Searcharxiv⌕ Search

arXiv subjects

Lucien Hardy

Publications and source records attributed to Lucien Hardy.

At least 37 records · Page 2Linked to original sources
arXiv2011

Reformulating and Reconstructing Quantum Theory

We provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of physical operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Sharpness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Sturdiness. Filters are non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilitieso follows. A more detailed abstract is provided in the paper.

quant-ph↗
arXiv2010

A formalism-local framework for general probabilistic theories including quantum theory

In this paper we consider general probabilistic theories that pertain to circuits which satisfy two very natural assumptions. We provide a formalism that is local in the following very specific sense: calculations pertaining to any region of spacetime employ only mathematical objects associated with that region. We call this "formalism locality". It incorporates the idea that space and time should be treated on an equal footing. Formulations that use a foliation of spacetime to evolve a state do not have this property nor do histories-based approaches. An operation (see figure on left) has inputs and outputs (through which systems travel). A circuit is built by wiring together operations such that we have no open inputs or outputs left over. A fragment (see figure on right) is a part of a circuit and may have open inputs and outputs. We show how each operation is associated with a certain mathematical object which we call a "duotensor" (this is like a tensor but with a bit more structure). In the figure on the left we show how a duotensor is represented graphically. We can link duotensors together such that black and white dots match up to get the duotensor corresponding to any fragment. The figure on the right is the duotensor for the above fragment. Links represent summing over the corresponding indices. We can use such duotensors to make probabilistic statements pertaining to fragments. Since fragments are the circuit equivalent of arbitrary spacetime regions we have formalism locality. The probability for a circuit is given by the corresponding duotensorial calculation (which is a scalar since there are no indices left over). We show how to put classical probability theory and quantum theory into this framework. [Note: the abstract in the paper has pictures.]

quant-ph↗
arXiv2010

Limited Holism and Real-Vector-Space Quantum Theory

Quantum theory has the property of "local tomography": the state of any composite system can be reconstructed from the statistics of measurements on the individual components. In this respect the holism of quantum theory is limited. We consider in this paper a class of theories more holistic than quantum theory in that they are constrained only by "bilocal tomography": the state of any composite system is determined by the statistics of measurements on pairs of components. Under a few auxiliary assumptions, we derive certain general features of such theories. In particular, we show how the number of state parameters can depend on the number of perfectly distinguishable states. We also show that real-vector-space quantum theory, while not locally tomographic, is bilocally tomographic.

quant-ph↗
arXiv2009

Foliable Operational Structures for General Probabilistic Theories

In this chapter a general mathematical framework for probabilistic theories of operationally understood circuits is laid out. Circuits are comprised of operations and wires. An operation is one use of an apparatus and a wire is a diagrammatic device for showing how apertures on the apparatuses are placed next to each other. Mathematical objects are defined in terms of the circuit understood graphically. In particular, we do not think of the circuit as sitting in a background time. Circuits can be foliated by hypersurfaces comprised of sets of wires. Systems are defined to be associated with wires. A closable set of operations is defined to be one for which the probability associated with any circuit built from this set is independent both of choices on other circuits and of extra circuitry that may be added to outputs from this circuit. States can be associated with circuit fragments corresponding to preparations. These states evolve on passing through circuit fragments corresponding to transformations. The composition of transformations is treated. A number of theorems are proven including one which rules out quaternionic quantum theory. The case of locally tomographic theories (where local measurements on a systems components suffice to determine the global state) is considered. For such theories the probability can be calculated for a circuit from matrices pertaining the operations that comprise that circuit. Classical probability theory and quantum theory are exhibited as examples in this framework.

quant-ph↗
arXiv2009

Entropy for theories with indefinite causal structure

Entropy is a concept that has traditionally been reliant on a definite notion of causality. However, without a definite notion of causality, the concept of entropy is not all lost. Indefinite causal structure results from combining probabilistic predictions and dynamical space-time. Combining the probabilistic nature of quantum theory and dynamical treatment space-time from general relativity is an approach to the problem of quantum gravity. The causaloid framework lays the mathematical groundwork to be able to treat indefinite causal structure. In this paper, we build on the causaloid mathematics and define a causally-unbiased entropy for an indefinite causal structure. In defining a causally-unbiased entropy, there comes about an emergent idea of causality in the form of a measure of causal connectedness, termed the Q factor.

gr-qc↗
arXiv2008

Formalism Locality in Quantum Theory and Quantum Gravity

We expect a theory of Quantum Gravity to be both probabilistic and have indefinite causal structure. Indefinite causal structure poses particular problems for theory formulation since many of the core ideas used in the usual approaches to theory construction depend on having definite causal structure. For example, the notion of a state across space evolving in time requires that we have some definite causal structure so we can define a state on a space-like hypersurface. We will see that many of these problems are mitigated if we are able to formulate the theory in a "formalism local" (or F-local) fashion. A formulation of a physical theory is said to be F-local if, in making predictions for any given arbitrary space-time region, we need only refer to mathematical objects pertaining to that region. This is a desirable property both on the grounds of efficiency and since, if we have indefinite causal structure, it is not clear how to select some other space-time region on which our calculations may depend. The usual ways of formulating physical theories (the time evolving state picture, the histories approach, and the local equations approach) are not F-local. We set up a framework for probabilistic theories with indefinite causal structure. This, the causaloid framework, is F-local. We describe how Quantum Theory can be formulated in the causaloid framework (in an F-local fashion). This provides yet another formulation of Quantum Theory. This formulation, however, may be particularly relevant to the problem of finding a theory of Quantum Gravity.

gr-qc↗
arXiv2007

Quantum gravity computers: On the theory of computation with indefinite causal structure

A quantum gravity computer is one for which the particular effects of quantum gravity are relevant. In general relativity, causal structure is non-fixed. In quantum theory non-fixed quantities are subject to quantum uncertainty. It is therefore likely that, in a theory of quantum gravity, we will have indefinite causal structure. This means that there will be no matter of fact as to whether a particular interval is timelike or not. We study the implications of this for the theory of computation. Classical and quantum computations consist in ivolving the state of the computer through a sequence of time steps. This will, most likely, not be possible for a quantum gravity computer because the notion of a time step makes no sense if we have indefinite causal structure. We show that it is possible to set up a model for computation even in the absence of definite causal structure by using a certain framework (the causaloid formalism) that was developed for the purpose of correlating data taken in this type of situation. Corresponding to a physical theory is a causaloid, Lambda (this is a mathematical object containing information about the causal connections between different spacetime regions). A computer is given by the pair {Lambda, S} where S is a set of gates. Working within the causaloid formalism, we explore the question of whether universal quantum gravity computers are possible. We also examine whether a quantum gravity computer might be more powerful than a quantum (or classical) computer. In particular, we ask whether indefinite causal structure can be used as a computational resource.

quant-ph↗
arXiv2006

Towards Quantum Gravity: A Framework for Probabilistic Theories with Non-Fixed Causal Structure

General relativity is a deterministic theory with non-fixed causal structure. Quantum theory is a probabilistic theory with fixed causal structure. In this paper we build a framework for probabilistic theories with non-fixed causal structure. This combines the radical elements of general relativity and quantum theory. The key idea in the construction is physical compression. A physical theory relates quantities. Thus, if we specify a sufficiently large set of quantities (this is the compressed set), we can calculate all the others. We apply three levels of physical compression. First, we apply it locally to quantities (actually probabilities) that might be measured in a particular region of spacetime. Then we consider composite regions. We find that there is a second level of physical compression for the composite region over and above the first level physical compression for the component regions. Each application of first and second level physical compression is quantified by a matrix. We find that these matrices themselves are related by the physical theory and can therefore be subject to compression. This is the third level of physical compression. This third level of physical compression gives rise to a new mathematical object which we call the causaloid. From the causaloid for a particular physical theory we can calculate verything the physical theory can calculate. This approach allows us to set up a framework for calculating probabilistic correlations in data without imposing a fixed causal structure (such as a background time). We show how to put quantum theory in this framework (thus providing a new formulation of this theory). We indicate how general relativity might be put into this framework and how the framework might be used to construct a theory of quantum gravity.

gr-qc↗
arXiv2005

Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity

Quantum theory is a probabilistic theory with fixed causal structure. General relativity is a deterministic theory but where the causal structure is dynamic. It is reasonable to expect that quantum gravity will be a probabilistic theory with dynamic causal structure. The purpose of this paper is to present a framework for such a probability calculus. We define an operational notion of space-time, this being composed of elementary regions. Central to this formalism is an object we call the causaloid. This object captures information about causal structure implicit in the data by quantifying the way in which the number of measurements required to establish a state for a composite region is reduced when there is a causal connection between the component regions. This formalism puts all elementary regions on an equal footing. It does not require that we impose fixed causal structure. In particular, it is not necessary to assume the existence of a background time. Remarkably, given the causaloid, we can calculate all relevant probabilities and so the causaloid is sufficient to specify the predictive aspect of a physical theory. We show how certain causaloids can be represented by suggestive diagrams and we show how to represent both classical probability theory and quantum theory by a causaloid. We do not give a causaloid formulation for general relativity though we speculate that this is possible. The work presented here suggests a research program aimed at finding a theory of quantum gravity. The idea is to use the causaloid formalism along with principles taken from the two theories to marry the dynamic causal structure of general relativity with the probabilistic structure of quantum theory.

gr-qc↗
arXiv2005

No Signalling and Quantum Key Distribution

Standard quantum key distribution protocols are provably secure against eavesdropping attacks, if quantum theory is correct. It is theoretically interesting to know if we need to assume the validity of quantum theory to prove the security of quantum key distribution, or whether its security can be based on other physical principles. The question would also be of practical interest if quantum mechanics were ever to fail in some regime, because a scientifically and technologically advanced eavesdropper could perhaps use post-quantum physics to extract information from quantum communications without necessarily causing the quantum state disturbances on which existing security proofs rely. Here we describe a key distribution scheme provably secure against general attacks by a post-quantum eavesdropper who is limited only by the impossibility of superluminal signalling. The security of the scheme stems from violation of a Bell inequality.

quant-ph↗
arXiv2004

Cheat Sensitive Quantum Bit Commitment

We define cheat sensitive cryptographic protocols between mistrustful parties as protocols which guarantee that, if either cheats, the other has some nonzero probability of detecting the cheating. We give an example of an unconditionally secure cheat sensitive non-relativistic bit commitment protocol which uses quantum information to implement a task which is classically impossible; we also describe a simple relativistic protocol.

quant-ph↗
arXiv2002

Substituting a qubit for an arbitrarily large number of classical bits

We show that a qubit can be used to substitute for an arbitrarily large number of classical bits. We consider a physical system S interacting locally with a classical field phi(x) as it travels directly from point A to point B. The field has the property that its integrated value is an integer multiple of some constant. The problem is to determine whether the integer is odd or even. This task can be performed perfectly if S is a qubit. On the otherhand, if S is a classical system then we show that it must carry an arbitrarily large amount of classical information. We identify the physical reason for such a huge quantum advantage, and show that it also implies a large difference between the size of quantum and classical memories necessary for some computations. We also present a simple proof that no finite amount of one-way classical communication can perfectly simulate the effect of quantum entanglement.

quant-ph↗
arXiv2002

Quantum nonlocality, Bell inequalities and the memory loophole

In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nth particle pair would be independent of measurement choices and outcomes for the first $(n-1)$ pairs. Models which violate this assumption exploit what we call the {\it memory loophole}. We focus on the strongest type of violation, which uses the {\it 2-sided} memory loophole, in which the hidden variables for pair $n$ can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the 2-sided memory loophole allows a systematic violation of the CHSH inequality when the data are analysed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the {\it simultaneous measurement loophole}, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearised CHSH inequality as well as other Bell-type inequalities.

quant-ph↗
arXiv2002

Nonlocality, Asymmetry, and Distinguishing Bipartite States

Entanglement is an useful resource because some global operations cannot be locally implemented using classical communication. We prove a number of results about what is and is not locally possible. We focus on orthogonal states, which can always be globally distinguished. We establish the necessary and sufficient conditions for a general set of 2x2 quantum states to be locally distinguishable, and for a general set of 2xn quantum states to be distinguished given an initial measurement of the qubit. These results reveal a fundamental asymmetry to nonlocality, which is the origin of ``nonlocality without entanglement'', and we present a very simple proof of this phenomenon.

quant-ph↗
arXiv2001

Why Quantum Theory?

The usual formulation of quantum theory is rather abstract. In recent work I have shown that we can, nevertheless, obtain quantum theory from five reasonable axioms. Four of these axioms are obviously consistent with both classical probability theory and quantum theory. The remaining axiom requires that there exists a continuous reversible transformation between any two pure states. The requirement of continuity rules out classical probability theory. In this paper I will summarize the main points of this new approach. I will leave out the details of the proof that these axioms are equivalent to the usual formulation of quantum theory (for these see quant-ph/0101012).

quant-ph↗
arXiv2001

Quantum Theory From Five Reasonable Axioms

The usual formulation of quantum theory is based on rather obscure axioms (employing complex Hilbert spaces, Hermitean operators, and the trace rule for calculating probabilities). In this paper it is shown that quantum theory can be derived from five very reasonable axioms. The first four of these are obviously consistent with both quantum theory and classical probability theory. Axiom 5 (which requires that there exists continuous reversible transformations between pure states) rules out classical probability theory. If Axiom 5 (or even just the word "continuous" from Axiom 5) is dropped then we obtain classical probability theory instead. This work provides some insight into the reasons quantum theory is the way it is. For example, it explains the need for complex numbers and where the trace formula comes from. We also gain insight into the relationship between quantum theory and classical probability theory.

quant-ph↗