arXiv subjects

Markus P. Mueller

Publications and source records attributed to Markus P. Mueller.

49 records · Page 3Linked to original sources

arXiv2017

Is the local linearity of space-time inherited from the linearity of probabilities?

The appearance of linear spaces, describing physical quantities by vectors and tensors, is ubiquitous in all of physics, from classical mechanics to the modern notion of local Lorentz invariance. However, as natural as this seems to the physicist, most computer scientists would argue that something like a "local linear tangent space" is not very typical and in fact a quite surprising property of any conceivable world or algorithm. In this paper, we take the perspective of the computer scientist seriously, and ask whether there could be any inherently information-theoretic reason to expect this notion of linearity to appear in physics. We give a series of simple arguments, spanning quantum information theory, group representation theory, and renormalization in quantum gravity, that supports a surprising thesis: namely, that the local linearity of space-time might ultimately be a consequence of the linearity of probabilities. While our arguments involve a fair amount of speculation, they have the virtue of being independent of any detailed assumptions on quantum gravity, and they are in harmony with several independent recent ideas on emergent space-time in high-energy physics.

gr-qc↗

arXiv2016

A generalization of majorization that characterizes Shannon entropy

We introduce a binary relation on the finite discrete probability distributions which generalizes notions of majorization that have been studied in quantum information theory. Motivated by questions in thermodynamics, our relation describes the transitions induced by bistochastic maps in the presence of additional auxiliary systems which may become correlated in the process. We show that this relation is completely characterized by Shannon entropy H, which yields an interpretation of H in resource-theoretic terms, and admits a particularly simple proof of a known characterization of H in terms of natural information-theoretic properties.

cs.IT↗

arXiv2015

Information-theoretic postulates for quantum theory

Why are the laws of physics formulated in terms of complex Hilbert spaces? Are there natural and consistent modifications of quantum theory that could be tested experimentally? This book chapter gives a self-contained and accessible summary of our paper [New J. Phys. 13, 063001, 2011] addressing these questions, presenting the main ideas, but dropping many technical details. We show that the formalism of quantum theory can be reconstructed from four natural postulates, which do not refer to the mathematical formalism, but only to the information-theoretic content of the physical theory. Our starting point is to assume that there exist physical events (such as measurement outcomes) that happen probabilistically, yielding the mathematical framework of "convex state spaces". Then, quantum theory can be reconstructed by assuming that (i) global states are determined by correlations between local measurements, (ii) systems that carry the same amount of information have equivalent state spaces, (iii) reversible time evolution can map every pure state to every other, and (iv) positivity of probabilities is the only restriction on the possible measurements.

quant-ph↗

arXiv2015

Stochastic independence as a resource in small-scale thermodynamics

It is well-known in thermodynamics that the creation of correlations costs work. It seems then a truism that if a thermodynamic transformation A->B is impossible, so will be any transformation that in sending A to B also correlates among them some auxiliary systems C. Surprisingly, we show that this is not the case for non-equilibrium thermodynamics of microscopic systems. On the contrary, the creation of correlations greatly extends the set of accessible states, to the point that we can perform on individual systems and in a single shot any transformation that would otherwise be possible only if the number of systems involved was very large. We also show that one only ever needs to create a vanishingly small amount of correlations (as measured by mutual information) among a small number of auxiliary systems (never more than three). The many, severe constraints of microscopic thermodynamics are reduced to the sole requirement that the non-equilibrium free energy decreases in the transformation. This shows that, in principle, reliable extraction of work equal to the free energy of a system can be performed by microscopic engines.

quant-ph↗

arXiv2015

Higher-order interference and single-system postulates characterizing quantum theory

We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the composition of many systems; and second, it is closer to experiment by having absence of higher-order interference as a postulate, which is currently the subject of experimental investigation. We give three postulates -- no higher-order interference, classical decomposability of states, and strong symmetry -- and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with 3-level octonionic quantum theory and ball state spaces of arbitrary dimension. Then we show that adding observability of energy as a fourth postulate yields complex quantum theory as the unique solution, relating the emergence of the complex numbers to the possibility of Hamiltonian dynamics. We also show that there may be interesting non-quantum theories satisfying only the first two of our postulates, which would allow for higher-order interference in experiments while still respecting the contextuality analogue of the local orthogonality principle.

quant-ph↗

arXiv2014

Entanglement and the three-dimensionality of the Bloch ball

We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this we impose two very natural assumptions: the continuity and reversibility of dynamics, and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories interacting dynamics is impossible. This result reveals that "existence of entanglement" is the requirement with minimal logical content which singles out quantum theory from our family of theories.

quant-ph↗

arXiv2013

Existence of an information unit as a postulate of quantum theory

Does information play a significant role in the foundations of physics? Information is the abstraction that allows us to refer to the states of systems when we choose to ignore the systems themselves. This is only possible in very particular frameworks, like in classical or quantum theory, or more generally, whenever there exists an information unit such that the state of any system can be reversibly encoded in a sufficient number of such units. In this work we show how the abstract formalism of quantum theory can be deduced solely from the existence of an information unit with suitable properties, together with two further natural assumptions: the continuity and reversibility of dynamics, and the possibility of characterizing the state of a composite system by local measurements. This constitutes a new set of postulates for quantum theory with a simple and direct physical meaning, like the ones of special relativity or thermodynamics, and it articulates a strong connection between physics and information.

quant-ph↗

arXiv2013

Three-dimensionality of space and the quantum bit: an information-theoretic approach

It is sometimes pointed out as a curiosity that the state space of quantum two-level systems, i.e. the qubit, and actual physical space are both three-dimensional and Euclidean. In this paper, we suggest an information-theoretic analysis of this relationship, by proving a particular mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (not assuming quantum theory in any way). Furthermore, suppose there are systems that carry "minimal amounts of direction information", interacting via some continuous reversible time evolution. We prove that this uniquely determines spatial dimension d=3 and quantum theory on two qubits (including entanglement and unitary time evolution), and that it allows observers to infer local spatial geometry from probability measurements.

quant-ph↗

arXiv2012

The black hole information problem beyond quantum theory

The origin of black hole entropy and the black hole information problem provide important clues for trying to piece together a quantum theory of gravity. Thus far, discussions on this topic have mostly assumed that in a consistent theory of gravity and quantum mechanics, quantum theory will be unmodified. Here, we examine the black hole information problem in the context of generalisations of quantum theory. In particular, we examine black holes in the setting of generalised probabilistic theories, in which quantum theory and classical probability theory are special cases. We compute the time it takes information to escape a black hole, assuming that information is preserved. We find that under some very general assumptions, the arguments of Page (that information should escape the black hole after half the Hawking photons have been emitted), and the black-hole mirror result of Hayden and Preskill (that information can escape quickly) need to be modified. The modification is determined entirely by what we call the Wootters-Hardy parameter associated with a theory. We find that although the information leaves the black hole after enough photons have been emitted, it is fairly generic that it fails to appear outside the black hole at this point -- something impossible in quantum theory due to the no-hiding theorem. The information is neither inside the black hole, nor outside it, but is delocalised. Our central technical result is an information decoupling theorem which holds in the generalised probabilistic framework.

hep-th↗

arXiv2012

The structure of reversible computation determines the self-duality of quantum theory

Predictions for measurement outcomes in physical theories are usually computed by combining two distinct notions: a state, describing the physical system, and an observable, describing the measurement which is performed. In quantum theory, however, both notions are in some sense identical: outcome probabilities are given by the overlap between two state vectors - quantum theory is self-dual. In this paper, we show that this notion of self-duality can be understood from a dynamical point of view. We prove that self-duality follows from a computational primitive called bit symmetry: every logical bit can be mapped to any other logical bit by a reversible transformation. Specifically, we consider probabilistic theories more general than quantum theory, and prove that every bit-symmetric theory must necessarily be self-dual. We also show that bit symmetry yields stronger restrictions on the set of allowed bipartite states than the no-signalling principle alone, suggesting reversible time evolution as a possible reason for limitations of non-locality.

quant-ph↗

arXiv2011

Deriving quantum theory from its local structure and reversibility

We investigate the class of physical theories with the same local structure as quantum theory, but a potentially different global structure. It has previously been shown that any bipartite correlations generated by such a theory must be simulatable in quantum theory, but that this does not hold for tripartite correlations. Here we explore whether imposing an additional constraint on this space of theories - that of dynamical reversibility - will allow us to recover the global quantum structure. In the particular case in which the local systems are identical qubits, we show that any theory admitting at least one continuous reversible interaction must be identical to quantum theory.

quant-ph↗

arXiv2011

A derivation of quantum theory from physical requirements

Quantum theory is usually formulated in terms of abstract mathematical postulates, involving Hilbert spaces, state vectors, and unitary operators. In this work, we show that the full formalism of quantum theory can instead be derived from five simple physical requirements, based on elementary assumptions about preparation, transformations and measurements. This is more similar to the usual formulation of special relativity, where two simple physical requirements -- the principles of relativity and light speed invariance -- are used to derive the mathematical structure of Minkowski space-time. Our derivation provides insights into the physical origin of the structure of quantum state spaces (including a group-theoretic explanation of the Bloch ball and its three-dimensionality), and it suggests several natural possibilities to construct consistent modifications of quantum theory.

quant-ph↗

arXiv2011

Absence of Thermalization in Nonintegrable Systems

We establish a link between unitary relaxation dynamics after a quench in closed many-body systems and the entanglement in the energy eigenbasis. We find that even if reduced states equilibrate, they can have memory on the initial conditions even in certain models that are far from integrable. We show that in such situations the equilibrium states are still described by a maximum entropy or generalized Gibbs ensemble, regardless of whether a model is integrable or not, thereby contributing to a recent debate. In addition, we discuss individual aspects of the thermalization process, comment on the role of Anderson localization, and collect and compare different notions of integrability.

quant-ph↗