arXiv subjects

Roger Colbeck

Publications and source records attributed to Roger Colbeck.

At least 37 records · Page 2Linked to original sources

arXiv2021

Device-independent randomness expansion against quantum side information

The ability to produce random numbers that are unknown to any outside party is crucial for many applications. Device-independent randomness generation does not require trusted devices and therefore provides strong guarantees of the security of the output, but it comes at the price of requiring the violation of a Bell inequality for implementation. A further challenge is to make the bounds in the security proofs tight enough to allow randomness expansion with contemporary technology. Although randomness has been generated in recent experiments, the amount of randomness consumed in doing so has been too high to certify expansion based on existing theory. Here we present an experiment that demonstrates device-independent randomness expansion. By developing a Bell test setup with a single-photon detection efficiency of around $84\%$ and by using a spot-checking protocol, we achieve a net gain of $2.57\times10^8$ certified bits with a soundness error $3.09\times10^{-12}$. The experiment ran for $19.2$ h, which corresponds to an average rate of randomness generation of $13,527$ bits per second. By developing the entropy accumulation theorem, we establish security against quantum adversaries. We anticipate that this work will lead to further improvements that push device-independence towards commercial viability.

quant-ph↗

arXiv2021

Introduction to UniversalQCompiler

We introduce an open source software package UniversalQCompiler written in Mathematica that allows the decomposition of arbitrary quantum operations into a sequence of single-qubit rotations (with arbitrary rotation angles) and controlled-NOT (C-NOT) gates. Together with the existing package QI, this allows quantum information protocols to be analysed and then compiled to quantum circuits. Our decompositions are based on Phys. Rev. A 93, 032318 (2016), and hence, for generic operations, they are near optimal in terms of the number of gates required. UniversalQCompiler allows the compilation of any isometry (in particular, it can be used for unitaries and state preparation), quantum channel, positive-operator valued measure (POVM) or quantum instrument, although the run time becomes prohibitive for large numbers of qubits. The resulting circuits can be displayed graphically within Mathematica or exported to LaTeX. We also provide functionality to translate the circuits to OpenQASM, the quantum assembly language used, for instance, by the IBM Q Experience.

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arXiv2021

Quantum Circuits for Sparse Isometries

We consider the task of breaking down a quantum computation given as an isometry into C-NOTs and single-qubit gates, while keeping the number of C-NOT gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.

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arXiv2021

Analysing causal structures using Tsallis entropies

Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realisable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional) independence between classical random variables and apply these to causal structures. We find that the number of independent constraints needed to characterise the causal structure is prohibitively high such that the computations required for the standard entropy vector method cannot be employed even for small causal structures. Instead, without solving the whole problem, we find new Tsallis entropic constraints for the triangle causal structure by generalising known Shannon constraints. Our results reveal new mathematical properties of classical and quantum Tsallis entropies and highlight difficulties of using Tsallis entropies for analysing causal structures.

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arXiv2020

Arbitrarily many independent observers can share the nonlocality of a single maximally entangled qubit pair

Alice and Bob each have half of a pair of entangled qubits. Bob measures his half and then passes his qubit to a second Bob who measures again and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality with the single Alice. This scenario was introduced in [Phys. Rev. Lett. 114, 250401 (2015)] where the authors mentioned evidence that when the Bobs act independently and with unbiased inputs then at most two of them can expect to violate the CHSH inequality with Alice. Here we show that, contrary to this evidence, arbitrarily many independent Bobs can have an expected CHSH violation with the single Alice. Our proof is constructive and our measurement strategies can be generalized to work with a larger class of two-qubit states that includes all pure entangled two-qubit states. Since violation of a Bell inequality is necessary for device-independent tasks, our work represents a step towards an eventual understanding of the limitations on how much device-independent randomness can be robustly generated from a single pair of qubits.

quant-ph↗

arXiv2020

On the insufficiency of entropic inequalities for detecting non-classicality in the Bell causal structure

Classical and quantum physics impose different constraints on the joint probability distributions of observed variables in a causal structure. These differences mean that certain correlations can be certified as non-classical, which has both foundational and practical importance. Rather than working with the probability distribution itself, it can instead be convenient to work with the entropies of the observed variables. In the Bell causal structure with two inputs and outputs per party, a technique that uses entropic inequalities is known that can always identify non-classical correlations. Here we consider the analogue of this technique in the generalization of this scenario to more outcomes. We identify a family of non-classical correlations in the Bell scenario with two inputs and three outputs per party whose non-classicality cannot be detected through the direct analogue of the previous technique. We also show that use of Tsallis entropy instead of Shannon entropy does not help in this case. Furthermore, we give evidence that natural extensions of the technique also do not help. More precisely, our evidence suggests that even if we allow the observed correlations to be post-processed according to a natural class of non-classicality non-generating operations, entropic inequalities for either the Shannon or Tsallis entropies cannot detect the non-classicality, and hence that entropic inequalities are generally not sufficient to detect non-classicality in the Bell causal structure. In addition, for the bipartite Bell scenario with two inputs and three outputs we find the vertex description of the polytope of non-signalling distributions that satisfy all of the CHSH-type inequalities, which is one of the main regions of investigation in this work.

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arXiv2020

Quantum Circuits for Isometries

We consider the decomposition of arbitrary isometries into a sequence of single-qubit and Controlled-NOT (C-NOT) gates. In many experimental architectures, the C-NOT gate is relatively 'expensive' and hence we aim to keep the number of these as low as possible. We derive a theoretical lower bound on the number of C-NOT gates required to decompose an arbitrary isometry from m to n qubits, and give three explicit gate decompositions that achieve this bound up to a factor of about two in the leading order. We also perform some bespoke optimizations for certain cases where m and n are small. In addition, we show how to apply our result for isometries to give decomposition schemes for arbitrary quantum operations and POVMs via Stinespring's theorem. These results will have an impact on experimental efforts to build a quantum computer, enabling them to go further with the same resources.

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arXiv2019

Smooth manifold structure for extreme channels

A quantum channel from a system $A$ of dimension $d_A$ to a system $B$ of dimension $d_B$ is a completely positive trace-preserving map from complex $d_A\times d_A$ to $d_B\times d_B$ matrices, and the set of all such maps with Kraus rank $r$ has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the $rd_B\times d_A$ dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a fixed rank $r$. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system $A$ to system $B$ of a fixed Kraus rank $r$ is a smooth submanifold of dimension $2rd_Ad_B-d_A^2-r^2$ of the set of all Choi-states of rank $r$. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from $A$ to $B$ arbitrarily well.

math-ph↗

arXiv2019

A framework for quantum-secure device-independent randomness expansion

A device-independent randomness expansion protocol aims to take an initial random seed and generate a longer one without relying on details of how the devices operate for security. A large amount of work to date has focussed on a particular protocol based on spot-checking devices using the CHSH inequality. Here we show how to derive randomness expansion rates for a wide range of protocols, with security against a quantum adversary. Our technique uses semidefinite programming and a recent improvement of the entropy accumulation theorem. To support the work and facilitate its use, we provide code that can generate lower bounds on the amount of randomness that can be output based on the measured quantities in the protocol. As an application, we give a protocol that robustly generates up to two bits of randomness per entangled qubit pair, which is twice that established in existing analyses of the spot-checking CHSH protocol in the low noise regime.

quant-ph↗

arXiv2019

Bell Inequalities From No-Signalling Distributions

A Bell inequality is a constraint on a set of correlations whose violation can be used to certify non-locality. They are instrumental for device-independent tasks such as key distribution or randomness expansion. In this work we consider bipartite Bell inequalities where two parties have $m_A$ and $m_B$ possible inputs and give $n_A$ and $n_B$ possible outputs, referring to this as the $(m_A, m_B, n_A, n_B)$ scenario. By exploiting knowledge of the set of extremal no-signalling distributions, we find all 175 Bell inequality classes in the (4, 4, 2, 2) scenario, as well as providing a partial list of 18277 classes in the (4, 5, 2, 2) scenario. We also use a probabilistic algorithm to obtain 5 classes of inequality in the (2, 3, 3, 2) scenario, which we confirmed to be complete, 25 classes in the (3, 3, 2, 3) scenario, and a partial list of 21170 classes in the (3, 3, 3, 3) scenario. Our inequalities are given in supplementary files. Finally, we discuss the application of these inequalities to the detection loophole problem, and provide new lower bounds on the detection efficiency threshold for small numbers of inputs and outputs.

quant-ph↗

arXiv2018

Analysing causal structures with entropy

A central question for causal inference is to decide whether a set of correlations fit a given causal structure. In general, this decision problem is computationally infeasible and hence several approaches have emerged that look for certificates of compatibility. Here we review several such approaches based on entropy. We bring together the key aspects of these entropic techniques with unified terminology, filling several gaps and establishing new connections regarding their relation, all illustrated with examples. We consider cases where unobserved causes are classical, quantum and post-quantum and discuss what entropic analyses tell us about the difference. This has applications to quantum cryptography, where it can be crucial to eliminate the possibility of classical causes. We discuss the achievements and limitations of the entropic approach in comparison to other techniques and point out the main open problems.

quant-ph↗

arXiv2018

Non-Shannon inequalities in the entropy vector approach to causal structures

A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.

quant-ph↗

arXiv2018

Keyring models: an approach to steerability

If a measurement is made on one half of a bipartite system, then, conditioned on the outcome, the other half has a new reduced state. If these reduced states defy classical explanation -- that is, if shared randomness cannot produce these reduced states for all possible measurements -- the bipartite state is said to be steerable. Determining which states are steerable is a challenging problem even for low dimensions. In the case of two-qubit systems a criterion is known for T-states (that is, those with maximally mixed marginals) under projective measurements. In the current work we introduce the concept of keyring models -- a special class of local hidden state models. When the measurements made correspond to real projectors, these allow us to study steerability beyond T-states. Using keyring models, we completely solve the steering problem for real projective measurements when the state arises from mixing a pure two-qubit state with uniform noise. We also give a partial solution in the case when the uniform noise is replaced by independent depolarizing channels.

quant-ph↗

arXiv2017

A system's wave function is uniquely determined by its underlying physical state

We address the question of whether the quantum-mechanical wave function $Ψ$ of a system is uniquely determined by any complete description $Λ$ of the system's physical state. We show that this is the case if the latter satisfies a notion of "free choice". This notion requires that certain experimental parameters---those that according to quantum theory can be chosen independently of other variables---retain this property in the presence of $Λ$. An implication of this result is that, among all possible descriptions $Λ$ of a system's state compatible with free choice, the wave function $Ψ$ is as objective as $Λ$.

quant-ph↗

arXiv2017

Quantum Circuits for Quantum Channels

We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of Controlled-NOT (C-NOT) gates required (single-qubit gates are free). We consider three different models. In the first, the Quantum Circuit Model (QCM), we consider sequences of single-qubit and C-NOT gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of C-NOT gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from m qubits to n qubits that uses at most one ancilla and has a low C-NOT count. We give explicit examples for small numbers of qubits that provide the lowest known C-NOT counts.

quant-ph↗

arXiv2016

Inability of the entropy vector method to certify nonclassicality in linelike causal structures

Bell's theorem shows that our intuitive understanding of causation must be overturned in light of quantum correlations. Nevertheless, quantum mechanics does not permit signalling and hence a notion of cause remains. Understanding this notion is not only important at a fundamental level, but also for technological applications such as key distribution and randomness expansion. It has recently been shown that a useful way to decide which classical causal structures could give rise to a given set of correlations is to use entropy vectors. These are vectors whose components are the entropies of all subsets of the observed variables in the causal structure. The entropy vector method employs causal relationships among the variables to restrict the set of possible entropy vectors. Here, we consider whether the same approach can lead to useful certificates of non-classicality within a given causal structure. Surprisingly, we find that for a family of causal structures that include the usual bipartite Bell structure they do not. For all members of this family, no function of the entropies of the observed variables gives such a certificate, in spite of the existence of nonclassical correlations. It is therefore necessary to look beyond entropy vectors to understand cause from a quantum perspective.

quant-ph↗

arXiv2013

Memory Attacks on Device-Independent Quantum Cryptography

Device-independent quantum cryptographic schemes aim to guarantee security to users based only on the output statistics of any components used, and without the need to verify their internal functionality. Since this would protect users against untrustworthy or incompetent manufacturers, sabotage or device degradation, this idea has excited much interest, and many device-independent schemes have been proposed. Here we identify a critical weakness of device-independent protocols that rely on public communication between secure laboratories. Untrusted devices may record their inputs and outputs and reveal information about them via publicly discussed outputs during later runs. Reusing devices thus compromises the security of a protocol and risks leaking secret data. Possible defences include securely destroying or isolating used devices. However, these are costly and often impractical. We propose other more practical partial defences as well as a new protocol structure for device-independent quantum key distribution that aims to achieve composable security in the case of two parties using a small number of devices to repeatedly share keys with each another (and no other party).

quant-ph↗

arXiv2013

The completeness of quantum theory for predicting measurement outcomes

The predictions that quantum theory makes about the outcomes of measurements are generally probabilistic. This has raised the question whether quantum theory can be considered complete, or whether there could exist alternative theories that provide improved predictions. Here we review recent work that considers arbitrary alternative theories, constrained only by the requirement that they are compatible with a notion of "free choice" (defined with respect to a natural causal order). It is shown that quantum theory is "maximally informative", i.e., there is no other compatible theory that gives improved predictions. Furthermore, any alternative maximally informative theory is necessarily equivalent to quantum theory. This means that the state a system has in such a theory is in one-to-one correspondence with its quantum-mechanical state (the wave function). In this sense, quantum theory is complete.

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