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arXiv:1008.1128 · Physical Review Letters

Entanglement Cost of Implementing Controlled-Unitary Operations

Akihito Soeda, Peter S. Turner, Mio Murao

Published Oct 25, 2011• 31 citations
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Abstract

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.

DOI
10.1103/physrevlett.107.180501

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It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank rrr, and showed that one of the protocol uses O(r)O(r)O(r) ebits of entanglement and O(r)O(r)O(r) bits of classical communication, while these two types of costs for the other protocol scale as O(rlog⁡r)O(r\log r)O(rlogr) but the actual values are smaller for all r<1100r<1100r<1100. Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively 111 ebit and some value between log⁡29−16/9\log_2 9 - 16/9log2​9−16/9 and log⁡23\log_2 3log2​3 ebits.
We prove, using a method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerical examples of Schmidt rank 2 unitaries, which can be deterministically implemented, by either SEP or LOCC, using an entangled resource of two qutrits with less than one ebit of entanglement.
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We analyze implementations of bipartite unitaries by means of local operations and classical communication (LOCC) assisted by shared entanglement. We employ concepts and techniques developed in the quantum Shannon theory to study an asymptotic scenario, in which two distant parties perform the same bipartite unitary on infinitely many pairs of inputs. We analyze minimum cost of entanglement and classical communication per copy. For two-round LOCC protocols, we derive a single-letter formula for the minimum cost of entanglement and classical communication, under an additional requirement that the error converges to zero faster than 1/n4, where n is the number of input pairs. The formula is given by the “Markovianizing cost” of a tripartite state associated with the unitary, which can be computed by a finite-step algorithm. We also derive a lower bound on the minimum cost of resources, which applies for protocols with arbitrary number of rounds.
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