Quantum Physics

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 distinction between a realist interpretation of quantum theory that is psi-ontic and one that is psi-epistemic is whether or not a difference in the quantum state necessarily implies a difference in the underlying ontic state. Psi-ontologists believe that it does, psi-epistemicists that it does not. This talk will address the question of whether the PBR theorem should be interpreted as lending evidence against the psi-epistemic research program. I will review the evidence in favour of the psi-epistemic approach and describe the pre-existing reasons for thinking that if a quantum state represents knowledge about reality then it is not reality as we know it, i.e., it is not the kind of reality that is posited in the standard hidden variable framework. I will argue that the PBR theorem provides additional clues for "what has to give" in the hidden variable framework rather than providing a reason to retreat from the psi-epistemic position. The first assumption of the theorem - that holistic properties may exist for composite systems, but do not arise for unentangled quantum states - is only appealing if one is already predisposed to a psi-ontic view. The more natural assumption of separability (no holistic properties) coupled with the other assumptions of the theorem rules out both psi-ontic and psi-epistemic models and so does not decide between them. The connection between the PBR theorem and other no-go results will be discussed. In particular, I will point out how the second assumption of the theorem is an instance of preparation noncontextuality, a property that is known not to be achievable in any ontological model of quantum theory, regardless of the status of separability (though not in the form posited by PBR). I will also consider the connection of PBR to the failure of local causality by considering an experimental scenario which is in a sense a time-inversion of the PBR scenario.

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

It is sometimes pointed out as a curiosity that the state space of quantum theory and actual physical space seem related in a surprising way: not only is space three-dimensional and Euclidean, but so is the Bloch ball which describes quantum two-level systems. In the talk, I report on joint work with Lluis Masanes, where we show how this observation can be turned into a mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (dropping quantum theory and complex amplitudes altogether). Furthermore, suppose there are systems that in some sense behave as “binary units of direction information”, interacting via some continuous reversible time evolution. We prove that this uniquely determines d=3 and quantum theory, and that it allows observers to infer local spatial geometry from probability measurements.

One of the most important open problems in physics is to reconcile quantum mechanics with our classical intuition. In this talk we look at quantum foundations through the lens of mathematical foundations and uncover a deep connection between the two fields. We show that Cantorian set theory is based on classical concepts incompatible with quantum experiments. Specifically, we prove that Zermelo-Fraenkel axioms of set theory (and the background classical logic) imply a Bell-type inequality. Consequently, quantum experiments violating Bell inequality cannot be described in the framework of classical set theory. This suggests that a non-Cantorian set theory could be a better framework to capture the elusive nature of quantum world. Finally, we discuss several possible options for a future logico-mathematical framework compatible with quantum experiments. 

We establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here.    Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity.  Since our main results are expressed in a diagrammatic language (the one of dagger-compact categories) they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory.    The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.   This is joint work with Ross Duncan, Aleks Kissinger and Quanlong (Harny) Wang. Paper: arXiv:1203.4988 - LiCS'12 

I present our work on inferring causality in the classical world and encourage the audience to think about possible generalizations to the quantum world. Statistical dependences between observed quantities X and Y indicate a causal relation, but it is a priori not clear whether X caused Y or Y caused X or there is a common cause of both. It is widely believed that this can only be decided if either one is able to do interventions on the system, or if X and Y are part of a larger set of variables. In the latter case, conditional statistical independences contain some information on causal directions, formalized by the Causal Markov Condition on directed acyclic graphs. Contrary to this belief, we have shown that empirical joint distributions of just two variables often indicate the causal direction. The observed asymmetry between cause and effect is, on the one hand, related to the thermodynamic arrow of time. On the other hand, it can be derived from a new principle that we have postulated: the Algorithmic Causal Markov Condition, which relates Kolmogorov complexity to causality.  
Literature: [1] Janzing, Schoelkopf: Causal inference using the algorithmic Markov condition, IEEE TIT 2010. 
[2] Daniusis, Janzing,...: Inferring deterministic causal relations, UAI 2010. 
[3] Janzing: On the entropy production of time-series with uni-directional linearity.Journ. Stat. Phys. 2010.

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.