text: It is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p-norms for any p > 1. In this work, we show that a weaker variant of multiplicativity nevertheless holds for these channels. For any constant p > 1, given a random quantum channel ( N ) (i.e. a channel whose Stinespring representation corresponds to a random subspace S), we show that with high probability the maximum output p-norm of ( N ^ n ) decays exponentially with n. The proof is based on relaxing the maximum output ∞-norm of ( N ) to the operator norm of the partial transpose of the projector onto S, then calculating upper bounds on this quantity using ideas from random matrix theory.
  @cite_5: We study the partial transposition W Γ=(id⊗t)W∈M dn (ℂ) of a Wishart matrix W∈M dn (ℂ) of parameters (dn,dm). Our main result is that, with d→∞, the law of mW Γ is a free difference of free Poisson laws of parameters m(n±1) 2. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half-line.
  @cite_22: Let W be a Wishart random matrix of size d^2 times d^2, considered as a block matrix with d times d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d tends to infinity. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on C^d tensor C^d obtained after partial tracing a random pure state over some ancilla of dimension alpha.d^2 is typically PPT when alpha>4 and typically non-PPT when alpha<4.
summary: As well as the final bound we obtain being of a similar order to that of Aubrun, the reader familiar with @cite_22 will notice that we need to prove some analogous combinatorial lemmas. However, it does not seem clear that the results given here could be obtained as a formal consequence of @cite_22 , or indeed vice versa; it would be interesting to determine whether this is the case. We also note two minor technical differences between this work and @cite_22 : Aubrun's result is only stated for @math (some recent work by Banica and Nechita removes this restriction @cite_5 ), and only for @math constant (so @math grows as a constant fraction of @math ). By contrast, here @math , @math and @math can be essentially arbitrary, although the bound we obtain becomes trivial if @math is too large as a fraction of @math .