arXiv subjects

Andrew J. P. Garner

Publications and source records attributed to Andrew J. P. Garner.

31 records · Page 2Linked to original sources

arXiv2018

Interferometric computation beyond quantum theory

There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer. There has been recent foundational interest in theories beyond quantum theory. Here, we present a generalized formulation of computation in the context of a many-armed interferometer, and explore how theories can differ from quantum theory and still perform distributed calculations in this set-up. We shall see that quaternionic quantum theory proves a suitable candidate, whereas box-world does not. We also find that a classical hidden variable model first presented by Spekkens [Phys. Rev. A, 75:3:32100, 2007] can also be used for this type of computation due to the epistemic restriction placed on the hidden variable.

quant-ph↗

arXiv2017

The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer

The patterns of fringes produced by an interferometer have long been important testbeds for our best contemporary theories of physics. Historically, interference has been used to contrast quantum mechanics to classical physics, but recently experiments have been performed that test quantum theory against even more exotic alternatives. A physically motivated family of theories are those where the state space of a two-level system is given by a sphere of arbitrary dimension. This includes classical bits, and real, complex and quaternionic quantum theory. In this paper, we consider relativity of simultaneity (that observers may disagree about the order of events at different locations) as applied to a two-armed interferometer, and show that this forbids most interference phenomena more complicated than those of complex quantum theory. If interference must depend on some relational property of the setting (such as path difference), then relativity of simultaneity will limit state spaces to standard complex quantum theory, or a subspace thereof. If this relational assumption is relaxed, we find one additional theory compatible with relativity of simultaneity: quaternionic quantum theory. Our results have consequences for current laboratory interference experiments: they have to be designed carefully to avoid rendering beyond-quantum effects invisible by relativity of simultaneity.

quant-ph↗

arXiv2017

Unbounded memory advantage in stochastic simulation using quantum mechanics

Simulating the stochastic evolution of real quantities on a digital computer requires a trade-off between the precision to which these quantities are approximated, and the memory required to store them. The statistical accuracy of the simulation is thus generally limited by the internal memory available to the simulator. Here, using tools from computational mechanics, we show that quantum processors with a fixed finite memory can simulate stochastic processes of real variables to arbitrarily high precision. This demonstrates a provable, unbounded memory advantage that a quantum simulator can exhibit over its best possible classical counterpart.

quant-ph↗

arXiv2017

The classical-quantum divergence of complexity in modelling spin chains

The minimal memory required to model a given stochastic process - known as the statistical complexity - is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this measure exhibits drastically different behaviour: it rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This demonstrates how complexity, as captured by the amount of memory required to model a process, can exhibit radically different behaviour when quantum processing is allowed.

quant-ph↗

arXiv2017

Thermodynamics of complexity and pattern manipulation

Many organisms capitalize on their ability to predict the environment to maximize available free energy, and reinvest this energy to create new complex structures. This functionality relies on the manipulation of patterns - temporally ordered sequences of data. Here, we propose a framework to describe pattern manipulators -- devices that convert thermodynamic work to patterns or vice versa - and use them to build a 'pattern engine' that facilitates a thermodynamic cycle of pattern creation and consumption. We show that the least heat dissipation is achieved by the provably simplest devices; the ones that exhibit desired operational behaviour while maintaining the least internal memory. We derive the ultimate limits of this heat dissipation, and show that it is generally non-zero and connected with the pattern's intrinsic crypticity - a complexity theoretic quantity that captures the puzzling difference between the amount of information the pattern's past behaviour reveals about its future, and the amount one needs to communicate about this past to optimally predict the future.

quant-ph↗

arXiv2017

Entropic equality for worst-case work at any protocol speed

We derive an equality for non-equilibrium statistical mechanics in finite-dimensional quantum systems. The equality concerns the worst-case work output of a time-dependent Hamiltonian protocol in the presence of a Markovian heat bath. It has has the form "worst-case work = penalty - optimum". The equality holds for all rates of changing the Hamiltonian and can be used to derive the optimum by setting the penalty to 0. The optimum term contains the max entropy of the initial state, rather than the von Neumann entropy, thus recovering recent results from single-shot statistical mechanics. Energy coherences can arise during the protocol but are assumed not to be present initially. We apply the equality to an electron box.

quant-ph↗

arXiv2017

Using quantum theory to reduce the complexity of input-output processes

All natural things process and transform information. They receive environmental information as input, and transform it into appropriate output responses. Much of science is dedicated to building models of such systems -- algorithmic abstractions of their input-output behavior that allow us to simulate how such systems can behave in the future, conditioned on what has transpired in the past. Here, we show that classical models cannot avoid inefficiency -- storing past information that is unnecessary for correct future simulation. We construct quantum models that mitigate this waste, whenever it is physically possible to do so. This suggests that the complexity of general input-output processes depends fundamentally on what sort of information theory we use to describe them.

quant-ph↗

arXiv2016

The ultimate physical limits to reversibility

We argue that if, in order to reverse an object's dynamics, we need to be able to keep track of it with enough precision, then there is an upper bound on the size of the object whose dynamics we can reverse - even using all the available resources of the universe. Using a bound from the holographic principle and treating the universe as the ultimate quantum computer, we show that if we want to reverse the dynamics of an object which has evolved since the beginning of time, its radius cannot exceed ten microns. If this computation is performed irreversibly, the object must be much smaller. This places a lower bound on the size of the smallest possible quantum measurement device.

quant-ph↗

arXiv2015

Introducing one-shot work into fluctuation relations

Two approaches to small-scale and quantum thermodynamics are fluctuation relations and one-shot statistical mechanics. Fluctuation relations (such as Crooks' Theorem and Jarzynski's Equality) relate nonequilibrium behaviors to equilibrium quantities such as free energy. One-shot statistical mechanics involves statements about every run of an experiment, not just about averages over trials. We investigate the relation between the two approaches. We show that both approaches feature the same notions of work and the same notions of probability distributions over possible work values. The two approaches are alternative toolkits with which to analyze these distributions. To combine the toolkits, we show how one-shot work quantities can be defined and bounded in contexts governed by Crooks' Theorem. These bounds provide a new bridge from one-shot theory to experiments originally designed for testing fluctuation theorems.

cond-mat.stat-mech↗

arXiv2014

Guaranteed energy-efficient bit reset in finite time

Landauer's principle states that it costs at least kTln2 of work to reset one bit in the presence of a heat bath at temperature T. The bound of kTln2 is achieved in the unphysical infinite-time limit. Here we ask what is possible if one is restricted to finite-time protocols. We prove analytically that it is possible to reset a bit with a work cost close to kTln2 in a finite time. We construct an explicit protocol that achieves this, which involves changing the system's Hamiltonian avoiding quantum coherences, and thermalising. Using concepts and techniques pertaining to single-shot statistical mechanics, we further develop the limit on the work cost, proving that the heat dissipated is close to the minimal possible not just on average, but guaranteed with high confidence in every run. Moreover we exploit the protocol to design a quantum heat engine that works near the Carnot efficiency in finite time.

quant-ph↗

arXiv2013

Particle exchange in post-quantum theories

In quantum theory, particles in three spatial dimensions come in two different types: bosons or fermions, which exhibit sharply contrasting behaviours due to their different exchange statistics. Could more general forms of probabilistic theories admit more exotic types of particles? Here, we propose a thought experiment to identify more exotic particles in general post-quantum theories. We consider how in quantum theory the phase introduced by swapping indistinguishable particles can be measured. We generalise this to post-quantum scenarios whilst imposing indistinguishability and locality principles. We show that our ability to witness exotic particle exchange statistics depends on which symmetries are admitted within a theory. These exotic particles can manifest unusual behaviour, such as non-abelianicity even in topologically simple three-dimensional space.

quant-ph↗

arXiv2013

How uncertainty enables non-classical dynamics

The uncertainty principle limits quantum states such that when one observable takes predictable values there must be some other mutually unbiased observables which take uniformly random values. We show that this restrictive condition plays a positive role as the enabler of non-classical dynamics in an interferometer. First we note that instantaneous action at a distance between different paths of an interferometer should not be possible. We show that for general probabilistic theories this heavily curtails the non-classical dynamics. We prove that there is a trade-off with the uncertainty principle, that allows theories to evade this restriction. On one extreme, non-classical theories with maximal certainty have their non-classical dynamics absolutely restricted to only the identity operation. On the other extreme, quantum theory minimises certainty in return for maximal non-classical dynamics.

quant-ph↗

arXiv2013

A general framework for phase and interference

Phase plays a crucial role in many quantum effects including interference. Phase is normally defined in terms of complex numbers that appear when representing quantum states as complex vectors. Here we give an operational definition whereby phase is instead defined in terms of measurement statistics. Our definition is phrased in terms of the operational framework known as generalised probabilistic theories, or the convex framework. The definition makes it possible to ask whether other theories in this framework can also have phase. We apply our definition to investigate phase and interference in several example theories: classical probability theory, a version of Spekken's toy model, quantum theory, and box-world. We find that phase is ubiquitous; any non-classical theory can be said to have non-trivial phase dynamics.

quant-ph↗