We conduct quasi-Monte Carlo numerical integrations in two very high (80 and 79)-dimensional domains -- the parameter spaces of rank-9 and rank-8 qutrit-qutrit (9 x 9) density matrices. We, then, estimate the ratio of the probability -- in terms of the Hilbert-Schmidt metric -- that a generic rank-9 density matrix has a positive partial transpose (PPT) to the probability that a generic rank-8 density matrix has a PPT (a precondition to separability/nonentanglement). Close examination of the numerical results generated -- despite certain large fluctuations -- indicates that the true ratio may, in fact, be 2. Our earlier investigation (eprint quant-ph/0410238) also yielded estimates close to 2 of the comparable ratios for qubit-qubit and qubit-qutrit pairs (the only two cases where the PPT condition fully implies separability). Therefore, it merits conjecturing (as Zyczkowski was the first to do) that such Hilbert-Schmidt (rank-NM/rank-(NM-1)) PPT probability ratios are 2 for all NM-dimensional quantum systems.
We construct entangled states with positive partial transposes using indecomposable positive linear maps between matrix algebras. We also exhibit concrete examples of entangled states with positive partial transposes arising in this way, and show that they generate extreme rays in the cone of all positive semi-definite matrices with positive partial transposes. They also have Schmidt numbers two.
We construct a class of $3\otimes 3$ entangled edge states with positive partial transposes using indecomposable positive linear maps. This class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes.
We study violations of n particle Bell inequalities (as developed by Mermin and Klyshko) under the assumption that suitable partial transposes of the density operator are positive. If all transposes with respect to a partition of the system into p subsystems are positive, the best upper bound on the violation is 2^((n-p)/2). In particular, if the partial transposes with respect to all subsystems are positive, the inequalities are satisfied. This is supporting evidence for a recent conjecture by Peres that positivity of partial transposes could be equivalent to existence of local classical mode…
We show that all quantum states that do not have a positive partial transpose are distillable via channels, which preserve the positivity of the partial transpose. The question whether NPT bound entanglement exist is therefore closely related to the connection between the set of separable superoperators and PPT-preserving maps.
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.
The partial transpose by which a subsystem's quantum state is solely transposed is of unique importance in quantum information processing from both fundamental and practical point of view. In this work, we present a practical scheme to realize a physical approximation to the partial transpose using local measurements on individual quantum systems and classical communication. We then report its linear optical realization and show that the scheme works with no dependence on local basis of given quantum states. A proof-of-principle demonstration of entanglement detection using the physical approx…
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, are 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 …
We construct tri-qubit genuinely entangled states which have positive partial transposes with respect to bi-partition of systems. These examples disprove a conjecture [L. Novo, T. Moroder and O. G\" uhne, Phys.Rev.A {88}, 012305 (2013)] which claims that PPT mixtures are necessary and sufficient for biseparability of three qubits.
We present a necessary and sufficient condition for a finite dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.
It is found that the problem of existence of bound entangled states with non-positive partial transpose (NPPT) has the intriguing relation to the Hilbert's 17th problem. More precisely, we compute the expectation value of the partially transposed Werner states by Schmidt rank-2 vectors for NPPT and 1-copy undistillable region. It is the positive polynomial but shown not to be expressed as the sum of squares of polynomials. A remedy for such pathological behavior as well as a similar but different approach to the problem is also mentioned.
In this letter we have analyzed an entangled state in $C^{3} \bigotimes C^{3}$ having a positive partial transposition and have shown that it is an edge state. Further we have constructed explicitly a witness operator $W$ which detects the entanglement.
Based on the matrix realignment and partial transpose, we develop an approach to entangling power and operator entanglement of quantum unitary operators. We demonstrate efficiency of the approach by studying several unitary operators on qudits, and indicate that these two matrix rearrangements are not only powerful for studying separability problem of quantum states, but also useful in studying entangling capabilities of quantum operators.
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.
It is known that, in an $(m \otimes n)$-dimensional quantum system, the maximum dimension of a subspace that contains only entangled states is (m-1)(n-1). We show that the exact same bound is tight if we require the stronger condition that every state with range in the subspace has non-positive partial transpose. As an immediate corollary of our result, we solve an open question that asks for the maximum number of negative eigenvalues of the partial transpose of a quantum state. In particular, we give an explicit method of construction of a bipartite state whose partial transpose has (m-1)(n-1…
We generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by X.-Y. Yu and H. Zhao [ Int. J. Theor. Phys.,54, 292, (2015)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.
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…
We show that any bipartite quantum state of rank four is distillable, when the partial transpose of the state has at least one negative eigenvalue, i.e., the state is NPT. For this purpose we prove that if the partial transpose of a two-qutrit NPT state has two non-positive eigenvalues, then the state is distillable. We further construct a parametrized two-qutrit NPT entangled state of rank five which is not 1-distillable, and show that it is not $n$-distillable for any given $n$ when the parameter is sufficiently small. This state has the smallest rank among all 1-undistillable NPT states. We…
In this Letter, we show that the fulfillment of uncertainty relations is a sufficient criterion for a quantum-mechanically permissible state. We specifically construct two pseudo-spin observables for an arbitrary non-positive Hermitian matrix whose uncertainty relation is violated. This method enables us to systematically derive separability conditions for all negative partial-transpose states in experimentally accessible forms. In particular, generalized entanglement criteria are derived from the Schrodinger-Robertson inequalities for bipartite continuous-variable states.
We express the positive partial transpose (PPT) separability criterion for symmetric states of multi-qubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental inte…
We show that positivity of {\it every} partial transpose of $N$-partite quantum states implies new inequalities on Bell correlations which are stronger than standard Bell inequalities by a factor of $2^{(N-1)/2}$. A violation of the inequality implies the system is in a bipartite distillable entangled state. It turns out that a family of $N$-qubit bound entangled states proposed by Dür {[Phys. Rev. Lett. {\bf 87}, 230402 (2001)]} violates the inequality for $N\geq 4$.
We prove that the minimal Renyi entropy of order 2 (RE2) output of a positive-partial-transpose(PPT)-inducing channel joint to an arbitrary other channel is equal to the sum of the minimal RE2 output of the individual channels. PPT-inducing channels are channels with a Choi matrix which is bound entangled or separable. The techniques used can be easily recycled to prove additivity for some non-PPT-inducing channels such as the depolarizing and transpose depolarizing channels, though not all known additive channels. We explicitly make the calculations for generalized Werner-Holevo channels as a…
Partial transpose is an important operation for quantifying the entanglement, here we study the (partial) transpose of any single (two-mode) operators. Using the Fock-basis expansion, it is found that the transposed operator of an arbitrary operator can be obtained by replacement of a^{†}(a) by a(a^{†}) instead of c-number within normal ordering form. The transpose of displacement operator and Wigner operator are studied, from which the relation of Wigner function, characteristics function and average values such as covariance matrix are constructed between density operator and transposed dens…
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.
Detection of entanglement in bipartite states is a fundamental task in quantum information. The first method to verify entanglement in mixed states was the partial-transpose criterion. Subsequently, numerous quantifiers for bipartite entanglement were introduced, among them concurrence and negativity. Surprisingly, these quantities are often treated as distinct or independent of each other. The aim of this contribution is to highlight the close relations between these concepts, to show the connections between seemingly independent results, and to present various estimates for the mixed-state c…
We address an open question about the existence of entangled continuous-variable (CV) Werner states with positive partial transpose (PPT). We prove that no such state exists by showing that all PPT CV Werner states are separable. The separability follows by observing that these CV Werner states can be approximated by truncating the states into a finite-dimensional convex mixture of product states. In addition, the constituents of the product states comprise a generalized non-Gaussian measurement which gives, rather surprisingly, a strictly tighter upper bound on quantum discord than photon cou…
We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT-operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this easily computable entanglement measure. As examples we discuss general Werner states and arbitrary bi-partite Gaussian states. Equipped with this result we then prove that for the anti-symmetric Werner state PPT-cost and PPT-entanglement of distillation coincide giving the first example of a truly mixed state for which entanglement ma…
We demonstrate that even under positive partial transpose preserving operations in an asymptotic setting GHZ and W states are not reversibly interconvertible. We investigate the structure of minimal reversible entanglement generating set (MREGS) for tri-partite states under positive partial transpose (ppt) preserving operations. We demonstrate that the set consisting of W and EPR states alone cannot be an MREGS. In this context we prove the surprising result that the relative entropy of entanglement can be strictly sub-additive for certain pure tri-partite states which is crucial to keep open …
The convex set of quantum states of a composite $K \times K$ system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For $K\ge 3$ this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive. The level density of the PPT states is shown to differ from the Marchenko-Pastur distribution, supported in [0,4] and corresponding asymptotically to the ent…
The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems concerning the partial transpose of permutation matrices. More specifically, we count the number of permutations matrices which are equal to their partial transpose and the number of permutation matrices whose partial transpose is still a permutation. We solve these problems also when restricted to symmetric permutation matrices only.
Motivated by link transformations of lattice gauge theory, a method for generating local unitary invariants, especially for a system of qubits, has been pointed out in an earlier work [M. S. Williamson {\it et. al.}, Phys. Rev. A {\bf 83}, 062308 (2011)]. This paper first points the equivalence of the so constructed transformations to the combined operations of partial transpose and realignment. This allows construction of local unitary invariants of any system, with subsystems of arbitrary dimensions. Some properties of the resulting operators and consequences for pure tripartite higher dimen…
We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.
We construct a large class of multipartite qudit states which are positive under the family of partial transpositions. The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure and hence generalizes a class of bipartite circulant states proposed recently by the authors. This class contains many well known examples of multipartite quantum states from the literature and gives rise to a huge family of completely new states.
A bipartite subspace $S$ is called strongly positive-partial-transpose-unextendible (PPT-unextendible) if for every positive integer $k$, there is no PPT operator supporting on the orthogonal complement of $S^{\otimes k}$. We show that a subspace is strongly PPT-unextendible if it contains a PPT-definite operator (a positive semidefinite operator whose partial transpose is positive definite). Based on these, we are able to propose a simple criterion for verifying whether a set of bipartite orthogonal quantum states is indistinguishable by PPT operations in the many copy scenario. Utilizing thi…
For the certification and benchmarking of medium-size quantum devices efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partial…
In quantum information theory, it is a fundamental problem to construct multipartite unextendible product bases (UPBs). We show that there exist two families UPBs in Hilbert space $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4$ by merging two different systems of an existing $7$-qubit UPB of size $11$. Moreover, a new family of $7$-qubit positive-partial-transpose (PPT) entangled states of rank $2^7-11$ is constructed. We analytically derive a geometric measure of entanglement of a special PPT entangled states. Also an upper bound a…
The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size $6$ and six UPBs of size $9$ in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size $9$ in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size $7$ in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a spec…
Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e. they have a high Schmidt number. Here we investigate $d\times d$ dimensional PPT states for $d\ge 4$ discussed recently by Sindici and Piani, and by generalizing their methods to the calculation of Schmidt numbers we s…
We present the PPT square conjecture introduced by M. Christandl. We prove the conjecture in the case $n=3$ as a consequence of the fact that two-qutrit PPT states have Schmidt at most two. Our result in Lemma 3 is independent from the proof found Müller-Hermes. Müller-Hermes announced that this conjecture is true for the states on $\mathbb{C}_3\otimes\mathbb{C}_3$ \cite{hermes} recently. The PPT square conjecture in the case $n\ge4$ is still open.
Using the Hilbert-Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EW ) for n (>2) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we gi…
This paper has been withdrawn by the author, due a crucial error in the optimization. For the last one month I have been trying to remove the error, but it seems to take a lot of time so I decided to withdraw this paper for the moment.
Genuinely entangled subspaces are a class of subspaces in the multipartite Hilbert spaces that are composed of only genuinely entangled states. They are thus an interesting object of study in the context of multipartite entanglement. Here we provide a construction of multipartite subspaces that are not only genuinely entangled but also fully non-positive-partial-transpose (NPT) in the sense that any mixed state supported on them has non-positive partial transpose across any bipartition. Our construction originates from the stabilizer formalism known for its use in quantum error correction. To …
We have studied mixed states in the system of three qubits with the property that all their partial transposes are positive, these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. In random numerical searches we find entangled PPT states with a large variety of rank combinations. For ranks equal to five or higher we find both extremal and nonextremal PPT states of nearly every rank combination, with the restriction that the square sum of the four ranks of an extremal PPT state can be at most 193. We have studied especially…
The absolute separability problem asks for a characterization of the quantum states $ρ\in M_m\otimes M_n$ with the property that $UρU^\dagger$ is separable for all unitary matrices $U$. We investigate whether or not it is the case that $ρ$ is absolutely separable if and only if $UρU^\dagger$ has positive partial transpose for all unitary matrices $U$. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range cri…
We construct a parameterized family of $n\otimes n$ PPT (positive partial transpose) states of corank one for each $n\ge 3$. With a suitable choice of parameters, we show that they are $n\otimes n$ PPT entangled edge states of corank one for $3\le n\le 1000$. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by $2n-3$, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first ex…
We construct $3\otimes 3$ PPT entangled edge states with maximal ranks, to complete the classification of $3\otimes 3$ PPT entangled edge states by their types. The ranks of the states and their partial transposes are 8 and 6, respectively. These examples also disprove claims in the literature.
The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of re…
Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for non-commuting observables form a subset of these inequalities. In addition, criterion of positivity under partial transposition (PPT) imposes distinct bounds on moments, violations of which signal entanglement. We present bounds on some novel sets of composite moments, consequent to positive partial transposition of the density operator and report their violation by entangled multiqubit states. In particular, we derive separability bounds on a multiqubit mom…
We show that all $2\otimes 4$ states with strong positive partial transposes (SPPT) are separable. We also construct a family of $2\otimes 5$ entangled SPPT states, so the conjecture on the separability of SPPT states are completely settled. In addition, we clarify the relation between the set of all $2\otimes d$ separable states and the set of all $2\otimes d$ SPPT states for the case of $d=3,4$.
