Quantum Information

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.

We propose a framework to describe and simulate a class of many-body quantum states. We do so by considering joint eigenspaces of sets of monomial unitary matrices, called "M-spaces"; a unitary matrix is monomial if precisely one entry per row and column is nonzero. We show that M-spaces encompass various important state families, such as all Pauli stabilizer states and codes, the AKLT model, Kitaev's anyon models, W states and several others. We furthermore demonstrate how basic properties of M-spaces can transparently be understood by manipulating their monomial stabilizer groups. Finally we show that a large subclass of M-spaces can be simulated efficiently classically with one unified method. [cf. M. Van den Nest, http://arxiv.org/abs/1108.0531]

Based on the joint work with Sergey Bravyi, IBM Watson.   We show that any topologically ordered local stabilizer model of spins in three dimensional lattices that lacks string logical operators can be used as a reliable quantum memory against thermal noise. It is shown that any local process creating a topologically charged particle separated from other particles by distance $R$, must cross an energy barrier of height $c \log R$. This property makes the model glassy. We devise an efficient decoding algorithm that should be used at the final read-out, and prove a lower bound on the memory time until which the fidelity between the outcome of the decoder and the initial state is close to 1. The memory time increases as $L^{\beta}$ where $L$ is the system size and $\beta$ the inverse temperature, as long as $L < L^\star \sim e^\beta$. Hence, the optimal memory time scales as $e^{\beta^2}$. Our bound applies when the system interacts with thermal bath via a Markovian master equation. We give an example of 3D local stabilizer codes that satisfies all of our assumptions. We numerically verify for this example that our bound is tight up to constants.