Multiple Access Channels

IEEE International Symposium on Information Theory - Proceedings

Current development in programmable analogue quantum simulators (AQS), whose physical implementation can be realised in the near-term compared to those of large-scale digital quantum computers, highlights the need for robust testing techniques in analogue platforms. Methods to properly certify or benchmark AQS should be efficiently scalable, and also provide a way to deal with errors from state preparation and measurement (SPAM). Up to now, attempts to address this combination of requirements have generally relied on model-specific properties. We put forward a new approach, applying a well-known digital noise characterisation technique called randomized benchmarking (RB) to the analogue setting. RB is a scalable experimental technique that provides a measure of the average error-rate of a gate-set on a quantum hardware, incorporating SPAM errors. We present the original form of digital RB, the necessary alterations to translate it to the analogue setting and introduce the analogue randomized benchmarking protocol (ARB). In ARB we measure the average error-rate per time evolution of a family of Hamiltonians and we illustrate this protocol with two case-studies of analogue models; classically simulating the system by incorporating several physically motivated noise scenarios. We find that for the noise models tested, the data fit with the theoretical predictions and we gain values for the average error rate for differing unitary sets. We compare our protocol with other relevant RB methods, where both advantages (physically motivated unitaries) and disadvantages (difficulty in reversing the time-evolution) are discussed.

Randomized Benchmarking in the Analogue Setting

Non-isometric quantum error correction in gravity

Archimedes' hat-box theorem states that uniform measure on a sphere projects to uniform measure on an interval. This fact can be used to derive Simpson's rule. We present various constructions of, and lower bounds for, numerical cubature formulas using moment maps as a generalization of Archimedes' theorem. We realize some well-known cubature formulas on simplices as projections of spherical designs. We combine cubature formulas on simplices and tori to make new formulas on spheres. In particular $S^n$ admits a 7-cubature formula (sometimes a 7-design) with $O(n^4)$ points. We establish a local lower bound on the density of a PI cubature formula on a simplex using the moment map. 
Along the way we establish other quadrature and cubature results of independent interest. For each $t$, we construct a lattice trigonometric $(2t+1)$-cubature formula in $n$ dimensions with $O(n^t)$ points. We derive a variant of the Moller lower bound using vector bundles. And we show that Gaussian quadrature is very sharply locally optimal among positive quadrature formulas.

Numerical cubature from Archimedes' hat-box theorem

Decoupling theorems are an important tool in quantum information theory where they are used as building blocks in a host of information transmission protocols. A decoupling theorem takes a bipartite quantum state shared between a system and a reference, applies some local operation on the system, and under suitable constraints, proves that the resulting state is close to a product state between the output system and the untouched reference. The theorem is said to be non-catalytic if it does not require an additional input of a quantum state, in tensor with the given input state, for the decoupling. Dupuis proved an important non-catalytic decoupling theorem where the operation on the system was a Haar random unitary followed by a fixed superoperator, unifying many decoupling results proved earlier. He also showed a concentration result for his decoupling theorem. In this work we give a new concentration result for non-catalytic decoupling by showing that, for suitably large t, a unitary chosen uniformly at random from an approximate t-design gives rise to a state close to a product state with probability $\exp\sim 1$. Our concentration, though exponential, is less than that of Dupuis but for many important applications it uses less random bits than Dupuis. In particular, we prove that approximate $|A_1|$-designs decouple a quantum system in the Fully Quantum Slepian Wolf (FQSW) theorem wherein a fixed superoperator traces out the subsystem $A_2$ from a system $A_1\otimes A_2$. This leads to a saving in the number of random bits to $O(|A_1| \log (|A_1| |A_2|))$ from $\Omega(|A_1|^2|A_2|^2\log (|A_1||A_2|))$ required by Haar random unitaries. Moreover, if $|A_1|=poly\log(|A_2|)$, efficient constructions of approximate $|A_1|$-designs exist. This further implies that approximate unitary $|A_1|$-designs achieve relative thermalisation in quantum thermodynamics with $exp$ high probability.

A concentration of measure result for non-catalytic decoupling via approximate unitary t-designs

We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard et al. [22]. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-shot setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting

We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation}, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard, Devetak and Hayden. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

Local pure states represent a fundamental resource in quantum information theory. In this work we obtain one-shot achievable bounds on the rates for local purity distillation, in the single-party and in the two-party cases. In both situations, local noisy operations are freely available, while in the two-party case also one-way classical communication can be used. In addition, in both situations local pure ancillas can be borrowed, as long as they are discounted from the final net rate of distillation. The one-shot rates that we obtain, written in terms of mutual information-like quantities, are shown to recover in the limit the asymptotic i.i.d. rates of Devetak [PRA, 2005], up to first order analysis.

One-shot purity distillation with local noisy operations and one-way classical communication

In this work, we prove a novel one-shot multi-sender decoupling theorem generalising Dupuis result. We start off with a multipartite quantum state, say on A1 A2 R, where A1, A2 are treated as the two sender systems and R is the reference system. We apply independent Haar random unitaries in tensor product on A1 and A2 and then send the resulting systems through a quantum channel. We want the channel output B to be almost in tensor with the untouched reference R. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a k-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard, Dein the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.

Aditya Nema

Papers

A concentration of measure result for non-catalytic decoupling via approximate unitary t-designs

Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting

Novel one-shot inner bounds for unassisted fully quantum channels via rate splitting

One-shot purity distillation with local noisy operations and one-way classical communication

One-shot multi-sender decoupling and simultaneous decoding for the quantum MAC.