Improved semidefinite programming upper bound on distillable entanglement

We show that the maximum fidelity obtained by a positive partial transpose (p.p.t.) distillation protocol is given by the solution to a certain semidefinite program. This gives a number of new lower and upper bounds on p.p.t. distillable entanglement (and thus new upper bounds on 2-locally distillable entanglement). In the presence of symmetry, the semidefinite program simplifies considerably, becoming a linear program in the case of isotropic and Werner states. Using these techniques, we determine the p.p.t. distillable entanglement of asymmetric Werner states and maximally correlated states. We conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1-local distillation.

The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics. The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve. In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties. We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension. We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity. As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned. This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point. We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement. Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes. We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.

We derive an upper bound on the one-way distillable entanglement of bipartite quantum states. To this end, we revisit the notion of degradable, conjugate degradable, and antidegrad-able bipartite quantum states [1]. We prove that for degradable and conjugate degradable states the one-way distillable entanglement is equal to the coherent information, and thus given by a single-letter formula. Furthermore, it is well-known that the one-way distillable entanglement of antidegradable states is zero. We use these results to derive an upper bound for arbitrary bipartite quantum states, which is based on a convex decomposition of a bipartite state into degradable and antidegradable states. This upper bound is always at least as good an upper bound as the entanglement of formation. Applying our bound to the qubit depolarizing channel, we obtain an upper bound on its quantum capacity that is strictly better than previously known bounds in the high noise regime. We also transfer the concept of approximate degradability [2] to quantum states and show that this yields another easily computable upper bound on the one-way distillable entanglement. Moreover, both methods of obtaining upper bounds on the one-way distillable entanglement can be combined into a generalized one.

Weak values of quantum observables are a powerful tool for investigating quantum phenomena. Some methods for measuring weak values in the laboratory require weak interactions and postselection, while others are deterministic, but require statistics over a number of experiments that grows linearly with the dimension of the measured system in the worst case over all possible observables. Here we propose a deterministic dimension-independent scheme for estimating weak values of arbitrary observables. The scheme is based on controlled operations and associates states and observables in the mathematical expression of the weak value to preparations devices and measurement devices in the experimental setup, respectively. Thanks to this feature, it provides insights into the relation between states of two identical quantum systems at a single moment of time and states of a single quantum system at two moments of time, also known as two-time states. Specifically, our scheme provides an alternative expression for two-time states and establishes a link between two-time states accessible through the controlled- scheme and bipartite quantum states with positive partial transpose. Published by the American Physical Society 2024

There are only two known kinds of zero-capacity channels. The first kind produces entangled states that have positive partial transpose, and the second one - states that are cloneable. We consider the family of 'hybrid' quantum channels, which lies in the intersection of the above classes of channels and investigate its properties. It gives rise to the first explicit examples of the channels, which create bound entangled states that have the property of being cloneable to the arbitrary finite number of parties. Hybrid channels provide the first example of highly cloneable binding entanglement channels, for which known superactivation protocols must fail - superactivation is the effect where two channels each with zero quantum capacity having positive capacity when used together. We give two methods to construct a hybrid channel from any binding entanglement channel. We also find the low-dimensional counterparts of hybrid states - bipartite qubit states which are extendible and possess two-way key.

I show that bound entanglement strongly influences the quantum entanglement processing of pure states: If N distant parties share appropriate bound entangled states with positive partial transpose, all N‐partite pure entangled states become inter‐convertible by stochastic local operations and classical communication (SLOCC) at the single copy level. This implies that the Schmidt rank of a bipartite pure entangled state can be increased, and that two incomparable tripartite entanglement of the GHZ and W type can be inter‐converted by the assistance of bound entanglement. Further, I propose the simplest experimental scheme for the demonstration of the corresponding bound‐entanglement‐assisted SLOCC. This scheme does not need quantum gates and is feasible for the current experimental technology of linear optics.

Revisiting the behavior of quantum correlations under nondissipative decoherence by means of the correlation matrix

I show that two distant parties can transform pure entangled states to arbitrary pure states by stochastic local operations and classical communication (SLOCC) at the single copy level, if they share bound entangled states. This is the effect of bound entanglement since this entanglement processing is impossible by SLOCC alone. A similar effect of bound entanglement exists in three qubits where two incomparable entangled states of GHZ (Greenberger, Horne, and Zeilinger) and W can be interconverted. In general, multipartite settings composed by N distant parties, all N-partite pure entangled states are interconvertible by SLOCC with the assistance of bound entangled states with positive partial transpose.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we develop a formalism for distilling a classical key from a quantum state in a systematic way, expanding on our previous work on a secure key from bound entanglement (Horodecki <etal/>, 2005). More detailed proofs, discussion, and examples are provided of the main results. Namely, we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory, with the only change being that instead of distilling Einstein–Podolsky–Rosen (EPR) pairs, the parties distill private states. The form of these general private states are given, and we show that there are a number of useful ways of expressing them. Some of the private states can be approximated by certain states, which are bound entangled. Thus, distillable entanglement is not a requirement for a private key. We find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel (PQC). We also find a general class of states, which have negative partial transpose (are NPT), but which appear to be bound entangled. The relative entropy distance is shown to be an upper bound on the rate of a key. This allows us to compute the <emphasis emphasistype="bold"><emphasis emphasistype="italic">exact</emphasis></emphasis> value of a distillable key for a certain class of private states. </para>

An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the classical and quantum capacities of the channel. Although our formula involves regularization, i.e. taking a limit over many copies of the channel, it reduces to a single-letter expression in the case of generalized dephasing channels. Analogous formulas are conjectured for the simultaneous public-private capacity of a quantum channel and for the simultaneously 1-way distillable common randomness and entanglement of a bipartite quantum state.

Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information. In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence. Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps. We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.

Quantum entanglement is the core resource in quantum information processing and quantum computing. It is a significant challenge to effectively characterize the entanglement of quantum states. Recently, elegant separability criterion is presented by Elben et al. (Phys Rev Lett 125:200501, 2020) based on the first three partially transposed (PT) moments of density matrices. Then, Yu et al. (Phys Rev Lett 127:060504, 2021) proposed two general powerful criteria based on the PT moments. In this paper, based on the realignment operations of matrices we propose entanglement detection criteria in terms of such realignment moments. We show by detailed example that the realignment moments can also be used to identify quantum entanglement.

The separability and entanglement of quantum mixed states in ${C}^{2}\ensuremath{\bigotimes}{C}^{3}\ensuremath{\bigotimes}{C}^{N}$ composite quantum systems are investigated. It is shown that all quantum states $\ensuremath{\rho}$ with positive partial transposes and rank $r(\ensuremath{\rho})&lt;~N$ are separable.

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 counting. These results uncover the presence of only negative partial transpose entanglement and illustrate the complexity of more general non-classical correlations in this paradigmatic class of genuine non-Gaussian quantum states.

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