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Boundederror quantum state identification and exponential separations in communication complexity
 In Proc. of the 38th Symposium on Theory of Computing (STOC
, 2006
"... We consider the problem of boundederror quantum state identification: given either state α0 or state α1, 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 o ..."
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We consider the problem of boundederror quantum state identification: given either state α0 or state α1, 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 are 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 boundederror state identification problem is O(ab). Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. 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 Ω(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 sharedrandomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to BarYossef et al., this shows that the quantum SMP model is incomparable with the classical sharedrandomness SMP model. 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 Ω((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.
On Weylcovariant channels
, 2006
"... Formalism of discrete noncommutative Fourier transform is developed and applied to the study of Weylcovariant channels. We then extend a result in [6] concerning a bound of the maximal output 2norm of a Weylcovariant channel. A class of channels which attain the bound is introduced, for which the ..."
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Formalism of discrete noncommutative Fourier transform is developed and applied to the study of Weylcovariant channels. We then extend a result in [6] concerning a bound of the maximal output 2norm of a Weylcovariant channel. A class of channels which attain the bound is introduced, for which the multiplicativity of the maximal output 2norm is proven. Complementary channels are described which share the multiplicativity properties with the Weylcovariant channels. 1 A noncommutative Fourier transform A state of finite quantum system is represented by a positive operator ρ of trace one (density operator) in a Hilbert space H of dimensionality d. The set of density operators in H is denoted S(H). A channel Φ is a completely positive (CP) tracepreserving (TP) map of the algebra B(H) of all operators in H. Although the TP condition is redundant in the context of our results, we shall impose it just for notational convenience. The maximal output pnorm of Φ is defined as where ‖ ‖p is the Schatten pnorm: ‖ρ‖p: = (trρ  p) 1 p. νp(Φ): = sup ‖Φ(ρ)‖p, (1.1) ρ∈S(H) 1 The current multiplicativity conjecture is that νp(Φ ⊗ Ω) = νp(Φ)νp(Ω), (1.2) for arbitrary channels Φ and Ω, and for p ∈ [1,2]. Note that the inequality νp(Φ ⊗ Ω) ≥ νp(Φ)νp(Ω) is straightforward. In this paper we consider the case p = 2, which is still an open problem (see [2],[7],[9] for some general results in this direction). Let us choose an orthonormal basis {ek;k = 0,...,d − 1} in H. Consider the additive cyclic group Zd and define an irreducible projective unitary representation of the group Z = Zd ⊕ Zd in H as z = (x,y) ↦ → Wz = U x V y, where x,y ∈ Zd, and U and V are the unitary operators such that