Fully Quantum Arbitrarily Varying Channels: Random Coding Capacity and Capacity Dichotomy

We consider a model of communication via a fully quantum jammer channel with quantum jammer, quantum sender and quantum receiver, which we dub quantum arbitrarily varying channel (QAVC). Restricting to finite dimensional user and jammer systems, we show, using permutation symmetry and a de Finetti reduction, how the random coding capacity (classical and quantum) of the QAVC is reduced to the capacity of a naturally associated compound channel, which is obtained by restricting the jammer to i.i.d. input states. Furthermore, we demonstrate that the shared randomness required is at most logarithmic in the block length, using a random matrix tail bound. This implies a dichotomy theorem: either the classical capacity of the QAVC is zero, and then also the quantum capacity is zero, or each capacity equals its random coding variant. I. FULLY QUANTUM AVC AND RANDOM CODES We consider a simple, fully quantum model of arbitrarily varying channel (QAVC). Namely, we have three agents, Alice (sender), Bob (receiver) and Jamie (jammer), each controlling a quantum system A, B and J , respectively. The channel is simply a completely positive and trace preserving (cptp) map N : L(A⊗J) −→ L(B), and we assume it to be memoryless on blocks of length l, i.e. N : L(A⊗J) −→ L(B), with A = A ⊗ · · · ⊗ A (l times), etc. However, crucially, neither Alice’s nor Jamie’s input states need to be tensor product or even separable states. We shall assume throughout that all three Hilbert spaces A, B and J have finite dimension, |A|, |B|, |J | < ∞. The previously introduced AVQC model of Ahlswede and Blinovsky [5], and more generally Ahlswede et al. [4], is obtained by channels N that first dephase the input J in a fixed basis, so that the choices of the jammer are effectively reduced to basis states |j〉〈j| of J and their convex combinations. Note that this generalises the classical AVC, which is simply a channel with input alphabet X×S and output alphabet Y , given by transition probabilities N(y|x, s), and such a channel can always be interpreted as a cptp map. This model has been considered in [19], [20], however in those works principally from the point of view that Jamie is helping Alice and Bob, passively, by providing a suitable input state to J . Contrary to the classical AVC and the AVQC considered in [5], [4], where the jammer effectively always selects a tensor product channel between Alice and Bob, the fact that we allow general quantum inputs on J, including entangled states, permits Jamie to induce non-classical correlations between the different channel systems. These correlations, as was observed in [19], [20], are not only highly nontrivial, but can also have a profound impact on the communication capacity of the channel between Alice and Bob. In the present context, however, Jamie is fundamentally an adversary. Define a (deterministic) classical code forN of block length l as a collection C = {(ρm, Dm) : m = 1, . . . ,M} of states ρm ∈ S(A) and POVM elements Dm ≥ 0 acting on B, such that ∑M m=1 Dm = 1 . Its rate is defined as 1 l logM , the number of bits encoded per channel use. Its error probability is defined as the average over uniformly distributed messages and with respect to a state σ on J: Perr(C, σ) := 1 M M ∑ m=1 Tr ( N(ρm ⊗ σ) ) (1 −Dm). For the transmission of quantum information, define a (deterministic) quantum code for N of block length l as a pair Q = (E ,D) of cptp maps E : L(C) −→ L(A) and D : L(B) −→ L(C). Its rate is 1 l logL, the number of qubits encoded per channel use, and the error is quantified, with respect to a state σ on J, as the “infidelity” F̂ (Q, σ) := 1− Tr ( (id⊗D ◦ N σ ◦ E)ΦL ) ·ΦL, with the maximally entangled state ΦL = 1 L ∑ ij |ii〉〈jj|. Here, we have introduced the channelsNσ : L(A) −→ L(B) defined by fixing the jammer’s state to σ, Nσ(ρ) := N (ρ⊗ σ). Note that we use the language of “deterministic” code, although in quantum information this is indistinguishable from stochastic encoders; it is meant to differentiate from “random” codes, which use shared correlation: A random classical [quantum] code for N of block length l consists of a random variable λ with a well-defined distribution and a family of deterministic codes Cλ [Qλ]. The error probability if (Cλ), always with respect to a state σ on J, is simply the expectation over λ, i.e. EλPerr(Cλ, σ). The error of the random quantum code is similarly EλF̂ (Qλ, σ). The operational interpretation of the random code model is that Alice and Bob share knowledge of the random variable λ, and use Cλ accordingly, but that Jamie is ignorant of it. This shared randomness is thus a valuable resource, which for random codes is considered freely available, whose amount, however, we would like to control at the same time. The capacities associated to these code concepts are defined as usual, as the maximum achievable rate as block length goes to infinity and the error goes to zero: Cdet(N ) := lim sup l→∞ 1 l logM s.t. sup σ Perr(C, σ) → 0, Crand(N ) := lim sup l→∞ 1 l logM s.t. sup σ EλPerr(Cλ, σ) → 0, Qdet(N ) := lim sup l→∞ 1 l logL s.t. sup σ F̂ (Q, σ) → 0, Qrand(N ) := lim sup l→∞ 1 l logL s.t. sup σ EλF̂ (Q, σ) → 0. If in the above error maximisations Jamie is restricted to tensor power states σ, the QAVC model becomes a compound channel: N σ⊗l = (Nσ), σ ∈ S(J). Its classical and quantum capacities are denoted C({Nσ}σ) and Q({Nσ}σ), respectively. II. RANDOM CODING CAPACITIES: FROM QAVC TO ITS COMPOUND CHANNEL By definition, (see also [7], [8] and [4]) Cdet(N ) ≤ Crand(N ) ≤ C({Nσ}σ), and Qdet(N ) ≤ Qrand(N ) ≤ Q({Nσ}σ). (1) Here, we show that for the random capacity, the rightmost inequalities are identities, by proving bounds in the opposite direction. For the quantum capacity, this was done in [19, Appendix A]. To present the argument, define the permutation operator U acting on the tensor power A as permuting the subsystems, for a permutation π ∈ Sl: U ( |α1〉|α2〉 · · · |αl〉 ) = |απ−1(1)〉|απ−1(2)〉 · · · |απ−1(l)〉, which extends uniquely by linearity. This is a unitary representation of the symmetric group, which is defined for any Hilbert space. The quantum channel obtained by the conjugation action of U is denoted Uπ(α) = UαU. Proposition 1 Let Q = (E ,D) be a quantum code for the compound channel {Nσ}σ∈S(J) at block length l of size L and with fidelity 1− ǫ, i.e. for all σ ∈ S(J), F̂ ( Q, σ ) = 1− Tr ( (id⊗D ◦ N σ ◦ E)ΦL ) ·ΦL ≤ ǫ. Then, the random quantum code (Qπ)π∈Sl with a uniformly distributed random permutation π of [l], defined by Qπ = (Uπ ◦ E ,D ◦ Uπ−1), has infidelity EπF̂ ( Qπ, σ ) ≤ ǫ ≤ ǫ(l+1) 2 for the QAVC N . Proposition 2 Let C = {(ρm, Dm) : m = 1, . . . ,M} be a code of block length l for the compound channel {Nσ}σ∈S(J) with error probability ǫ, i.e. for all σ ∈ S(J), Perr(C, σ ) = 1 M M ∑ m=1 Tr ( N σ (ρm)(1 −Dm) ) ≤ ǫ. Then, the random code (Cπ)π∈Sl with a uniformly distributed random permutation π of [l], defined by Cπ := {(U ρmU , UDmU ) : m = 1, . . . ,M}, has error probability ǫ ≤ ǫ(l+ 1) 2 for the QAVC N . Proof We only prove Proposition 2, since Proposition 1 has been argued in [19, Appendix A], with analogous proofs. For an arbitrary state ζ on J, the error probability of the random code (Cπ)π∈Sl can be written as EπPerr(Cπ , ζ)