Quantum Information

Optimally encoding classical information in a quantum system is one of the oldest and most fundamental challenges of quantum information theory. Holevo’s bound places a hard upper limit on such encodings, while the Holevo-Schumacher-Westmoreland (HSW) theorem addresses the question of how many classical messages can be “packed” into a given quantum system. In this article, we use Sen’s recent quantum joint typicality results to prove a one-shot multiparty quantum packing lemma generalizing the HSW theorem. The lemma is designed to be easily applicable in many network communication scenarios. As an illustration, we use it to straightforwardly obtain quantum generalizations of well-known classical coding schemes for the relay channel: multihop, coherent multihop, decode-forward, and partial decode-forward. We provide both finite blocklength and asymptotic results, the latter matching existing formulas. Given the key role of the classical packing lemma in network information theory, our packing lemma should help open the field to direct quantum generalization.

In this talk, I will discuss some interesting connections between Hamiltonian complexity, error correction, and quantum circuits. First, motivated by the Quantum PCP Conjecture, I will describe a construction of a family of local Hamiltonians where the complexity of ground states — even when subject to large amounts of noise — is superpolynomial (under plausible complexity assumptions). The construction is simple, making use of the well-known Feynman-Kitaev circuit Hamiltonian construction.

Next, I will describe how this complexity result can be repurposed to construct local approximate error correcting codes with (nearly) linear distance and (nearly) constant rate. By contrast, it remains an open problem to construct quantum low-density parity check (QLDPC) stabilizer codes that achieve similarly good parameters.

An interesting component of our code construction is the Brueckmann-Terhal spacetime circuit Hamiltonian construction, and a novel analysis of its spectral gap. Our codes are ground spaces of non-commuting  — but frustration-free -- local Hamiltonians. We believe that our results present additional evidence that the study of approximate error correction and non-stabilizer codes may be much richer than previously thought.

This talk will have more questions than answers, and is based on two papers:

·  https://arxiv.org/abs/1802.07419 (joint with Chinmay Nirkhe and Umesh Vazirani)

·  https://arxiv.org/abs/1811.00277 (joint work with Thom Bohdanowicz, Elizabeth Crosson, and Chinmay Nirkhe)

In the usual paradigm of quantum error correction, the information to be protected can be encoded in a system of abstract qubits or modes.  But how does this work for physical information, which cannot be described in this way?  Just as direction information cannot be conveyed using a sequence of words if the parties involved do not share a reference frame, physical quantum information cannot be conveyed using a sequence of qubits or modes without a shared reference frame.  Covariant quantum error correction is a procedure for protecting such physical information against noise in such a way that the encoding and decoding operations transform covariantly with respect to an external symmetry group. In this talk, we'll study covariant QEC, and we will see that there do not exist finite dimensional quantum codes that are covariant with respect to continuous symmetries.  Conversely, we'll see that there do exist finite codes for finite groups, and continuous variable (CV) codes for continuous groups.  This leads to a CV method of circumventing the Eastin-Knill theorem.  By relaxing our requirements to allow for only approximate error correction and covariance, we'll find a fundamental tension between a code's ability to approximately correct errors and covariance with respect to a continuous symmetry.  In this way, we'll arrive at an approximate version of the Eastin-Knill theorem, and we'll end by learning what covariant QEC tells us about continuous symmetries in AdS/CFT, among other applications. 

Von Neuman entropy. Classical data compression. Shannon's source coding theorem. Quantum data compression and Schumacher compression. Shannon's channel compression theorem. Mutual information.