arXiv subjects

Máté Farkas

Publications and source records attributed to Máté Farkas.

At least 19 recordsLinked to original sources

arXiv2026

$k$-fold unbiased measurements and maximal incompatibility

Mutually unbiased bases capture perfect complementarity between two quantum measurements. Extensions beyond pairwise unbiasedness have been proposed, but essentially no non-trivial higher-order constructions are known. We introduce $k$-fold unbiased measurements ($k$-UMs), extending the $k$-fold unbiased bases notion of [arXiv:1706.04446] from rank-one basis measurements to arbitrary-rank projective measurements, and show that this higher-rank setting supports a much richer theory. We develop the notions of algebraic and spectral $k$-UMs and prove that they coincide for rank-one measurements and triples of measurements (3-UMs). We establish strong no-go results for higher-order rank-one constructions and three-outcome 3-UMs, but obtain infinitely many higher-rank triples using Hadamard matrices and Clifford algebras. We then give these structures an exact operational interpretation in terms of measurement incompatibility. For 3-UMs with any number of outcomes, we determine their generalised incompatibility robustness exactly and construct an explicit joint measurement for their noisy versions at the compatibility threshold. Finally, we implement the symmetry reduction of the sum-of-squares hierarchy recently introduced in [New J. Phys. 28, 064509 (2026)] and give numerical evidence that the noise thresholds arising from the $k$-UM analysis may characterise the asymptotic behaviour of this hierarchy. In particular, with very high precision, we numerically show that our constructed four-outcome 3-UMs are among the most incompatible triples of four-outcome measurements.

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arXiv2026

Maximal global device-independent randomness from projective measurements in every dimension

Device-independent random number generation (DIQRNG) is the most secure form of generating private randomness using quantum physical processes. Its strength lies in producing numbers that are impossible to predict by any eavesdropper restricted by the laws of quantum theory. Moreover, security is proven solely from observed measurement statistics, without the need to characterise or trust the devices used in random number generation. Implementing DIQRNG is, however, costly, as it requires high-quality entangled systems. It is therefore important to make the best use of available resources. In this work, we show that using projective measurements -- which are most readily implementable experimentally -- one can certify $2\log(d)$ bits of device-independent randomness from a bipartite system of local dimension $d$ for every $d \ge 2$, thus reaching the theoretically maximum possible rate of DIQRNG. We provide explicit protocols reaching $2\log(d)$ bits based on mutually unbiased bases. Furthermore, we compute numerical bounds on the rate for the case of imperfect implementations, showing that our protocols are robust to experimental noise.

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arXiv2025

Self-testing tilted strategies for maximal loophole-free nonlocality

The degree of experimentally attainable nonlocality, as gauged by the loophole-free or effective violation of Bell inequalities, remains severely limited due to inefficient detectors. We address an experimentally motivated question: Which quantum strategies attain the maximal loophole-free nonlocality in the presence of inefficient detectors? For any Bell inequality and any specification of detection efficiencies, the optimal strategies are those that maximally violate a tilted version of the Bell inequality in ideal conditions. In the simplest scenario, we demonstrate that the quantum strategies that maximally violate the doubly-tilted versions of Clauser-Horne-Shimony-Holt inequality are unique up to local isometries. We utilize a Jordan's lemma and Gröbner basis-based proof technique to analytically derive self-testing statements for the entire family of doubly-tilted CHSH inequalities and numerically demonstrate their robustness. These results enable us to reveal the insufficiency of even high levels of the Navascués--Pironio--Acín hierarchy to saturate the maximum quantum violation of these inequalities.

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arXiv2025

Extending quantum correlations to arbitrary distances via parallel repetition of routed Bell tests

Applications such as Device-Independent Quantum Key Distribution (DIQKD) require loophole-free certification of long-distance quantum correlations. However, these distances remain severely constrained by detector inefficiencies and unavoidable transmission losses. To overcome this challenge, we consider parallel repetitions of the recently proposed routed Bell experiments, where transmissions from the source are actively directed either to a nearby or a distant measurement device. We analytically show that the threshold detection efficiency of the distant device--needed to certify non-jointly-measurable measurements, a prerequisite of secure DIQKD--decreases exponentially, optimally, and robustly, following $η^*=1/2^N$, with the number $N$ of parallel repetitions.

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arXiv2025

Simple and general bounds on quantum random access codes

Random access codes are a type of communication task that is widely used in quantum information science. The optimal average success probability that can be achieved through classical strategies is known for any random access code. However, only a few cases are solved exactly for quantum random access codes. In this paper, we provide bounds for the fully general setting of n independent variables, each selected from a d-dimensional classical alphabet and encoded in a D-dimensional quantum system subject to an arbitrary quantum measurement. The bound recovers the exactly known special cases, and we demonstrate numerically that even though the bound is not tight overall, it can still yield a good approximation.

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arXiv2024

Indistinguishability in general probabilistic theories

The existence of indistinguishable quantum particles provides an explanation for various physical phenomena we observe in nature. We lay out a path for the study of indistinguishable particles in general probabilistic theories (GPTs) via two frameworks: the traditional GPT framework and the diagrammatic framework of process theories. In the first approach we define different types of indistinguishable particle by the orbits of symmetric states under transformations. In the diagrammatic approach, we find a decomposition of the symmetrised state space using two key constructions from category theory: the biproduct completion and the Karoubi envelope. In both cases for pairs of indistinguishable particles in quantum theory we recover bosons and fermions.

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arXiv2024

Maximal device-independent randomness in every dimension

Random numbers are used in a wide range of sciences. In many applications, generating unpredictable private random numbers is indispensable. Device-independent quantum random number generation is a framework that makes use of the intrinsic randomness of quantum processes to generate numbers that are fundamentally unpredictable according to our current understanding of physics. While device-independent quantum random number generation is an exceptional theoretical feat, the difficulty of controlling quantum systems makes it challenging to carry out in practice. It is therefore desirable to harness the full power of the quantum degrees of freedom (the dimension) that one can control. It is known that no more than $2 \log(d)$ bits of private device-independent randomness can be extracted from a quantum system of local dimension $d$. In this paper we demonstrate that this bound can be achieved for all dimensions $d$ by providing a family of explicit protocols. In order to obtain our result, we develop new certification techniques that can be of wider interest in device-independent applications for scenarios in which complete certification ('self-testing') is impossible or impractical.

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arXiv2024

Maximal intrinsic randomness of a quantum state

One of the most counterintuitive aspects of quantum theory is its claim that there is 'intrinsic' randomness in the physical world. Quantum information science has greatly progressed in the study of intrinsic, or secret, quantum randomness in the past decade. With much emphasis on device-independent and semi-device-independent bounds, one of the most basic questions has escaped attention: how much intrinsic randomness can be extracted from a given state $ρ$, and what measurements achieve this bound? We answer this question for three different randomness quantifiers: the conditional min-entropy, the conditional von Neumann entropy and the conditional max-entropy. For the first, we solve the min-max problem of finding the projective measurement that minimises the maximal guessing probability of an eavesdropper. The result is that one can guarantee an amount of conditional min-entropy $H^*_{\textrm{min}}=-\log_2 P_{\textrm{guess}}^{*}(ρ)$ with $P_{\textrm{guess}}^{*}(ρ)=\frac{1}{d}(\textrm{tr} \sqrtρ)^2$ by performing suitable projective measurements. For the conditional von Neumann entropy, we find that the maximal value is $H^{*}= \log_{2}d-S(ρ)$, with $S(ρ)$ the von Neumann entropy of $ρ$, while for the conditional max-entropy, we find the maximal value $H^{*}_\textrm{max}=\log_{2}d + \log_{2}λ_{\textrm{max}}(ρ)$, where $λ_{\textrm{max}}(ρ)$ is the largest eigenvalue of $ρ$. Optimal values for $H^{*}_{\textrm{min}}$, $H^{*}$ and $H^{*}_\textrm{max}$ are achieved by measuring in any basis that is unbiased to the eigenbasis of $ρ$, as well as by other, less intuitive, measurements.

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arXiv2023

An invertible map between Bell non-local and contextuality scenarios

We present an invertible map between correlations in any bipartite Bell scenario and behaviours in a family of contextuality scenarios. The map takes local, quantum and non-signalling correlations to non-contextual, quantum and contextual behaviours, respectively. Consequently, we find that the membership problem of the set of quantum contextual behaviours is undecidable, the set cannot be fully realised via finite dimensional quantum systems and is not closed. Finally, we show that neither this set nor its closure is the limit of a sequence of computable supersets, due to the result MIP*=RE.

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arXiv2023

Upper bounds on key rates in device-independent quantum key distribution based on convex-combination attacks

The device-independent framework constitutes the most pragmatic approach to quantum protocols that does not put any trust in their implementations. It requires all claims, about e.g. security, to be made at the level of the final classical data in hands of the end-users. This imposes a great challenge for determining attainable key rates in device-independent quantum key distribution (DIQKD), but also opens the door for consideration of eavesdropping attacks that stem from the possibility of a given data being just generated by a malicious third-party. In this work, we explore this path and present the convex-combination attack as an efficient, easy-to-use technique for upper-bounding DIQKD key rates. It allows verifying the accuracy of lower bounds on key rates for state-of-the-art protocols, whether involving one-way or two-way communication. In particular, we demonstrate with its help that the currently predicted constraints on the robustness of DIQKD protocols to experimental imperfections, such as the finite visibility or detection efficiency, are already very close to the ultimate tolerable thresholds.

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arXiv2023

Unbounded device-independent quantum key rates from arbitrarily small non-locality

Device-independent quantum key distribution allows for proving the security of a shared cryptographic key between two distant parties with potentially untrusted devices. The security proof is based on the measurement outcome statistics (correlation) of a Bell experiment, and security is guaranteed by the laws of quantum theory. While it is known that the observed correlation must be Bell non-local in order to prove security, recent results show that Bell non-locality is in general not sufficient for standard device-independent quantum key distribution. In this work, we show that conversely, there is no lower bound on the amount of non-locality that is sufficient for device-independent quantum key distribution. Even more so, we show that from certain correlations that exhibit arbitrarily small non-locality, one can still extract unbounded device-independent key rates. Therefore, a quantitative relation between device-independent key rates and Bell non-locality cannot be drawn in general. Our main technique comprises a rigorous connection between self-testing and device-independent quantum key distribution, applied to a recently discovered family of Bell inequalities with arbitrarily many measurement outcomes.

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arXiv2023

Mutually Unbiased Measurements, Hadamard Matrices, and Superdense Coding

Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalization of MUBs called mutually unbiased measurements (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterization of MUMs that are direct sums of MUBs. We then proceed to construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for these construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that -- in stark contrast with MUBs -- the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with d outcomes defines a d-dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.

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arXiv2023

Contextuality and inductive bias in quantum machine learning

Generalisation in machine learning often relies on the ability to encode structures present in data into an inductive bias of the model class. To understand the power of quantum machine learning, it is therefore crucial to identify the types of data structures that lend themselves naturally to quantum models. In this work we look to quantum contextuality -- a form of nonclassicality with links to computational advantage -- for answers to this question. We introduce a framework for studying contextuality in machine learning, which leads us to a definition of what it means for a learning model to be contextual. From this, we connect a central concept of contextuality, called operational equivalence, to the ability of a model to encode a linearly conserved quantity in its label space. A consequence of this connection is that contextuality is tied to expressivity: contextual model classes that encode the inductive bias are generally more expressive than their noncontextual counterparts. To demonstrate this, we construct an explicit toy learning problem -- based on learning the payoff behaviour of a zero-sum game -- for which this is the case. By leveraging tools from geometric quantum machine learning, we then describe how to construct quantum learning models with the associated inductive bias, and show through our toy problem that they outperform their corresponding classical surrogate models. This suggests that understanding learning problems of this form may lead to useful insights about the power of quantum machine learning.

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arXiv2022

Characterizing and quantifying the incompatibility of quantum instruments

Incompatibility of quantum devices is one of the cornerstones of quantum theory, and the incompatibility of quantum measurements and channels has been linked to quantum advantage in certain information theoretic tasks. In this work, we focus on the less well-explored question of the incompatibility of quantum instruments, that is, devices that describe the measurement process in its entirety, accounting for both the classical measurement outcome and the quantum post-measurement state. In particular, we focus on the recently introduced notion of parallel compatibility of instruments, which has been argued to be a natural notion of instrument compatibility. We introduce -- similarly to the case of measurements and channels -- the incompatibility robustness of quantum instruments and derive universal bounds on it. We then prove that post-processing of quantum instruments is a free operation for parallel compatibility. Last, we provide families of instruments for which our bounds are tight, and families of compatible indecomposable instruments.

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arXiv2022

Three numerical approaches to find mutually unbiased bases using Bell inequalities

Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old conjecture, known as Zauner's conjecture, stating that there exist at most three. Here we tackle Zauner's conjecture numerically through the construction of Bell inequalities for every pair of integers $n,d \ge 2$ that can be maximally violated in dimension $d$ if and only if $n$ MUBs exist in that dimension. Hence we turn Zauner's conjecture into an optimisation problem, which we address by means of three numerical methods: see-saw optimisation, non-linear semidefinite programming and Monte Carlo techniques. All three methods correctly identify the known cases in low dimensions and all suggest that there do not exist four mutually unbiased bases in dimension six, with all finding the same bases that numerically optimise the corresponding Bell inequality. Moreover, these numerical optimisers appear to coincide with the "four most distant bases" in dimension six, found through numerically optimising a distance measure in [P. Raynal, X. Lü, B.-G. Englert, Phys. Rev. A, 83 062303 (2011)]. Finally, the Monte Carlo results suggest that at most three MUBs exist in dimension ten.

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arXiv2022

On the compatibility of quantum instruments

Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. While the incompatibility of measurements and quantum channels is well-studied, the incompatibility of quantum instruments has not been explored in much detail. In this work, we revise a notion of instrument compatibility introduced in the literature that we call traditional compatibility. Then, we introduce the new notion of parallel compatibility, and show that these two notions are inequivalent. Then, we argue that the notion of traditional compatibility is incomplete, and prove that while parallel compatibility captures measurement and channel compatibility, traditional compatibility does not. Hence, we propose parallel compatibility as the conceptually complete definition of compatibility of quantum instruments.

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arXiv2021

Bell nonlocality is not sufficient for the security of standard device-independent quantum key distribution protocols

Device-independent quantum key distribution is a secure quantum cryptographic paradigm that allows two honest users to establish a secret key, while putting minimal trust in their devices. Most of the existing protocols have the following structure: first, a bipartite nonlocal quantum state is distributed between the honest users, who perform local measurements to establish nonlocal correlations. Then, they announce the implemented measurements and extract a secure key by post-processing their measurement outcomes. We show that no protocol of this form allows for establishing a secret key when implemented on any correlation obtained by measuring local projective measurements on certain entangled nonlocal states, namely on a range of entangled two-qubit Werner states. To prove this result, we introduce a technique for upper-bounding the asymptotic key rate of device-independent quantum key distribution protocols, based on a simple eavesdropping attack. Our results imply that either different reconciliation techniques are needed for device-independent quantum key distribution in the large-noise regime, or Bell nonlocality is not sufficient for this task.

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arXiv2021

Characterising and bounding the set of quantum behaviours in contextuality scenarios

The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, to circumvent the inadequacy of pure states and projective measurements in contextuality scenarios, we present a novel unitary operator based semidefinite relaxation technique. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities and identifying novel maximally contextual quantum strategies. To further illustrate the versatility of these relaxations, we demonstrate monogamy of preparation contextuality in a tripartite setting, and present a secure semi-device independent quantum key distribution scheme powered by quantum advantage in parity oblivious random access codes.

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