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arXiv:quant-ph/0110011 · IEEE Conference on Computational Complexity (CCC)

Extracting quantum entanglement (general entanglement purification protocols)

Andris Ambainis, Adam Smith, Ke Yang

Published Jun 25, 2003• 11 citations
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Abstract

We study the problem of extracting Einstein-Podolsky-Rosen (EPR) pairs from a general source of entanglement. Suppose Alice and Bob share a bipartite state /spl rho/ which is "reasonably close" to perfect EPR pairs. The only information Alice and Bob possess is a lower bound on the fidelity of /spl rho/ and a maximally entangled state. They wish to "purify" /spl rho/ using local operations and classical communication, and output a state that is arbitrarily close to EPR pairs. We prove that, on average, Alice and Bob cannot increase the fidelity of the input state significantly. On the other hand, there exist protocols that may fail with a small probability, and otherwise will output states arbitrarily close to EPR pairs with very high probability. These protocols come from the "purity-testing protocols" of H. Barnum et al. (2001).

DOI
10.1109/ccc.2002.1004345

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We study the problem of extracting Einstein-Podolsky-Rosen (EPR) pairs from a general source of entanglement. Suppose Alice and Bob share a bipartite state /spl rho/ which is "reasonably close" to perfect EPR pairs. The only information Alice and Bob possess is a lower bound on the fidelity of /spl rho/ and a maximally entangled state. They wish to "purify" /spl rho/ using local operations and classical communication, and output a state that is arbitrarily close to EPR pairs. We prove that, on average, Alice and Bob cannot increase the fidelity of the input state significantly. On the other hand, there exist protocols that may fail with a small probability, and otherwise will output states arbitrarily close to EPR pairs with very high probability. These protocols come from the "purity-testing protocols" of H. Barnum et al. (2001).
We study the problem of extracting EPR pairs from a general source of entanglement. Suppose Alice and Bob share a bipartite state R which is reasonably to perfect EPR pairs. The only information Alice and Bob possess is a lower bound on the fidelity of R and a maximally entangled state. They wish to purify R using local operations and classical communication and output a state that is arbitrarily close to EPR pairs. We prove that on average, Alice and Bob cannot increase the fidelity of the input state significantly. On the other hand, there exist that may fail with a small probability, and otherwise will output states arbitrarily close to EPR pairs with very high probability. These come from the purity-testing protocols of Barnum et al.
Bipartite entanglement purification is the conversion of copies of weakly entangled pairs shared between two separated parties into a smaller number of strongly entangled shared pairs using only local operations and classical communication. Choosing between different entanglement purification protocols generally involves weighing up a trade-off between the ratio of strongly entangled pairs produced to weakly entangled pairs consumed, which we call the rate of the protocol, and the degree of the entanglement of the strongly entangled pairs, typically measured by the fidelity of those pairs to maximally entangled states. By randomly choosing a protocol according to a probability distribution over a list of protocols for each pair we want to produce, we can achieve rates and fidelities not achieved by any of the original protocols. Here, we show how to choose this distribution to maximize the rate at which we produce qubit pairs with a given fidelity to a Bell state or, equivalently, to maximize the fidelity to a Bell state of the qubit pairs produced at a given rate. We investigate both the asymptotic case, where the number of initial pairs goes to infinity, and the finite-size regime, where protocols are restricted to a finite number of weakly entangled pairs.
Quantum entanglement distillation protocols are LOCC protocols between Alice and Bob that convert imperfect EPR pairs, or, in general, partially entangled bipartite states into perfect or near-perfect EPR pairs. The classical communication complexity of these protocols is the minimal amount of classical communication needed for the conversion. In this paper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., where the inputs are mixed states and/or prepared adversarially. We study 3 models of imperfect EPR pairs. In the measure-r model, r out of n EPR pairs are measured by an adversary; in the depolarization model, Bob's share of qubits underwent a depolarization channel; in the fidelity model, the only information Alice and Bob possess is the fidelity of the shared state. For the measure-r model and the depolarization model, we prove tight and almost-tight bounds on the outcome of LOCC protocols that don't use communication. For the fidelity model, we prove a lower bound on the communication complexity that matches the upper bound given by Ambainis, Smith, and Yang [ASY02].
In order to perform faithfully quantum communication protocols such as quantum teleportation, superdense coding, quantum key distribution, and quantum secret sharing, we need the entanglement purification process to make a maximal entanglement out of many copies of the partially entangled pairs by using local operations and classical communication. In this dissertation,\nwe analyze the efficiency of several entanglement purification protocols for qubit states and qudit states, and present a new efficient entanglement purification protocol. Furthermore, we show that our method can be applied to the existing best protocol\nso as to improve the efficiency.
Quantum entanglement distillation protocols are LOCC protocols that, over ``imperfect'' EPR pairs shared between Alice and Bob as input, output perfect or near-perfect EPR pairs. In this paper, we study the communication complexity of these protocols, that is, the minimal number of (classical) bits needed for Alice and Bob. To our knowledge, it is the first paper that studies entanglemend distillation protocols in the resource-bounded case. We consider 3 models for imperfect EPR pairs. In the measure-rrr model, rrr out of nnn EPR pairs are measured by an adversary (and thus become ``corrupted''); in the depolarization model, the qubits of Bob underwent a depolarization channel; in the fidelity model, the only information Alice and Bob possess is the fidelity of the shared state. For the first 2 models, we prove tight and almost-tight bounds on the base case, i.e., how well Alice and Bob can do if they are not allowed to communicate. For the fidelity model, we prove a very tight lower bound (up to an additive constant) on the communication complexity. This lower bound implies the optimality of the Random Hash protocol in [ASY02].