21 research outputs found
Asymptotic Compressibility of Entanglement and Classical Communication in Distributed Quantum Computation
We consider implementations of a bipartite unitary on many pairs of unknown
input states by local operation and classical communication assisted by shared
entanglement. We investigate to what extent the entanglement cost and the
classical communication cost can be compressed by allowing nonzero but
vanishing error in the asymptotic limit of infinite pairs. We show that a lower
bound on the minimal entanglement cost, the forward classical communication
cost, and the backward classical communication cost per pair is given by the
Schmidt strength of the unitary. We also prove that an upper bound on these
three kinds of the cost is given by the amount of randomness that is required
to partially decouple a tripartite quantum state associated with the unitary.
In the proof, we construct a protocol in which quantum state merging is used.
For generalized Clifford operators, we show that the lower bound and the upper
bound coincide. We then apply our result to the problem of distributed
compression of tripartite quantum states, and derive a lower and an upper bound
on the optimal quantum communication rate required therein.Comment: Section II and VIII adde
Exact Exponent for Atypicality of Random Quantum States
We study the properties of the random quantum states induced from the
uniformly random pure states on a bipartite quantum system by taking the
partial trace over the larger subsystem. Most of the previous studies have
adopted a viewpoint of "concentration of measure" and have focused on the
behavior of the states close to the average. In contrast, we investigate the
large deviation regime, where the states may be far from the average. We prove
the following results: First, the probability that the induced random state is
within a given set decreases no slower or faster than exponential in the
dimension of the subsystem traced out. Second, the exponent is equal to the
quantum relative entropy of the maximally mixed state and the given set,
multiplied by the dimension of the remaining subsystem. Third, the total
probability of a given set strongly concentrates around the element closest to
the maximally mixed state, a property that we call conditional concentration.
Along the same line, we also investigate an asymptotic behavior of coherence of
random pure states in a single system with large dimensions.Comment: Minor changes. References added. Comments are welcom