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arXiv:quant-ph/0311095 · Physical Review A

Distilling multipartite pure states from a finite number of copies of multipartite mixed states

Ping‐Xing Chen, Cheng-Zu Li

Published Jan 14, 2004• 3 citations
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Abstract

This paper addresses the question of the distillation of entanglement from a finite number of multipartite mixed states. It is shown that, if one can distill a pure entangled state from n copies of a mixed state ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}},$ there must be at least a subspace in the whole Hilbert space of all the copies such that the projection of ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}}^{\ensuremath{\bigotimes}n}$ onto the subspace is a pure entangled state. We also show that the purification of entanglement or distillation of entanglement can be carried out by local joint projective measurements with the help of classical communication and local general positive operator valued measurements on a single particle, in principle. Finally, we discuss experimental realizability of the entanglement purification.

DOI
10.1103/physreva.69.012308

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This paper addresses the question of the distillation of entanglement from a finite number of multipartite mixed states. It is shown that, if one can distill a pure entangled state from n copies of a mixed state \ensuremathσABC\ensuremath⋅\ensuremath⋅\ensuremath⋅,{\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}},\ensuremathσABC\ensuremath⋅\ensuremath⋅\ensuremath⋅​, there must be at least a subspace in the whole Hilbert space of all the copies such that the projection of \ensuremathσABC\ensuremath⋅\ensuremath⋅\ensuremath⋅\ensuremath⨂n{\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}}^{\ensuremath{\bigotimes}n}\ensuremathσABC\ensuremath⋅\ensuremath⋅\ensuremath⋅\ensuremath⨂n​ onto the subspace is a pure entangled state. We also show that the purification of entanglement or distillation of entanglement can be carried out by local joint projective measurements with the help of classical communication and local general positive operator valued measurements on a single particle, in principle. Finally, we discuss experimental realizability of the entanglement purification.
It is well known that any entangled mixed state in $2\ensuremath{\bigotimes}2$ systems can be purified via infinite copies of the mixed state. But, can one distill a pure maximally entangled state from finite copies of a mixed state in any bipartite system by local operation and classical communication? This is more meaningful in practical application. We give a necessary and sufficient condition for this distillability. This condition requires that there exist distillable subspaces. According to this condition, one can judge easily whether a mixed state is distillable or not. We also analyze some properties of distillable subspaces, and discuss the most efficient purification protocol. Finally, we discuss the distillable enanglement of a two-qubit system for the case of finite copies.
Distillation and purification of entanglement is an important field of quantum information both in theory and in experiment. This paper presents some main conclusions in this field, and discusses mainly the distillation of entanglement from finite copies of a mixed state. By using the conceptions of quasiseparable state and distillable subspace, we show a necessary condition and a sufficient and necessary condition of the distillability from finite copies of a mixed state. We also get the most efficient protocols of entanglement distillation for the case of finite copies. Finally we consider the generalization of these results to the case of infinite copies of a mixed state.
Known entanglement purification protocols for mixed states use collective measurements on several copies of the state in order to increase the entanglement of some of them. We address the question of whether it is possible to purify the entanglement of a state by processing each copy separately. While this is possible for pure states, we show that this is impossible, in general, for mixed states. The importance of this result both conceptually and for experimental realization of purification is discussed. We also give explicit invariants of an entangled state of two qubits under local actions and classical communication.
In this work, we study how to distill two typical multipartite entangled noisy states, the amplitude-damped W state and the amplitude-damped Greenberger-Horne-Zeilinger state. Unlike the currently existing multipartite entanglement purification schemes, our scheme is a single-copy-based scheme, i.e., only one noisy state is needed in each distillation round, and the complicated multilateral controlled-not (cnot) operations are replaced by measurements of local positive-operator-valued measures (POVMs). The results show that local POVM measurements on one copy of the noisy state can project the initial state onto a new state with higher entanglement and nonlocality. In this scheme, although the final state can violate the multipartite Bell inequality, the corresponding initial noisy state does not violate it, which means that this distillation scheme can reveal the hidden genuine multipartite nonlocality of these initial states. For the case of the amplitude-damped W state, if the POVM parameters are appropriately chosen, the fidelity of the output state can approach 1.01.01.0 for all the initial states with fidelity bigger than zero. So this single-copy-based multipartite entanglement distillation scheme has a wide range of distillable fidelities, and it is much simpler than the two-copy- or multiple-copy-based schemes. Furthermore, if the POVM measurement is carried out only on some member qubits rather than all of them, the distillation scheme can succeed too, which will greatly decrease the implementation complexity of the scheme. So this multipartite entanglement distillation scheme may be implementable in some physical system.
A mixed quantum state $\ensuremath{\rho}$ shared between two parties is said to be distillable if, by means of a protocol involving only local quantum operations and classical communication, the two parties can transform some number of copies of $\ensuremath{\rho}$ into a single shared pair of qubits having high fidelity with the maximally entangled state $|{\ensuremath{\varphi}}^{+}\ensuremath{\rangle}=(|00\ensuremath{\rangle}+|11\ensuremath{\rangle})/\sqrt{2}$. In this Letter it is proved that there exist states that are distillable, but for which an arbitrarily large number of copies is required before any distillation procedure can produce a shared pair of qubits with even a small amount of entanglement. Specifically, for every positive integer nnn there exists a state $\ensuremath{\rho}$ that is distillable, but, given nnn or fewer copies of $\ensuremath{\rho}$, every distillation procedure outputting a single shared pair of qubits outputs those qubits in a separable (i.e., unentangled) state.
Abstract This chapter discusses how entangled two states are. One obvious way is to consider the distillation procedure, in which the entanglement of any state is quantified by how many fully entangled qubits can be distilled from an entangled state. However, it is not easy to see how to distil singlets out of n copies of a general mixed state. Hence, it is necessary to first see what can be done with pure states. This chapter discusses the distillation of multiple copies of a pure state, analogy between quantum entanglement and the Carnot cycle, properties of entanglement measures, entanglement of pure states, entanglement of mixed states, measures of entanglement derived from relative entropy, relationship between classical information and entanglement, and links between entanglement and thermodynamics.