Quantum Information

Entanglement in quantum many-body systems is typically fragile to interactions with the environment. Generic unital quantum channels, for example, have the maximally mixed state with no entanglement as their unique steady state. However, we find that for a unital quantum channel that is `strongly symmetric', i.e. it preserves a global on-site symmetry, the maximally mixed steady state in certain symmetry sectors can be highly entangled. For a given symmetry, we analyze the entanglement and correlations of the maximally mixed state in the invariant sector (MMIS), and show that the entanglement of formation and distillation are exactly computable and equal for any bipartition. For all Abelian symmetries, the MMIS is separable, and for all non-Abelian symmetries, the MMIS is entangled. Remarkably, for non-Abelian continuous symmetries described by compact semisimple Lie groups (e.g. SU(2)), the bipartite entanglement of formation for the MMIS scales logarithmically ∼logN with the number of qudits N.

Quantum key distribution protocols can be based on quantum error correcting codes, where the structure of the code determines the post processing protocol applied to a raw key produced by BB84 or a similar scheme.  Luo and Devetak showed that basing a similar protocol on entanglement-assisted quantum error-correcting codes (EAQECCs) leads to quantum key expansion (QKE) protocols, where some amount of previously shared secret key is used as a seed in the post-processing stage to produce a larger secret key. One of the promising aspects of EAQECCs is that they can be constructed from classical linear codes that don't satisfy the dual-containing property, which among other things allows the use of low density parity-check (LDPC) codes with girth greater than 4, for which the iterative decoding algorithm has better performance.  We looked into QKE based on a family of EAQECCs generated by classical finite geometry (FG) LDPC codes.  Very efficient iterative decoders exist for these codes, and they were shown by Hsieh, Yen and Hau to produce quantum LDPC codes that require very little entanglement.  We modify the original QKE protocol to detect bad code blocks without the consumption of secret key when the protocol fails.  This allows us to greatly reduce the bit error rate of the key, at the cost of a minor reduction in the net key production rate, but without increasing the consumption rate of pre-shared key.  Numerical simulations for the family of FG LDPC codes show that this improved QKE protocol has a good net key production rate even at relatively high error rates, for appropriate code choices.

In quantum information theory, random techniques have proven to be very useful. For example, many questions related to the problem of the additivity of entropies of quantum channels rely on fine properties of concentration of measure.  In this talk, I will show that very different techniques of random matrix theory can complement quite efficiently more classical random techniques. I will spend some time on discussing the Weingarten calculus approach, and the operator norm approach. Both techniques have been initially used in free probability theory, and I will give some new applications of these techniques to quantum information theory.

In this talk I will sketch a project which aims at the design of systematic and efficient procedures to infer quantum models from measured data. Progress in experimental control have enabled an increasingly fine tuned probing of the quantum nature of matter, e.g., in superconducting qubits. Such experiments have shown that we not always have a good understanding of how to model the experimentally performed measurements via POVMs. It turns out that the ad hoc postulation of POVMs can lead to inconsistencies. For example, when doing asymptotic state tomography via linear inversion, one sometimes recovers density operators which are significantly not positive semidefinite. Assuming the asymptotic regime, we suggest an alternative procedure where we do not make a priori assumptions on the quantum model, i.e., on the Hilbert space dimension, the prepared states or the measured POVMs. In other words, we simultaneously estimate the dimension of the underlying Hilbert space, the quantum states and the POVMs. We are guided by Occam's razor, i.e., we search for the minimal quantum model consistent with the data.

It is widely known in the quantum information community that the states that satisfy strong subadditivity of entropy with equality have the form of quantum Markov chain. Based on a recent strengthening of strong subadditivity of entropy, I will describe how such structure can be exploited in the studies of gapped quantum many-body system. In particular, I will describe a diagrammatic trick to i) give a quantitative statement about the locality of entanglement spectrum ii) perturbatively bound changes of topological entanglement entropy under generic perturbation.

A "one-time program" for a channel C is a hypothetical cryptographic primitive by which a user may evaluate C on only one input state of her choice.  (Think Mission Impossible: "this tape will self-destruct in five seconds.")  One-time programs cannot be achieved without extra assumptions such as secure hardware; it is known that one-time programs can be constructed for classical channels using a very basic hypothetical hardware device called a "one-time memory".   Our main result is the construction of a one-time program for any quantum channel specified by a circuit, assuming the same basic one-time memory devices used for classical channels.  The construction achieves universal composability -- the strongest possible security -- against any quantum adversary.  It employs a technique for computation on authenticated quantum data and we present a new authentication scheme called the "trap" scheme for this purpose.   Finally, we observe that there is a pathological class of channels that admit trivial one-time programs without any hardware assumptions whatsoever.  We characterize these channels, assuming an interesting conjecture on the invertible (or decoherence-free) subspaces of an arbitrary channel.   Joint work with Anne Broadbent and Douglas Stebila. http://arxiv.org/abs/1211.1080

We study the robustness of quantum information stored in the degenerate ground space of a local, frustration-free Hamiltonian with commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier separating the distinct ground states under local transformations would protect the information from thermal fluctuations. On the other hand, local topological order would shield the ground space from static perturbations. Here we demonstrate that local topological order implies a constant energy barrier, thus inhibiting thermal stability. Joint work with David Poulin. arXiv:1209.5750

Self-testing a multipartite quantum state means verifying the existence of the state based on the outcomes of unknown or untrusted measurements. This concept is important in device-independent quantum cryptography. There are some previously known results on self-testing which involve nonlocal binary XOR games such as the CHSH test and the GHZ paradox.  In our work we expand on these results.  We provide a general criterion which, when satisfied, guarantees that a given nonlocal binary XOR game is a robust self-test.  The error term in this result is quadratic, which is the best possible.  In my talk I will explain the conceptual basis for the criterion and offer some examples.  This is joint work with Yaoyun Shi (arXiv:1207.1819).

The minimal dimension of the Hilbert space that hosts states of an entangled pair of photons can be extremely high. The process of spontaneous parametric down-conversion (SPDC) is a possible way of producing highly entangled photon pairs, in both the spatial and temporal parts of the wave function. However, the most common approximations that are used in the analytical treatment of SPDC hinder the possibility of noticing further structures of the single joint modes. We used a more general formalism, showing that the entangled modes are still eigenfunctions of the orbital angular momentum, but the radial modes are far from the usual ones and they show novel interesting features that might be explained by introducing an additional quantum number. The problem of dealing with SPDC states has two faces: we need to know with enough confidence what state are created, and we need to know with enough confidence what states we are projecting on, upon measurement. We tried to approach both these problems together, and we showed that high dimensional entanglement shields the amount of information that can be stored in a photon from imperfect measurements. In my talk I will present both these aspects of high-dimensionally entangled states of photon pairs.

Winter's measurement compression theorem stands as one of the most important, yet perhaps less well-known coding theorems in quantum information theory. Not only does it make an illuminative statement about measurement in quantum theory, but it also underlies several other general protocols used for entanglement distillation or local purity distillation. The theorem provides for an asymptotic decomposition of any quantum measurement into an "extrinsic" source of noise, classical noise in the measurement that is independent of the actual outcome, and "intrinsic" quantum noise that can be due in part to the nonorthogonality of quantum states. This decomposition leads to an optimal protocol for a sender to 1) simulate many instances of a quantum measurement acting on many copies of some state and 2) send the outcomes of the measurements to a receiver using as little classical communication as possible while still having a faithful simulation. The protocol assumes that the parties have access to some amount of common randomness, which is a strictly weaker resource than classical communication.   In this talk, we provide a full review of Winter's measurement compression theorem, detailing the information processing task, providing examples for understanding it, overviewing Winter's achievability proof, and detailing a new approach to its single-letter converse theorem. We then overview the Devetak-Winter theorem on classical data compression with quantum side information, providing new proofs of the achievability and converse parts of this theorem. From there, we outline a new protocol that we call "measurement compression with quantum side information," a protocol announced in prior work on trade-offs in quantum Shannon theory. This protocol has several applications, including its part in the "classically-assisted state redistribution" protocol, which is the most general protocol on the static side of the quantum information theory tree, and its role in reducing the classical communication cost in a task known as local purity distillation. We finally outline a connection between this protocol and recent work on entropic uncertainty relations in the presence of quantum memory.   This is joint with Patrick Hayden, Francesco Buscemi, and Min-Hsiu Hsieh.