Channels as Views of Quantum Systems

We end this chapter by replacing the conditional expectations in GCMI by quantum channels in specific circumstances in which it remains well-defined, positive, and monotonic. Inspired by the duality between the Heisenberg and Schr¨odinger pictures of quantum mechanics, we may think of degradations to a quantum state instead as degradations to an observer’s access to that state. For some classes of state-modifying channels, we can modify the conditional expectations instead of the state itself. This yields an SSA-type inequality and generalized mutual information for pairs of channels.

Theorem 3.5. Let S, T ⊆ S ∨ T ⊆ form a commuting square so that S ∨ T is a factor. Let ρS∨T be a density, and Φ : S1(S ∨ T ) → S1(S ∨ T ) and Ψ : S1(S ∨ T ) → S1(S ∨ T ) be quantum channels such that:

1. [ES, Φ] = [ET, Ψ] = [EST, Φ] = [ES∨T, Ψ] = 0.

2. ETΦ = ΦET = ET, and ESΨ = ΨES = ES.

Figure 3.2: A physical system’s mixed state corresponds to density ρ. A typical observer interacts with the system through layers of equipment and perception, which may impart locality, noise, basis restrictions, and other limitations. To account for imperfect access, we model the observer’s perceived state by Φ(ρ). Volt- meter image CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=41945. Photon Source Image by Alyssa Haroldsen - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=32775941.

Let ΦS= Φ ◦ ES, and ΨT = Ψ ◦ T . Then

I(ΦS : ΨT)ρ≡ H(ΦS(ρ)) + H(ΨT(ρ)) − H(Φ ◦ Ψ(ES∨T(ρ))) − H(ES∩T(ρ))

≥ −2 log(F (Φ ◦ Ψ(ρ), RΦS(ρ),ET ◦ ΨT(ρ))) ≥ 0.

(3.32)

I(ΦS : ΨT)ρ is monotonic under the transformations Φ → Θ ◦ Φ and Ψ → Γ ◦ Ψ such that Θ ◦ Φ and Γ ◦ Ψ

still obey conditions 1-3. In particular,

I(ΦS : ΨT)ρ− I(Θ ◦ ΦS : ΨT)ρ≥ −2 log(F (ES∨T(Φ ◦ Ψ(ρ)), RES(Φ◦Ψ(ρ)),Θ◦ Θ(ES∨T(Φ ◦ Ψ(ρ))))), (3.33)

and similarly for Ψ → Γ ◦ Ψ, where R is a universal recovery map in the sense of [57, 56].

Remark 3.1. Let Φ and Ψ be any pair of quantum channels. We may write Φ = Φ ◦ ES, and Ψ = Ψ ◦ ET,

where ES and ET are the conditional expectations onto the respective injective envelopes [83] of the output

spaces of Φ† and Ψ†. The injective envelope gives the smallest such subalgebras. We may then check if Φ, Ψ, S, and T satisfy the conditions of Theorem 3.5. In general, we will get the strongest results from minimal S and T .

Remark 3.2. We may extend Theorem 3.5 by applying another channel Ω : S1(M ) → S1(M ) to ρ before

all others. This simply replaces ρ → Ω(ρ).

Example 3.1. (Uncertainty Relation with Memory and Noise) Let XAand ZAbe a pair of mutually

unbiased bases with shift generators XA and ZA, such that XAZA = eiθZAXA. Let Φ(ρ) = (1 − p)ρ +

see that Φ ◦ Ψ(ρ) =((1 − p)(1 − q)ρ + p(1 − q)XAρXA +(1 − p)qZAρZA+ pqZAXAρXAZA  = Ψ ◦ Φ(ρ). (3.34)

By substituting p or q with 1/2, [EZA, Φ] = [EXA, Ψ] = 0. We also have that EXAΦ = ΦEXA = EXA, and

EZAΨ = ΨEZA = EZA. Hence Φ and Ψ satisfy the conditions of Theorem 3.5. Let B be an auxiliary or

memory system, and ρAB _{be the full system-memory state. Then}

H(ΦZA|B)ρ+ H(ΨXB|B)ρ− H(Φ ◦ Ψ(ρ))

≥ log |A| − 2 log F (Φ ◦ Ψ(ρ), RΦ◦E_ZA(ρ),E_XA ◦ Ψ ◦ EXA(ρ))

(3.35)

via the Petz recovery map, where H(ΦZA|B)ρ = H(ΦZA ⊗ ˆ1

B_{(ρ)) − H(B)}

ρ. Physically, Φ is a dephasing

in the X basis, which appears as random shift noise in the Z basis but has no effect on an already fully X -dephased state or X -basis measurement. Hence Φ and Ψ each play the role of partially dephasing noise applied to the Z and X bases respectively.

While I(ΦS : ΨT)ρ is a very general positive and monotonic bipartite information, it requires many

assumptions on Φ and Ψ, including that they factor through subalgebras and commute. In [24], Gao, Junge and I derive inequalities on channels without these assumptions. In general, the absence of a commuting square implies the existence of at least some cases with non-positive GCMI. While [24] is more geared toward proving the most general inequality possible, this section focuses on the cases for which we still have a potential operational interpretation in terms of bipartite, individual operations. It naturally leads to Chapter 4, which focuses on the operational interpretations of subalgebraic mutual information. It is through the operational picture that we will see why Theorem 3.5 must hold (see Remark 4.1), so we forgo the proof for now.

3.4.1

What Makes Conditional Expectations Special?

Ultimately, the proof that I(S : T )ρ≥ 0 and is monotonic under a reasonable set of operations (as discussed

in Chapter 4) comes down to the fact that

which follows from the commuting square condition and Lemma 3.5. After that, we may apply data pro- cessing either to relate the two terms, or to the first term in either, such as when ρ undergoes a channel Φ that commutes with ES and is absorbed by ET. A key consequence of Lemma 3.5 is that for any conditional

expectation E ,

H(E (ρ)) − H(ρ) = D(ρkE (ρ)) .

In words, the entropy difference between ρ and E (ρ) is equal to the relative entropy. In general,

H(Φ(ρ)) − H(ρ) 6= D(ρkΦ(ρ))

for an arbitrary channel Φ, and we must require the input and output spaces of Φ to be the same for the latter to even make sense. Intuitively, a conditional expectation is a special degradation of a quantum state that introduces no rotation of bases, so it naturally guarantees that ρ and E (ρ) remain comparable. A mathematical notion of “rotation free” might be that Φ = Φ† up to normalization under the inner product with respect to the partial trace, which we might interpret further as Φ = Rˆ_1/d,Φ, where Rˆ_1/d,Φ is the Petz

map. A conditional expectation is always its own Petz map. While necessary, this condition is not entirely sufficient. The depolarizing channel (see for instance Equation (7.9)) is its own Petz recovery map, but it does not obey the chain rule of relative entropy. We require idempotence as well as self-adjointness, which imposes that the channel be a projector. To fully obtain the chain expansion as in Lemma 3.5, we actually need that the channel restrict to a subalgebra and be a conditional expectation.

In the course of this project, Junge and I spent substantial time trying to generalize to a meaningful notion of I(Φ : Ψ) for an arbitrary pair of channels Φ and Ψ. A major issue we encountered is the ambiguity in what would constitute a meaningful notion of an intersection or a union between the outputs of two channels. When the channels are conditional expectations to subalgebras given explicitly by generators in a larger joint algebra, it is clear at least in finite dimension how to compute the joint algebra as that generated by the union of generators, and an intersection algebra via a literal intersection of elements. A particular hint came from the equivalence to commutant algebras as in Theorem 3.4, in which it was necessary to use this minimal joint system (not a larger algebra) and maximal intersection (not a smaller algebra). Another hint appeared in deriving the results of Chapter 4, where choosing e.g. a larger-than-necessary joint algebra may break monotonicity of some natural operations. Hence we may guess that if there is a reasonable generalization to other channels, we would expect the joint system to be minimal in some sense, the intersection to be maximal, and that there is some notion of complementarity under which the complements of these would exchange roles.

One exception to the restriction that we consider conditional expectations is the mutual information, as

I(A : B)ρ= D(ρkρA⊗ ρB) = H(ρA⊗ ρB) − H(ρAB) (3.36)

for a bipartite density ρAB _{on A ⊗ B. While there is no conditional expectation, and in fact no channel}

mapping ρAB _{→ ρ}A_{⊗ ρ}B_{, the mutual information still reduces to a single entropy difference. The product}

states are not even a convex set. This may hint that there is a generalization beyond subalgebras that does not rely on quantum channels. This is a potential area of future research.