Dave Touchette

Smooth Entropy Bounds on One-Shot Quantum State Redistribution

In quantum state redistribution as introduced by Luo and Devetak and Devetak and Yard, there are four systems of interest: the A system held by Alice; the B system held by Bob; the C system that is to be transmitted from Alice to Bob; and the R system that holds a purification of the state in the ABC registers. We give upper and lower bounds on the amount of quantum communication and entanglement required to perform the task of quantum state redistribution in a one-shot setting. Our bounds are in terms of the smooth conditional minand max-entropy, and the smooth max-information. The protocol for the upper bound has a clear structure, building on the work of Oppenheim: it decomposes the quantum state redistribution task into two simpler coherent state merging tasks by introducing a coherent relay. In the independent and identical (i.i.d.) asymptotic limit our bounds for the quantum communication cost converge to the quantum conditional mutual information I(C; R|B), and our bounds for the total cost converge to the conditional entropy H(C|B). This yields an alternative proof of optimality of these rates for quantum state redistribution in the i.i.d. asymptotic limit. In particular, we obtain a strong converse for quantum state redistribution, which even holds when allowing for feedback.

Near-Optimal Bounds on Bounded-Round Quantum Communication Complexity of Disjointness

We prove a near optimal round-communication trade off for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of a#x03A9;(n/r) on the communication required for computing disjointness of input size n, which is optimal up to logarithmic factors. The previous best lower bound was a#x03A9;(n/r2) due to Jain, Radha krishnan and Sen. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any boolean function f is at most 2O(QIC(f)), where QIC(f) is the prior-free quantum information complexity of f (with error 1/3).

The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting

In the context of two-party interactive quantum communication protocols, we study a recently defined notion of quantum information cost (QIC), which possesses most of the important properties of its classical analogue. Although this definition has the advantage to be valid for fully quantum inputs and tasks, its interpretation for classical tasks remained rather obscure. Also, the link between this new notion and other notions of information cost for quantum protocols that had previously appeared in the literature was not clear, if existent at all. We settle both these issues: for quantum communication with classical inputs, we provide an alternate characterization of QIC in terms of information about the input registers, avoiding any reference to the notion of a purification of the classical input state. We provide an exact operational interpretation of this alternative characterization as the sum of the cost of transmitting information about the classical inputs and the cost of forgetting information about these inputs. To obtain this characterization, we prove a general lemma, the Information Flow Lemma, assessing exactly the transfer of information in general interactive quantum processes. Furthermore, we clarify the link between QIC and IC of classical protocols by simulating quantumly classical protocols. Finally, we apply these concepts to argue that any quantum protocol that does not forget information solves Disjointness on n-bits in Omega (n) communication, completely losing the quadratic quantum speedup. This provides a specific sense in which forgetting information is a necessary feature of interactive quantum protocols. We also apply these concepts to prove that QIC at zero-error is exactly n for the Inner Product function, and n (1 - o(1)) for a random Boolean function on n+n bits.

Exponential separation of quantum communication and classical information

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha. As another application of the techniques that we develop, a simple proof for an optimal trade-off between Alices and Bobs communication is given, even when allowing pre-shared entanglement, while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/2O (b) bits to Bob. We also present a classical protocol achieving this bound.

Capacity approaching coding for low noise interactive quantum communication

We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with n messages, designed for noiseless qudit channels (where d is arbitrary), our main result is a simulation method that fails with probability less than 2−Θ (nє) and uses a qudit channel n (1 + Θ (√є)) times, of which an є fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the √є term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Perhaps surprisingly, this outperforms the best known overhead of 1 + O(√є loglog1/є) in the corresponding classical model, which is also conjectured to be optimal [Haeupler, FOCS’14]. Our work also improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCS’14] for low є.