lower bounds for several interesting POVMs acting on multipartite systems, such as the set of local POVMS, POVMs which can be implemented using local operations and classical communication (LOCC), separable POVMs, and finally POVMs for which every bipartition results in a measurement having positive partial transpose (PPT). In particular, our results show that a scheme of Terhal et. al. for hiding data against local operations and classical communication [31] has the best possible dimensional dependence.