By A. Harrow
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Extra resources for Appl of Coherent Communication to Quantum Information Theory [thesis]
However, in the case without feedback, the best we can do is a tradeoff curve between cbits and rbits, with Eq. 30) representing the case of unlimited randomness consumption. The full tradeoff will be given by an RI of the following form a [c → c] + b [c c] ≥ N : ρXA′ where (a, b) range over some convex set CR(N ). It can be shown[Wyn75, BW05] that (a, b) ∈ CR(N ) iff there exist channels N 1 : XA′ → WC ′ , N 2 : WC ′ → YB such that N = N 2 ◦ N 1 and a ≥ I(XA ; WC )ω , b ≥ I(XA YB ; WC )ω , where WC ′ →WC ′ WC ω XA WC YB := N 2 ◦ ∆ XA′ →XA′ XA ◦ N1 ◦ ∆ (ρXA′ ).
Another difficulty with resources is that relative resources are only guaranteed to work properly when given the right sort of input state. Here we show that using shared randomness, some of the standard relative resources can be “absolutized,” removing the restriction to a particular input state. CHAPTER 1. QUANTUM SHANNON THEORY 33 Nl ... N2 N1 m+l−1 ... Nl ... ... Nl N2 N1 N2 N1 m ... Nl ... l N2 N1 Nl ... ... Nl N2 N1 2 N2 N1 1 1 ... 2 m Figure 1-1: The sliding protocol. We would like to simulate P, which uses N1 , .
This theorem was proven in [Win04], and is a generalization ′ of the CRST. 34) where ′ σ RAXB = T(ψ RA ) and |ψ ψ|RXA ⊇ ρXA . Moreover, given a modified remote instrument which also provides Alice with a copy of the instrument output, XB →XA XB ◦ T, T′ = ∆ the RI still holds: ′ I(R; XB )σ [c → c] + H(XB |R)σ [c c] ≥ T′ : ρA . 35) Only this latter RI is known to be optimal (up to a trivial substitution of [c → c] for [c c]); indeed ′ a[c → c] + b[c c] ≥ T′ : ρA . 36) iff a ≥ I(R; XB )σ and a + b ≥ H(XB )σ .
Appl of Coherent Communication to Quantum Information Theory [thesis] by A. Harrow