Robustness of entanglement

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Abstract

In the quest of completely describing entanglement in the general case of a finite number of parties sharing a physical system of finite dimensional Hilbert space a new entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurence of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of binary systems, and analytical bounds for mixed states of binary systems. Specific results are obtained mainly for the qubit-qubit system. As byproducts local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced and arguments for considering convexity a necessary condition of any entanglement magnitude are put forward.

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Entanglement of Formation of an Arbitrary State of Two Qubits

William Wootters (1997)

The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state is defined as the minimum average entanglement of an ensemble of pure states that represents the given mixed state. An earlier paper [Phys. Rev. Lett. 78, 5022 (1997)] conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula to be true for a special class of mixed states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing pure-state decompositions.

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Mixed State Entanglement and Quantum Error Correction

William Wootters, John Smolin, David P. DiVincenzo … (1996)

Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state \(|\xi\rangle\) can be transmitted at some rate Q through a noisy channel \(\chi\) without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state \(\hat{M}(\chi)\) (obtained by sharing halves of EPR pairs through a channel \(\chi\)) yields a QECC on \(\chi\) with rate \(Q=D\), and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts \(D_1(M)\) and \(D_2(M)\) that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for \(E(M)\) when \(M\) is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic \(Q\) (or \(D\)) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.