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arXiv:2106.16216 · Physical Review A

Absolutely entangled sets of pure states for bipartitions and multipartitions

Baichu Yu, Pooja Jayachandran, Adam Burchardt, Yu Cai, Nicolas Brunner, Valerio Scarani

Published Sep 17, 2021• 0 citations
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Abstract

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 $N$ states such that $N\ensuremath{-}3$ (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition $d={d}_{1}{d}_{2}$, 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.

DOI
10.1103/physreva.104.032414

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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.
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We propose a straightforward method to determine the maximal entanglement of pure states using the criterion of maximal I-concurrence, a measure of entanglement. The square of concurrence for a bipartition X∣X′X|X^\primeX∣X′ of a pure state is defined as EX∣X′2=2[1−tr(ρX2)]E^2_{X| X ^\prime}=2[1-tr({ρ_X}^2)]EX∣X′2​=2[1−tr(ρX​2)]. From this, we can infer that the concurrence EX∣X′E_{X| X ^\prime}EX∣X′​ reaches its maximum when tr(ρX2)tr({ρ_X}^2)tr(ρX​2) is minimized. Using this approach, we identify numerous Absolutely Maximally Entangled (AME) pure states that exhibit maximal entanglement across all possible bipartitions. Conditions are derived for pure states to achieve maximal mixedness in all bipartitions, revealing that any pure state with an odd number of subsystem coefficients does not meet the AME criterion. Furthermore, we obtain equal maximal multipartite entangled pure states across all bipartitions using our maximal concurrence criterion.
Abstract A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a 2 × 3 × 3 × 3 system that shows maximal entanglement across every bipartition.
Abstract Genuine entanglement is the strongest form of multipartite entanglement. Genuinely entangled pure states contain entanglement in every bipartition and as such can be regarded as a valuable resource in the protocols of quantum information processing. A recent direction of research is the construction of genuinely entangled subspaces (GESs)—the class of subspaces consisting entirely of genuinely entangled pure states. In this paper we present methods of construction of such subspaces including those of maximal possible dimension. The approach is based on the composition of bipartite entangled subspaces and quantum channels of certain types. The examples include maximal subspaces for systems of three qubits, four qubits, three qutrits. We also provide lower bounds on two entanglement measures for mixed states, the concurrence and the convex-roof extended negativity, which are directly connected with the projection on GESs.