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arXiv:2006.07165 · Physical Review A

Entanglement for any definition of two subsystems

Yu Cai, Baichu Yu, Pooja Jayachandran, Nicolas Brunner, Valerio Scarani, Jean-Daniel Bancal

Published May 26, 2021• 4 citations
arXivHTMLPDF

Abstract

The notion of entanglement of quantum states is usually defined with respect to a fixed bipartition. Indeed, a global basis change can always map an entangled state to a separable one. The situation is, however, different when considering a set of states. In this work we define the notion of an ``absolutely entangled set'' of quantum states: for any possible choice of global basis, at least one of the states in the set is entangled. Hence, for all bipartitions, i.e., any possible definition of the subsystems, the set features entanglement. We present examples of such sets, including sets with minimal size. Moreover, we propose a quantitative measure for absolute set entanglement. To lower bound this quantity, we develop a method based on polynomial optimization to perform convex optimization over unitaries, which is of independent interest.

DOI
10.1103/physreva.103.052432

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The notion of entanglement of quantum states is usually defined with respect to a fixed bipartition. Indeed, a global basis change can always map an entangled state to a separable one. The situation is, however, different when considering a set of states. In this work we define the notion of an ``absolutely entangled set'' of quantum states: for any possible choice of global basis, at least one of the states in the set is entangled. Hence, for all bipartitions, i.e., any possible definition of the subsystems, the set features entanglement. We present examples of such sets, including sets with minimal size. Moreover, we propose a quantitative measure for absolute set entanglement. To lower bound this quantity, we develop a method based on polynomial optimization to perform convex optimization over unitaries, which is of independent interest.
Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on Cd1⊗Cd2\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}Cd1​⊗Cd2​ for any nonprime dimension d=d1×d2d=d_1\times d_2d=d1​×d2​. Moreover, based on the structure of the orthogonal product basis in C2⊗Cn\mathbb{C}^2\otimes \mathbb{C}^nC2⊗Cn, we obtain another construction of absolutely entangled set with 2n+12n+12n+1 elements in C2⊗Cn\mathbb{C}^2\otimes \mathbb{C}^nC2⊗Cn.
A set of quantum states is said to be absolutely entangled when at least one state in the set remains entangled for any definition of subsystems, i.e., for any choice of the global reference frame. In this work we investigate the properties of absolutely entangled sets (AESs) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of NNN states such that $N\ensuremath{-}3$ (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition d=d1d2d={d}_{1}{d}_{2}d=d1​d2​, we prove that sets of $N>\ensuremath{\lfloor}({d}_{1}+1)({d}_{2}+1)/2\ensuremath{\rfloor}$ states are AESs with Haar measure 1. Then, we define AESs for multipartitions. We derive a general lower bound on the number of states in an AES for a given multipartition, and also construct explicit examples. In particular, we exhibit an AES with respect to any possible multipartitioning of the total system.
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
The unambiguous detection and quantification of entanglement is a hot topic of scientific research, though it is limited to low dimensions or specific classes of states. Here we identify an additional class of quantum states, for which bipartite entanglement measures can be efficiently computed, providing rigorous results. Such states are written in arbitrary $d\ifmmode\times\else\texttimes\fi{}d$ dimensions, where each basis state in the subsystem A is paired with only one state in B. This class, to which we refer as pair basis states, is remarkably relevant in many physical situations, including quantum optics. We find that negativity is a necessary and sufficient measure of entanglement for mixtures of states written in the same pair basis. We also provide analytical expressions for a tight lower-bound estimation of the entanglement of formation, a central quantity in quantum information.
An absolutely entangled set (AES) is a collection of states for which at least one member remains entangled under any choice of global unitary transformation. We develop a semi-device-independent framework for certifying this property from dimensionally bounded prepare-and-measure correlations. A witness value above the maximum attainable with separable preparations of a given dimension rules out every non-AES realization. To upperbound this limit, we construct a semidefinite-programming (SDP) hierarchy tailored to separable preparations. For a two-qubit random-access-code witness, the SDP upper bound and the heuristic lower bound coincide to the reported precision, yielding a concrete AES certification. We also formulate a general construction of witnesses for extended preparation sets containing target pure-state AESs. Examples based on equiangular tight frames, a minimal AES, and mutually unbiased bases further illustrate the scope of the method. In summary, our results establish a systematic and experimentally feasible approach to the certification of AESs in semi-device-independent quantum information protocols.
The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one cannot expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher-order structure characterizing that is needed. In this paper we make some steps in this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. K\"ok\'enyesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It also turns out that, physically, this structure is related to the local operations and classical communication convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy. Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some ``unified'' notion of entanglement.