By A. Harrow
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Extra info for Appl of Coherent Communication to Quantum Information Theory [thesis]
This formulation ensures that we work with well-defined resources instead of using the natural-seeming, but incorrect idXA →XB : ρXA B (which violates Eqns. 11)). Of course, with no extra resource cost Alice could keep a copy of XA . Instrument compression theorem. 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 .
2. Immediate from the definitions. 3. Let k = ⌊zm⌋ and k ′ = ⌊wm⌋, where m is a parameter we will choose later. For any δ and sufficiently large n (depending on δ and m), ∗ α⌊zn(1+2δ)⌋ ≥ (α⌊⌊zn(1+δ)⌋/k⌋ )⊗k , ∗ ′ α⌊wn(1+2δ)⌋ ≥ (α⌊⌊wn(1+δ)⌋/k′ ⌋ )⊗k , ∗ ′ α⌊(z+w)n(1+2δ)⌋ ≥ (α⌊⌊(z+w)n(1+δ)⌋/(k+k′ )⌋ )⊗(k+k ) , ∗ (α⌊⌊zn(1−δ)⌋/k⌋ )⊗k ≥ α⌊zn(1−2δ)⌋ , ′ ∗ (α⌊⌊wn(1−δ)⌋/k′ ⌋ )⊗k ≥ α⌊wn(1−2δ)⌋ , ′ ∗ (α⌊⌊(z+w)n(1−δ)⌋/(k+k′ )⌋ )⊗(k+k ) ≥ α⌊(z+w)n(1−2δ)⌋ . Observe: |zn − kn/m| ≤ |⌊zn⌋ − ⌊kn/m⌋| ≤ |⌊zn⌋ − k⌊n/m⌋| ≤ |⌊⌊zn⌋/k⌋ − ⌊n/m⌋| ≤ n/m n/m + 1 n/m + k + 2 n/(km) + 2 + 2/k.
For example, if we were to explore unitary and/or bidirectional resources more carefully, then we would need to reexamine our treatments of depth and of relative resources. 16 we (1) always simulate the depth-1 version of the output resource, (2) are allowed to use a depth-k version of the input resource where k depends only on the target inefficiency and not the target error. 10). However, it is possible that a different model of resources would allow protocols which deal with depth differently.
Appl of Coherent Communication to Quantum Information Theory [thesis] by A. Harrow