Quantum Capacity #

This focuses on defining and proving theorems about the quantum capacity, the maximum asymptotic rate at which quantum information can be coherently transmitted. The precise definition is not consistent in the literature, see Capacity_doc for a note on what has been used and how that was used to arrive at the following definition:

A channel A Emulates another channel B if there are D and E such that D∘A∘E = B.
A channel A εApproximates channel B (of the same dimensions) if the for every state ρ, the fidelity F(A(ρ), B(ρ)) is at least 1-ε.
A channel A AchievesRate R:ℝ if for every ε>0, n copies of A emulates some channel B such that log2(dimout(B))/n ≥ R, and that B is εApproximately the identity.
The quantumCapacity of the channel A is the supremum of the achievable rates, i.e. sSup { R : ℝ | AchievesRate A R }.

The most basic facts:

emulates_self: Every channel emulates itself.
emulates_trans: If A emulates B and B emulates C, then A emulates C. (That is, emulation is an ordering.)
εApproximates A B ε is equivalent to the existence of some δ (depending ε and dims(A)) so that |A-B| has diamond norm at most δ, and δ→0 as ε→0.
achievesRate_0: Every channel achievesRate 0. So, the set of achievable rates is Nonempty.
If a channel achievesRate R₁, it also every achievesRate R₂ every R₂ ≤ R₁, i.e. it is an interval extending left towards -∞. Achievable rates are ¬BddBelow.
bddAbove_achievesRate: A channel C : dimX → dimY cannot achievesRate R with R > log2(min(dimX, dimY)). Thus, the interval is BddAbove.

The nice lemmas we would want:

Capacity of a replacement channel is zero.
Capacity of an identity channel is log2(D).
Capacity is superadditive under tensor products. (That is, at least additive. Showing that it isn't exactly additive, unlike classical capacity which is additive, is a much harder task.)
Capacity of a kth tensor power is exactly k times the capacity of the original channel.
Capacity does not decrease under tensor sums.
Capacity does not increase under composition.

Then, we should show that our definition is equivalent to some above. Most, except (3), should be not too hard to prove.

Then the LSD theorem establishes that the single-copy coherent information is a lower bound. This is stated in coherentInfo_le_quantumCapacity. The corollary, that the n-copy coherent information converges to the capacity, is quantumCapacity_eq_piProd_coherentInfo.

TODO #

The only notion of "capacity" here currently is "quantum capacity" in the usual sense. But there are several non-equal capacities relevant to quantum channels, see e.g. Watrous's notes for a list:

Quantum capacity (quantumCapacity)
Quantum 1-shot capacity
Entanglement-assisted classical capacity
Qss, the quantum side-channel capacity
Holevo capacity, aka Holevo χ. The Holevo–Schumacher–Westmoreland theorem as a major theorem
Entanglement-assisted Holevo capacity
Entanglement-assisted quantum capacity
One- and two-way distillable entanglement

And other important theorems like superdense coding, nonadditivity, superactivation