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arXiv:1005.5236 · Journal of Physics A Mathematical and Theoretical

Entanglement cost of two-qubit orthogonal measurements

Somshubhro Bandyopadhyay, Ramij Rahaman, William K. Wootters

Published Oct 19, 2010• 34 citations
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Abstract

The ‘entanglement cost’ of a bipartite measurement is the amount of shared entanglement two participants need to use up in order to carry out the given measurement by means of local operations and classical communication. We numerically investigate the entanglement cost of generic orthogonal measurements on two qubits. Our results strongly suggest that for almost all measurements of this kind, the entanglement cost is strictly greater than the average entanglement of the eigenstates associated with the measurements, implying that the nonseparability of a two-qubit orthogonal measurement is generically distinct from the nonseparability of its eigenstates.

DOI
10.1088/1751-8113/43/45/455303

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The ‘entanglement cost’ of a bipartite measurement is the amount of shared entanglement two participants need to use up in order to carry out the given measurement by means of local operations and classical communication. We numerically investigate the entanglement cost of generic orthogonal measurements on two qubits. Our results strongly suggest that for almost all measurements of this kind, the entanglement cost is strictly greater than the average entanglement of the eigenstates associated with the measurements, implying that the nonseparability of a two-qubit orthogonal measurement is generically distinct from the nonseparability of its eigenstates.
For certain joint measurements on a pair of spatially separated particles, we ask how much entanglement is needed to carry out the measurement exactly. For a class of orthogonal measurements on two qubits with partially entangled eigenstates, we present upper and lower bounds on the entanglement cost. The upper bound is based on a recent result by Berry [Phys. Rev. A 75, 032349 (2007)]. The lower bound, based on the entanglement production capacity of the measurement, implies that for almost all measurements in the class we consider, the entanglement required to perform the measurement is strictly greater than the average entanglement of its eigenstates. On the other hand, we show that for any complete measurement in $d\ifmmode\times\else\texttimes\fi{}d$ dimensions that is invariant under all local generalized Pauli operations, the cost of the measurement is exactly equal to the average entanglement of the states associated with the outcomes.
We analyze the amount of classical communication required to reproduce the statistics of local projective measurements on a general pair of entangled qubits, |ΨAB⟩=p |00⟩+1−p |11⟩ (with 1/2≤p≤1 ). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when 2p(1−p)2p−1log⁡(p1−p)+2(1−p)≤1 , approximately 0.835≤p≤1 , we present a classical protocol that requires only a single bit of communication. The latter model even allows a perfect classical simulation with an average communication cost that approaches zero in the limit where the degree of entanglement approaches zero ( p→1 ). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.
We investigate the minimum entanglement cost of the deterministic implementation of two-qubit controlled-unitary operations using local operations and classical communication (LOCC). We show that any such operation can be implemented by a three-turn LOCC protocol, which requires at least 1 ebit of entanglement when the resource is given by a bipartite entangled state with Schmidt number 2. Our result implies that there is a gap between the minimum entanglement cost and the entangling power of controlled-unitary operations. This gap arises due to the requirement of implementing the operations while oblivious to the identity of the inputs.
We quantify the one-shot entanglement cost of an arbitrary bipartite state, that is, the minimum number of singlets needed by two distant parties to create a single copy of the state up to a finite accuracy, by using local operations and classical communication only. This analysis, in contrast to the traditional one, pertains to scenarios of practical relevance, in which resources are finite and transformations can be achieved only approximately. Moreover, it unveils a fundamental relation between two well-known entanglement measures, namely, the Schmidt number and the entanglement of formation. Using this relation, we are able to recover the usual expression of the entanglement cost as a special case.
We investigate the amount of communication that must augment classical local hidden variable models in order to simulate the behavior of entangled quantum systems. We consider the scenario where a bipartite measurement is given from a set of possibilities and the goal is to obtain exactly the same correlations that arise when the actual quantum system is measured. We show that, in the case of a single pair of qubits in a Bell state, a constant number of bits of communication is always sufficient---regardless of the number of measurements under consideration. We also show that, in the case of a system of nnn Bell states, a constant times 2n{2}^{n}2n bits of communication is necessary.
We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state.