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arXiv:quant-ph/0103005 · Physical Review A

Mixed state entanglement: Manipulating polarization-entangled photons

Rob Thew, William J. Munro

Published Jul 18, 2001• 23 citations

Abstract

There has recently been much discussion regarding entanglement transformations in terms of local filtering operations and whether the optimal entanglement for an arbitrary two-qubit state could be realized. We introduce an experimentally realizable scheme for manipulating the entanglement of an arbitrary state of two polarization-entangled qubits. This scheme is then used to provide some perspective to the mathematical concepts inherent in this field with respect to a laboratory environment. Specifically, we look at how to extract enhanced entanglement from systems with a fixed rank, and, in the case where the rank of the density operator for the state can be reduced, show how the state can be made arbitrarily close to a maximally entangled pure state. In this context we also discuss bounds on entanglement in mixed states.

DOI: 10.1103/physreva.64.022320

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There has recently been much discussion regarding entanglement transformations in terms of local filtering operations and whether the optimal entanglement for an arbitrary two-qubit state could be realized. We introduce an experimentally realizable scheme for manipulating the entanglement of an arbitrary state of two polarization-entangled qubits. This scheme is then used to provide some perspective to the mathematical concepts inherent in this field with respect to a laboratory environment. Specifically, we look at how to extract enhanced entanglement from systems with a fixed rank, and, in the case where the rank of the density operator for the state can be reduced, show how the state can be made arbitrarily close to a maximally entangled pure state. In this context we also discuss bounds on entanglement in mixed states.

Two methods for creating arbitrary two-photon polarization pure states are introduced. Based on these, four schemes for creating two-photon polarization mixed states are proposed and analyzed. The first two schemes can synthesize completely arbitrary two-qubit mixed states, i.e., control all 15 free parameters: scheme I requires several sets of crystals, while scheme II requires only a single set, but relies on decohering the pump beam. Additionally, we describe two further schemes which are much easier to implement. Although the total capability of these is still being studied, we show that they can synthesize all two-qubit Werner states, maximally entangled mixed states, Collins-Gisin states, and arbitrary Bell-diagonal states.

The optimal entanglement manipulation for a single copy of mixed states of two qubits is to transform it to a Bell diagonal state. In this paper we derive an explicit form of the local operation that can realize such a transformation. The result obtained is universal for arbitrary entangled two-qubit states and it discloses that the corresponding local filter is not unique for density matrices with rank

n=2

and can be exclusively determined for that with

n=3

and 4. As illustrations, a four-parameter family of mixed states are explored, the local filter as well as the transformation probability are given explicitly, which verify the validity of the general result.

We establish a quantitative relation between local spin polarization and quantum entanglement in two-qubit systems by mathematically proving a closed-form upper bound on the concurrence at fixed local polarization magnitudes. The bound shows how the maximal concurrence is jointly constrained by the two local polarizations. We further demonstrate that this bound is saturated by pure states in certain cases with identical polarizations. As a concrete physical application, we consider the parity-violating process

e^+e^- \to Z^0 \to q\bar{q}

, which generates final-state spin polarization. We show that the maximal concurrence is attained in specific kinematic regions and is significantly reduced relative to the unpolarized case. These results establish a general, process-independent framework connecting local polarization, maximal entanglement, and pure states.

We propose a deterministic remote state preparation scheme for photon polarization qubit states, where entanglement, local operations, and classical communication are used. By consuming one maximally entangled state and two classical bits, an arbitrary (either pure or mixed) qubit state can be prepared deterministically at a remote location. We experimentally demonstrate the scheme by remotely preparing

12

pure states and

6

mixed states. The fidelities between the desired and achieved states are all higher than

0.99

and have an average of

0.9947

.