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arXiv:quant-ph/0703033 · Physical Review A

Multipartite unlockable bound entanglement in the stabilizer formalism

Guoming Wang, Mingsheng Ying

Published May 24, 2007• 18 citations

Abstract

We find an interesting relationship between multipartite bound entangled states and the stabilizer formalism. We prove that, if a set of commuting operators from the generalized Pauli group on $n$ qudits satisfy certain constraints, then the maximally mixed state over the subspace stabilized by them is an unlockable bound entangled state. Moreover, the properties of this state, such as symmetry under permutations of parties, undistillability, and unlockability, can be easily explained from the stabilizer formalism without tedious calculation. In particular, the four-qubit Smolin state [Smolin, Phys. Rev. A 63, 032306 (2001)] and its recent generalization to even numbers of qubits [Bandyopadhyay et al., Phys. Rev. A 71, 062317 (2005); Augusiak et al., ibid 73, 012318 (2006)] can be viewed as special examples of our results. Finally, we extend our results to arbitrary multipartite systems in which the dimensions of all parties may be different.

DOI: 10.1103/physreva.75.052332

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We find an interesting relationship between multipartite bound entangled states and the stabilizer formalism. We prove that, if a set of commuting operators from the generalized Pauli group on

n

qudits satisfy certain constraints, then the maximally mixed state over the subspace stabilized by them is an unlockable bound entangled state. Moreover, the properties of this state, such as symmetry under permutations of parties, undistillability, and unlockability, can be easily explained from the stabilizer formalism without tedious calculation. In particular, the four-qubit Smolin state [Smolin, Phys. Rev. A 63, 032306 (2001)] and its recent generalization to even numbers of qubits [Bandyopadhyay et al., Phys. Rev. A 71, 062317 (2005); Augusiak et al., ibid 73, 012318 (2006)] can be viewed as special examples of our results. Finally, we extend our results to arbitrary multipartite systems in which the dimensions of all parties may be different.

A class of unlockable bound entangled states and their applications are presented. They can be considered as quasi generalized Smolin states [Phys. Rev. A 63 (2001) 032306], which are the states of N + N qubits. No pure entanglement can be distilled from this class of states by local quantum operations and classical communications. However, if certain parties group together, they become distillable. Although they are bound entangled states, they could be used to achieve some non-trivial tasks, such as quantum secret sharing shown in the study.

I present a four-party unlockable bound entangled state, that is, a four-party quantum state which cannot be written in a separable form and from which no pure entanglement can be distilled by local quantum operations and classical communication among the parties, and yet when any two of the parties come together in the same laboratory they can perform a measurement which enables the other two parties to create a pure maximally entangled state between them without coming together. This unlocking ability can be viewed in two ways, either as a determination of which Bell state is shared in the mixture or as a kind of quantum teleportation with cancellation of Pauli operators.

We investigate the entanglement properties in the symmetric subspace of N-partite d-dimensional systems (qudits). As it happens already for bipartite diagonal symmetric states, also in the multipartite case the local dimension d plays a crucial role. Here, we demonstrate that there is no bound entanglement for d = 3, 4 and N = 3. Using different techniques, we present strong analytical evidence that no bound entanglement exist for any N if d ≤ 4. Interestingly, bound entanglement of diagonal symmetric states exist for any number of parties, N ≥ 2, and local dimensions d ≥ 5.

How entangled is a randomly chosen bipartite stabilizer state? We show that if the number of qubits each party holds is large, the state will be close to maximally entangled with probability exponentially close to 1. We provide a similar tight characterization of the entanglement present in the maximally mixed state of a randomly chosen stabilizer code. Finally, we show that typically very few Greenberger-Horne-Zeilinger states can be extracted from a random multipartite stabilizer state via local unitary operations. Our main tool is a concentration inequality which bounds deviations from the mean of random variables which are naturally defined on the Clifford group.

We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for

n

-qubit states, we express the condition that all

k

-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all

k

vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group

{\mathcal{P}}_{n}

that is isomorphic to

{\mathbb{Z}}_{2}^{n}

. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed

k

for fixed

n

. In particular, the ratio

k/n

can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed

n

-qubit reductions with

k

. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though

k>1

is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.