Quantum Information

It is certainly possible to express ordinary quantum mechanics in the framework of a real vector space: by adopting a suitable restriction on all operators--Stueckelberg’s rule--one can make the real-vector-space theory exactly equivalent to the standard complex theory.  But can we achieve a similar effect without invoking such a restriction?  In this talk I explore a model within real-vector-space quantum theory in which the role of the complex phase is played by a separate physical system called the ubit (for “universal rebit”).  The ubit is a single binary real-vector-space quantum object that is allowed to interact with everything in the world.  It also rotates in its two-dimensional state space.  In the limit of infinitely fast rotation, one recovers standard quantum theory.  When the rotation rate is large but not infinite, one finds small deviations from the standard theory.  Here I describe a few such deviations that we have seen numerically and explained analytically.

When two independent analog signals, $X$ and $Y$ are added together giving $Z=X+Y$, the entropy of $Z$, $H(Z)$, is not a simple function of the entropies $H(X)$ and $H(Y)$, but rather depends on the details of $X$ and $Y$'s distributions.  Nevertheless, the entropy power inequality (EPI), which states that $e^{2H(Z)} \geq e^{2H(X)} + e^{2H(Y)}$, gives a very tight restriction on the entropy of $Z$. This inequality has found many applications in information theory and statistics.  The quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter. The purpose of this talk is to give an outline of the proof of two separate generalizations of the entropy power inequality to the quantum regime.  These inequalities provide strong new upper bounds for the classical capacity of quantum additive noise channels, including quantum analogues of the additive white Gaussian noise channels.   Our proofs are similar in spirit to standard classical proofs of the EPI, but some new quantities and ideas are needed in the quantum setting. Specifically, we find a new quantum de Bruijin identity relating entropy production under diffusion to a divergence-based quantum Fisher information. Furthermore, this Fisher information exhibits certain convexity properties in the context of beam splitters.   This is joint work with Graeme Smith.

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.

Joint work with Earl Campbell (FU-Berlin) and Hussain Anwar (UCL)   Magic state distillation is a key component of some high-threshold schemes for fault-tolerant quantum computation [1], [2]. Proposed by Bravyi and Kitaev [3] (and implicitly by Knill [4]), and improved by Reichardt [4], Magic State Distillation is a method to broaden the vocabulary of a fault-tolerant computational model, from a limited set of gates (e.g. the Clifford group or a sub-group[2])  to full universality, via the preparation of mixed ancilla qubits which may be prepared without fault tolerant protection.    Magic state distillation schemes have a close relation with quantum error correcting codes, since a key step in such protocols [5] is the projection onto a code subspace. Bravyi and Kitaev proposed two protocols; one based upon the 5-qubit code, the second derived from a punctured Reed-Muller code. Reed Muller codes are a very important family of classical linear code. They gained much interest [6] in the early years of quantum error correction theory, since their properties make them well-suited to the formation of quantum codes via the CSS-construction [7]. Punctured Reed-Muller codes (loosely speaking, Reed-Muller codewords with a bit removed) in particular lead to quantum codes with an unusual property, the ability to implement non-Clifford gates transversally [8].    Most work in fault-tolerant quantum computation focuses on qubits, but fault tolerant constructions can be generalised to higher dimensions [9]  - particularly readily for prime dimensions. Recently, we presented the first magic state distillation protocols [10] for non-binary systems, providing explicit protocols for the qutrit case (complementing a recent no-go theorem demonstrating bound states for magic state distillation in higher dimensions [11]). In this talk, I will report on more recent work [12], where the properties of punctured Reed-Muller codes are employed to demonstrate Magic State distillation protocols for all prime dimensions. In my talk, I will give a technical account of this result and present numerical investigations of the performance of such a protocol in the qutrit case. Finally, I will discuss the potential for application of these results to fault-tolerant quantum computation.     This will be a technical talk, and though some concepts of linear codes and quantum codes will be briefly revised, I will assume that listeners are familiar with quantum error correction theory (e.g. the stabilizer formalism and the CSS construction) for qubits.   [1] E. Knill. Fault-tolerant postselected quantum computation: schemes, quant-ph/0402171 [2] R. Raussendorf, J. Harrington and K. Goyal, Topological fault-tolerance in cluster state quantum computation, arXiv:quant-ph/0703143v1 [3] S. Bravyi and A. Kitaev. Universal quantum computation based on a magic states distillation, quant- ph/0403025 [4] B. W. Reichardt, Improved magic states distillation for quantum universality, arXiv:quant-ph/0411036v1 [5] E.T. Campbell and D.E. Browne, On the Structure of Protocols for Magic State Distillation, arXiv:0908.0838
[6] A. Steane, Quantum Reed Muller Codes, arXiv:quant-ph/9608026 [7] Nielsen and Chuang, Quantum Information and Computation, chapter 10 [8] E. Knill, R. Laflamme, and W. Zurek, Threshold accuracy for quantum computation, quant-ph/9610011 [9] D. Gottesman, Fault-Tolerant Quantum Computation with Higher-Dimensional Systems, quant-ph/9802007 [10] H. Anwar, E.T Campbell and D.E. Browne, Qutrit Magic State Distillation, arXiv:1202.2326 [11] V. Veitch, C. Ferrie, J. Emerson, Negative Quasi-Probability Representation is a Necessary Resource for Magic State Distillation, arXiv:1201.1256v3 [12] H. Anwar, E.T Campbell and D.E. Browne, in preparation

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]

We study the problem of reconstructing an unknown matrix M, of rank r and dimension d, using O(rd poly log d) Pauli measurements. This has applications to compressed sensing methods for quantum state tomography.  We give a solution to this problem based on the restricted isometry property (RIP), which improves on previous results using dual certificates. In particular, we show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r RIP. This implies that M can be recovered from a fixed ("universal") set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.