We consider the problem of bounded-error quantum state identification: given either state α_0 or state α_1, we are required to output `0', `1' or `?' ("don't know"), such that conditioned on outputting `0' or `1', our guess is correct with high probability. The goal is to maximize the probability of not outputting `?'. We prove a direct product theorem: if we're given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is O(ab). Our proof is based on semidefinite programming duality and may be of wider interest.
Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. Both are shown in the strongest possible sense. First, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared randomness, but needs Omega(n^{1/3}) communication if the parties don't share randomness, even if communication is quantum. This shows the optimality of Yao's recent exponential simulation of shared-randomness protocols by quantum protocols without shared randomness. Second, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared entanglement, but needs Omega((n/log n)^{1/3}) communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys you much more than quantum communication does.
Hankel Submatrices with Bounded Word Length: Dimension Tradeoffs in Probabilistic and Quantum Automata
Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Arbitrary Polynomial Separations in Trainable Quantum Machine Learning
Institute for Quantum Information and Matter（量子信息与物质研究所）；Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, CA, USA（沃尔特·伯克理论物理研究所，卡内基梅隆大学，帕萨迪纳，CA，USA）；MIT Center for Theoretical Physics, Cambridge, MA, USA（麻省理工学院理论物理中心，剑桥，MA，USA）；Department of Physics, CU Boulder, Boulder, CO, USA（科罗拉多大学博尔德分校物理系）
