Partial Transpose Research Articles

Measurement-device-independent nonlinear entanglement witnesses

Entanglement witnesses (EWs) are one of the most effective methods to detect entanglement. It is known that nonlinear EW provide better entanglement detection than their linear counterparts, in that the former detect a strictly larger subset of entangled states than the latter. Whether linear or nonlinear, the method is measurement-device dependent, so that imperfect measurements may cause false certification of entanglement in a shared state. Measurement-device-independent EW provide an escape from such measurement dependence of the entanglement detection for linear EW. Here we present measurement-device-independent nonlinear EW for non-positive partial transpose entangled states as well as for bound entangled states with positive partial transpose. Although the witness considered herein does not detect a larger set of entangled states than other nonlinear EW, it is more efficient in that it never leads to a false detection corresponding to wrong measurements. The constructed measurement-device-independent nonlinear EW certify the entanglement of the same sets of entangled states as their device-dependent parents do, and therefore are better than the linear EW, device-independent or otherwise.

Stability of PPT in equilibrium states

We use simple spectral perturbation theory to show that the positive partial transpose property is stable under bounded perturbations of the Hamiltonian, for equilibrium states in infinite dimensions. The result holds provided the temperature is high enough, or equivalently, provided the perturbation is small enough.

Finite temperature negativity Hamiltonians of the massless Dirac fermion

The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation of mixed-state entanglement. However, so far, it has only been studied for the mixed-state density matrices corresponding to subsystems of globally pure states. Here, we consider as a genuine example of a mixed state the one-dimensional massless Dirac fermions in a system at finite temperature and size. As subsystems, we consider an arbitrary set of disjoint intervals. The structure of the corresponding negativity Hamiltonian resembles the one for the entanglement Hamiltonian in the same geometry: in addition to a local term proportional to the stress-energy tensor, each point is non-locally coupled to an infinite but discrete set of other points. However, when the lengths of the transposed and non-transposed intervals coincide, the structure remarkably simplifies and we retrieve the mild non-locality of the ground state negativity Hamiltonian. We also conjecture an exact expression for the negativity Hamiltonian associated to the twisted partial transpose, which is a Hermitian fermionic matrix. We finally obtain the continuum limit of both the local and bi-local operators from exact numerical computations in free-fermionic chains.

Extraction of product and higher moment weak values: Applications in quantum state reconstruction and entanglement detection

Weak values of product observables or higher moments of an observable are informationally significant because of their ability to solve some paradoxes, realise strange quantum effects, reconstruct density matrices, etc. In this work, we demonstrate that pairwise orthogonal post-selections can be used to obtain higher moment weak values. By measuring only local weak values (defined as single system weak values in a multipartite scenario), product weak values can be obtained. As applications, we use product and higher moment weak values to reconstruct quantum states showing advantages over previous works in terms of number of required measurement operators and experimental feasibility. Additionally, a necessary separability criteria is given using product weak values to detect entanglement. For some classes of entangled states, positive partial transpose (PPT) criteria is achieved by cleverly choosing product observables and post selections. Robustness of our method against inappropriate choices of observables and noisy post-selections is also discussed.

Partial transpose moments, principal minors and entanglement detection

Recently, it has been shown that locally randomized measurements can be employed to get partial transpose moments of a density matrix [Elben A., {\it et al.} Phys. Rev. Lett. {\bf 125}, 200501 (2020)]. Consequently, two general entanglement detection methods were proposed based on partial transpose moments of a density matrix [Yu X-D., {\it et al.} Phys. Rev. Lett. {\bf 127}, 060504 (2021)]. In this context, a natural question arises that how partial transpose moments are related with entanglement and with well known idea of principal minors. In this work, we analytically demonstrate that for qubit-qubit quantum systems, partial transpose moments can be expressed as simple functions of principal minors. We expect this relation to exist for every bipartite quantum systems. In addition, we have extended the idea of PT-moments for tripartite qubit systems and have shown that PT-moments can only detect the whole range of being NPT for $GHZ$ and $W$ states mixed with white noise.

Negativity vs. purity and entropy in witnessing entanglement

In this paper, we explore the value of measures of mixedness in witnessing entanglement. While all measures of mixedness may be used to witness entanglement, we show that all such entangled states must have a negative partial transpose (NPT). Where the experimental resources needed to determine this negativity scale poorly at high dimension, we compare different measures of mixedness over both Haar-uniform and uniform-purity ensembles of joint quantum states at varying dimension to gauge their relative success at witnessing entanglement. In doing so, we find that comparing joint and marginal purities is overwhelmingly (albeit not exclusively) more successful at identifying entanglement than comparing joint and marginal von Neumann entropies, in spite of requiring fewer resources. We conclude by showing how our results impact the fundamental relationship between correlation and entanglement and related witnesses.

The Communication Value of a Quantum Channel

There are various ways to quantify the communication capabilities of a quantum channel. In this work we introduce the communication value (cv) of quantum channel, which describes the optimal probability of guessing the channel input from its output. By connecting to prior work on zero-error channel simulation, we show that the cv and its entanglement-assisted variant also offer dual interpretations as the classical communication cost for perfectly simulating different aspects of a channel using non-signaling resources. Our study involves characterizing the communication value as a generalized conditional min-entropy over the cone of separable operators. Using this characterization, we evaluate the cv for all qubit channels and higher-dimensional channels with certain symmetries. We find that the any entanglement-breaking channel has multiplicative cv when used in parallel with any other channel; the same is shown to hold for Pauli channels and partially depolarizing channels. In contrast, the cv is found to be non-multiplicative for a subset of the well-known Werner-Holevo channels. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT).

The SWAP imposter: Bidirectional quantum teleportation and its performance

Bidirectional quantum teleportation is a fundamental protocol for exchanging quantum information between two parties. Specifically, two individuals make use of a shared resource state as well as local operations and classical communication (LOCC) to swap quantum states. In this work, we concisely highlight the contributions of our companion paper [A. U. Siddiqui and M. M. Wilde, arXiv:2010.07905 (2020)]. We develop two different ways of quantifying the error of nonideal bidirectional teleportation by means of the normalized diamond distance and the channel infidelity. We then establish that the values given by both metrics are equal for this task. Additionally, by relaxing the set of operations allowed from LOCC to those that completely preserve the positivity of the partial transpose, we obtain semidefinite programing lower bounds on the error of nonideal bidirectional teleportation. We evaluate these bounds for some key examples—isotropic states and when there is no resource state at all. In both cases, we find an analytical solution. The second example establishes a benchmark for classical versus quantum bidirectional teleportation. Another example that we investigate consists of two Bell states that have been sent through a generalized amplitude damping channel. For this scenario, we find an analytical expression for the error, as well as a numerical solution that agrees with the former up to numerical precision.

Bipartite entanglement and the arrow of time

We provide a different perspective on the close relationship between entanglement and time. Our main focus is on bipartite entanglement, where this connection is foreshadowed both in the positive partial transpose criterion due to Peres [A. Peres, Phys. Rev. Lett. 77, 1413 (1996)] and in the classification of quantum within more general nonsignaling bipartite correlations [M. Frembs and A. D\"oring, Phys. Rev. A 106, 062420 (2022)]. Extracting the relevant common features, we identify a necessary and sufficient condition for bipartite entanglement in terms of a compatibility condition with respect to time orientations in local observable algebras, which express the dynamics in the respective subsystems. As an outlook and a program for future work, we discuss the relevance of the latter in the broader context of von Neumann algebras and the thermodynamical notion of time naturally arising within the latter.

Detection of polarization-spatial classical optical entanglement in partially coherent light fields using intensity measurements.

Detection of polarization-spatial classical optical entanglement through implementation of partial transpose on measured intensities is explored. A sufficient criterion for polarization-spatial entanglement in partially coherent light fields based on intensities measured at various orientations of the polarizer, as implied through partial transpose, is outlined. Detection of polarization-spatial entanglement using the outlined method is demonstrated experimentally through a Mach-Zehnder interferometer setup.

Fully non-positive-partial-transpose genuinely entangled subspaces

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 this end, we first introduce a couple of criteria allowing to assess whether any state from a given non-trivial stabilizer subspace is genuinely multipartite entangled. We then use these criteria to construct genuinely entangled stabilizer subspaces for any number of parties and arbitrary local dimension and conjecture them to be of maximal dimension achievable within the stabilizer formalism. At the same time, we prove that every genuinely entangled subspace is fully NPT in the above sense, which implies a quite surprising fact that no genuinely entangled stabilizer subspace can support PPT entangled states.

On adjacency and Laplacian cospectral non-isomorphic signed graphs

Let Γ = (G,σ) be a signed graph, where σ is the sign function on the edges of G. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.

Quantifying the performance of approximate teleportation and quantum error correction via symmetric 2-PPT-extendible channels

The ideal realization of quantum teleportation relies on having access to a maximally entangled state; however, in practice, such an ideal state is typically not available and one can instead only realize an approximate teleportation. With this in mind, we present a method to quantify the performance of approximate teleportation when using an arbitrary resource state. More specifically, after framing the task of approximate teleportation as an optimization of a simulation error over one-way local operations and classical communication (LOCC) channels, we establish a semi-definite relaxation of this optimization task by instead optimizing over the larger set of two-PPT-extendible channels. The main analytical calculations in our paper consist of exploiting the unitary covariance symmetry of the identity channel to establish a significant reduction of the computational cost of this latter optimization. Next, by exploiting known connections between approximate teleportation and quantum error correction, we also apply these concepts to establish bounds on the performance of approximate quantum error correction over a given quantum channel. Finally, we evaluate our bounds for various examples of resource states and channels.

Exact entanglement cost of quantum states and channels under positive-partial-transpose-preserving operations

This paper establishes single-letter formulas for the exact entanglement cost of simulating quantum channels under free quantum operations that completely preserve positivity of the partial transpose (PPT). First, we introduce the $\kappa$-entanglement measure for point-to-point quantum channels, based on the idea of the $\kappa$-entanglement of bipartite states, and we establish several fundamental properties for it, including amortization collapse, monotonicity under PPT superchannels, additivity, normalization, faithfulness, and non-convexity. Second, we introduce and solve the exact entanglement cost for simulating quantum channels in both the parallel and sequential settings, along with the assistance of free PPT-preserving operations. In particular, we establish that the entanglement cost in both cases is given by the same single-letter formula, the $\kappa$-entanglement measure of a quantum channel. We further show that this cost is equal to the largest $\kappa$-entanglement that can be shared or generated by the sender and receiver of the channel. This formula is calculable by a semidefinite program, thus allowing for an efficiently computable solution for general quantum channels. Noting that the sequential regime is more powerful than the parallel regime, another notable implication of our result is that both regimes have the same power for exact quantum channel simulation, when PPT superchannels are free. For several basic Gaussian quantum channels, we show that the exact entanglement cost is given by the Holevo--Werner formula [Holevo and Werner, Phys. Rev. A 63, 032312 (2001)], giving an operational meaning of the Holevo-Werner quantity for these channels.

A note on positive partial transpose blocks

<abstract><p>In this article, we study the class of positive partial transpose blocks. We introduce several inequalities related to this class with an emphasis on comparing the main diagonal and off-diagonal components of a $ 2 \times 2 $ positive partial transpose block.</p></abstract>

Singular value inequalities on 2 ? 2 block accretive partial transpose matrices

Singular value inequalities on 2 ? 2 block accretive partial transpose matrices

Operational entanglement detection based on $$\Lambda $$-moments

We introduce $\Lambda$-moments with respect to any positive map $\Lambda$. We show that these $\Lambda$-moments can effectively characterize the entanglement of unknown quantum states without theirs prior reconstructions. Based on $\Lambda$-moments necessary and sufficient separability criteria, as well as necessary optimized criteria are presented, which include the ones in [\href{https://link.aps.org/doi/10.1103/PhysRevLett.127.060504}{Phys. Rev. Lett. \textbf{127}, 060504 (2021)}] as special cases. Detailed example is given to show that our criteria can detect bound entanglement that can not be identified by positive partial transpose criterion, with the explicit measurement operators to experimentally measure the corresponding $\Lambda$-moments.

Inertia of two-qutrit entanglement witnesses

Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We investigate the inertia of bipartite EWs constructed by the partial transpose of NPT states. Furthermore, we find out most of the inertia of the partial transpose of the two-qutrit bipartite NPT states. As an application, we extend our results to high-dimensional states.

On positive and positive partial transpose matrices

A block matrix $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ is positive partial transpose (PPT) if both $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ and $\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]$ are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.

Entanglement in phase-space distribution for an anisotropic harmonic oscillator in noncommutative space

The bipartite Gaussian state, corresponding to an anisotropic harmonic oscillator in a noncommutative space (NCS), is investigated with the help of Simon’s separability condition (generalized Peres-Horodecki criterion). It turns out that, to exhibit the entanglement between the noncommutative co-ordinates, the parameters (mass and frequency) have to satisfy a unique constraint equation. We have considered the most general form of an anisotropic oscillator in NCS, with both spatial and momentum noncommutativity. The system is transformed to the usual commutative space (with usual Heisenberg algebra) by a well-known Bopp shift. The system is transformed into an equivalent simple system by a unitary transformation, keeping the intrinsic symplectic structure (\(Sp(4,{\mathbb {R}})\)) intact. Wigner quasiprobability distribution is constructed for the bipartite Gaussian state with the help of a Fourier transformation of the characteristic function. It is shown that the identification of the entangled degrees of freedom is possible by studying the Wigner quasiprobability distribution in phase space. We have shown that the co-ordinates are entangled only with the conjugate momentum corresponding to other co-ordinates.