The characterization of possible transformations between quantum states and the quantification of entanglement are central for understanding quantum resources. Explicit results are, however, scarce since they require complicated optimizations over transformation protocols or state decompositions. We introduce a method to optimize functions of the entanglement spectrum of bipartite states over mixed-state decompositions with arbitrary accuracy. Our results are based on a convergent semidefinite programming hierarchy of optimizations over entanglement spectra using a natural polynomial basis and Schur-Weyl duality. Our approach can also be used to determine non-polynomial measures, such as the entanglement of formation, and to characterize the dimensionality of entanglement. Moreover, extracting the decomposition allows us to formulate criteria to determine state transformations under local operations and classical communication (LOCC). In particular, we provide a hierarchy of certificates of LOCC non-convertibility from pure to mixed states in any dimension and a heuristic approach for LOCC convertibility.
The characterization of possible transformations between quantum states and the quantification of entanglement are central for understanding quantum resources. Explicit results are, however, scarce since they require complicated optimizations over transformation protocols or state decompositions. We introduce a method to optimize functions of the entanglement spectrum of bipartite states over mixed-state decompositions with arbitrary accuracy. Our results are based on a convergent semidefinite programming hierarchy of optimizations over entanglement spectra using a natural polynomial basis and Schur-Weyl duality. Our approach can also be used to determine non-polynomial measures, such as the entanglement of formation, and to characterize the dimensionality of entanglement. Moreover, extracting the decomposition allows us to formulate criteria to determine state transformations under local operations and classical communication (LOCC). In particular, we provide a hierarchy of certificates of LOCC non-convertibility from pure to mixed states in any dimension and a heuristic approach for LOCC convertibility.
The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions Efcr(ρ)≥Efcr(σ) for all convex roof entanglement measures Efcr is insufficient to guarantee the existence of an LOCC transformation ΛL(⋅) from ρ to σ; and (ii) The inequalities ∑jpjE(φj)≥∑lqlE(ψl) for all entanglement monotones E are not sufficient to ensure the existence of an LOCC transformation from {pj,∣φj⟩} to {ql,∣ψl⟩}.
The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit states cannot be obtained from nor transformed to any state (i.e., they are isolated), which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that generic n-qubit symmetric states are also isolated, we consider particular families for which we can determine to be, on the contrary, endowed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. In the course of the study of LOCC transformations, we also characterize the local symmetries of symmetric states.
Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n\ensuremath{\ge}5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality.
For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays.
The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from m such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2 and relax to a semidefinite program of size O(m) for arbitrary m. The dual solutions are polynomial minorants of x on [0,1], and two explicit families, binomial and L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.
Abstract We advance on the conversion of bipartite quantum states via local operations and classical communication (LOCC) for infinite-dimensional systems. We introduce δ-LOCC convertibility based on the observation that any pure state can be approximated by a state with finite-support Schmidt coefficients. We show that δ-LOCC convertibility of bipartite states is fully characterized by a majorization relation between the sequences of squared Schmidt coefficients, providing a novel extension of Nielsen’s theorem for infinite-dimensional systems. Hence, our definition is equivalent to the one of ε-LOCC convertibility (Owari et al 2008 Quantum Inf. Comput. 8 0030), but deals with states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss the notions of optimal common resource and optimal common product in this scenario. The optimal common product always exists, whereas the optimal common resource depends on the existence of a common resource. This highlights a distinction between the resource-theoretic aspects of finite versus infinite-dimensional systems. Our results rely on the order-theoretic properties of majorization for infinite sequences, applicable beyond the LOCC convertibility problem.
In this study, we enhance the understanding of entanglement transformations and their quantification by extending the concept of Schmidt vector from pure to mixed bipartite states, exploiting the lattice structure of majorization. The Schmidt vector of a bipartite mixed state is defined using two distinct methods: as a concave roof extension of Schmidt vectors of pure states, or equivalently, from the set of pure states that can be transformed into the mixed state through local operations and classical communication (LOCC). We demonstrate that the Schmidt vector fully characterized separable and maximally entangled states. Furthermore, we prove that the Schmidt vector is monotonic and strongly monotonic under LOCC, giving necessary conditions for conversions between mixed states. Additionally, we extend the definition of the Schmidt rank from pure states to mixed states as the cardinality of the support of the Schmidt vector and show that it is equal to the Schmidt number introduced in previous work [Terhal and Horodecki, Phys. Rev. A 61, 040301(R) (2000)]. Finally, we introduce a family of entanglement monotones by considering concave and symmetric functions applied to the Schmidt vector.