By A. Harrow
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Extra resources for Appl of Coherent Communication to Quantum Information Theory [thesis]
XA →XA XB ρXA B + H(XA |B)ρ [c → c] ≥ ρXA B + ∆ ≥ H(XA )ρ [c c]. ≥ ≥ : ρXA B ρXA XB B ρXA XB The first inequality is by classical compression with quantum side information (Eq. 31, part 2, and the fourth by common randomness concentration (Eq. 29)). 5 Discussion This chapter has laid the foundations of a formal approach to quantum Shannon theory in which the basic elements are asymptotic resources and protocols mapping between them. Before presenting applications of this approach in the next three chapters, we pause for a moment to discuss the limitations of our formalism and possible ways it may be extended.
26) Were it not for the o [c → c] term, we would have the equality φAB = H(B)φ [q q]. However, it turns out that the o[c → c] term cannot be avoided[HL04, HW03]. This means that the strongest equality we can state has a sublinear amount of classical communication on both sides: H(B)φ [q q] + o [c → c] = φAB + o [c → c]. 27) Note how Eq. 27) states the converse in a form that is in some ways stronger than Eq. 24), since it implies the transformation is not only optimal, but also asymptotically reversible.
At the other extreme we might consider the case when we are allowed an unlimited amount of some resource, typically when proving converse theorems. 34 (∞ terms). We write α + ∞γ ≥ β if for any δ > 0, there exists k such that for any ǫ > 0 there exists n1 , n2 and a ǫ-valid protocol P satisfying P[(α⌊n1 /k⌋ + γ⌊n2 /k⌋ )×k ] − β⌊(1−δ)n⌋ ≤ ǫ. 3. GENERAL RESOURCE INEQUALITIES This means that we can use an amount of γ that increases arbitrarily quickly with n. Note that ∞γ cannot be defined as a resource, since it violates Eq.
Appl of Coherent Communication to Quantum Information Theory [thesis] by A. Harrow