arXiv subjects

Santiago Llorens

Publications and source records attributed to Santiago Llorens.

9 recordsLinked to original sources

arXiv2026

Counterexamples to the fractional coloring conjecture for triply efficient shadow tomography

Fractional graph colorings are useful for the Shadow tomography of Pauli observables. In practice, it is desirable that any experimentally interesting set of Pauli operators has a small fractional chromatic number $χ_{f}$ for its anticommutation graph. Conjecture 13 in King, Gosset, Kothari, and Babbush [PRX Quantum 6, 010336 (2025)] states that if $B_ε(\varrho)$ is the set of Pauli observables having expectation value magnitude at least $ε$ in some given quantum state $\varrho$, then the fractional chromatic number of the anticommutation graph $G$ induced by $B_ε(\varrho)$ is $O(ε^{-2})$. In other words, it asserts that there exists a constant $C$ such that $χ_{f} \cdot ε^2 \leq C$ on all states and graphs. If the conjecture were true, it would imply that there exists a triply efficient Pauli shadow tomography algorithm for {\it any} subset $S$ of Pauli observables, provided that there is also an efficient fractional coloring algorithm for the set $B_ε$. Here we show that the conjecture is false by constructing a family of states and observables for which no finite $C$ satisfying the bound exists. We also give a more general construction relying on the commutation index or $β$ number of a graph. The key ingredient in the proofs can be seen as an instance of the amplification trick, where fractional chromatic numbers, $β$ numbers, and expectation values are amplified through lexicographic graph products.

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arXiv2026

Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions

The state of a finite-dimensional quantum system is described by a density matrix that can be decomposed into a real diagonal, a real off-diagonal and and an imaginary off-diagonal part. The latter plays a peculiar role. While it is intuitively clear that some of the imaginary coordinates cannot have the same extension as their real counterparts the precise relation is not obvious. We give a complete characterization of the constraints in terms of tight inequalities for real and imaginary Bloch-type coordinates. Our description entails a three-dimensional Bloch ball-type model for the state space. We uncover a surprising qualitative difference for the state-space boundaries in even and odd dimensions.

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arXiv2026

Combining moment matrices, symmetric extension, and Lovász theta: $Φ_{\text{E8}}$ is entangled

We solve an open problem in entanglement theory posed by Yu et al., {\it Nature Communications 12, 1012 (2021)}. The problem is to show, via an entanglement witness, that the $14$-qubit state $Φ_{\text{E8}}$ is entangled. Inspired by a method from quantum codes, we combine symmetric extension with moment matrices to prove that $Φ_{\text{E8}}$ is entangled. The proof has the form of a rational infeasibility certificate for a semidefinite program, yielding an explicit entanglement witness. Our approach unifies and extends several earlier methods that involve the Lovász theta number of the Pauli anti-commutativity graph, promising scalability and flexibility in further applications.

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arXiv2026

Exact identification of unknown unitary processes

The accurate identification of faulty hardware is a fundamental requirement for reliable quantum information processing. We address this problem in a quantum setting, where a series of $n$ devices is intended to apply the same unitary operation, but $k$ malfunctioning devices among them apply a different, unknown unitary action. Under the assumption of complete ignorance regarding the specific unitary transformation applied, we model our hypotheses using representation-theoretic tools and study the zero-error protocol for identifying these faulty devices. We derive the optimal success probability for the single- and two-anomaly scenarios, demonstrating that it is independent of the total number of devices in the series. Furthermore, we present a simple protocol that makes use of ancillary systems that achieves this optimal limit. Notably, this protocol offers significant operational advantages, such as allowing us to test each device independently. Finally, we extend our analysis to the general scenario in which both the number of anomalies and the local dimension of the systems are arbitrary, evaluating our protocol's performance and conjecturing its global optimality in the general case.

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arXiv2026

Quantum state exclusion for group-generated ensembles of pure states

Quantum state exclusion is the task of determining which states from a given set a system was not prepared in. We provide a complete solution to optimal quantum state exclusion for arbitrary sets of pure states generated by finite groups, establishing necessary and sufficient conditions for perfect (zero-error conclusive) exclusion. When perfect exclusion is impossible, we introduce two natural extensions: minimum-error and unambiguous exclusion. For both, we derive the optimal protocols and present analytical expressions for the corresponding failure probabilities and measurements, providing additional insight into how quantum states encode information.

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arXiv2026

Quantum Advantage in Identifying the Parity of Permutations with Certainty

We establish a sharp quantum advantage in determining the parity (even/odd) of an unknown permutation applied to any number $n \ge 3$ of particles. Classically, this is impossible with fewer than $n$ labels, being that the success is limited to random guessing. Quantum mechanics does it with certainty with as few as $\lceil \sqrt{n}\, \rceil$ distinguishable states per particle, thanks to entanglement. Below this threshold, not even quantum mechanics helps: both classical and quantum success are limited to random guessing. For small $n$, we provide explicit expressions for states that ensure perfect parity identification. We also assess the minimum entanglement these states need to carry, finding it to be close to maximal, and even maximal in some cases. The task requires no oracles or contrived setups and provides a simple, rigorous example of genuine quantum advantage.

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arXiv2025

Quantum Edge Detection

This paper introduces quantum edge detection, aimed at locating boundaries of quantum domains where all particles share the same pure state. Focusing on the 1D scenario of a string of particles, we develop an optimal protocol for quantum edge detection, efficiently computing its success probability through Schur-Weyl duality and semidefinite programming techniques. We analyze the behavior of the success probability as a function of the string length and local dimension, with emphasis in the limit of long strings. We present a protocol based on square root measurement, which proves asymptotically optimal. Additionally, we explore a mixed quantum change point detection scenario where the state of particles transitions from known to unknown, which may find practical applications in detecting malfunctions in quantum devices

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arXiv2024

Identification of malfunctioning quantum devices

We consider the problem of correctly identifying a malfunctioning quantum device that forms part of a network of $N$ such devices, which can be considered as the quantum analogue of classical anomaly detection. In the case where the devices in question are sources assumed to prepare identical quantum pure states, with the faulty source producing a different anomalous pure state, we show that the optimal probability of successful identification requires a global quantum measurement. We also put forth several local measurement strategies -- both adaptive and non-adaptive, that achieve the same optimal probability of success in the limit where the number of devices to be checked are large. In the case where the faulty device performs a known unitary operation we show that the use of entangled probes provides an improvement that even allows perfect identification for values of the unitary parameter that surpass a certain threshold. Finally, if the faulty device implements a known qubit channel we find that the optimal probability for detecting the position of rank-one and rank-two Pauli channels can be achieved by product state inputs and separable measurements for any size of network, whereas for rank-three and general amplitude damping channels optimal identification requires entanglement with N qubit ancillas.

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arXiv2024

Quantum multi-anomaly detection

A source assumed to prepare a specified reference state sometimes prepares an anomalous one. We address the task of identifying these anomalous states in a series of $n$ preparations with $k$ anomalies. We analyze the minimum-error protocol and the zero-error (unambiguous) protocol and obtain closed expressions for the success probability when both reference and anomalous states are known to the observer and anomalies can appear equally likely in any position of the preparation series. We find the solution using results from association schemes theory, thus establishing a connection between graph theory and quantum hypothesis testing. In particular, we use the Johnson association scheme which arises naturally from the Gram matrix of this problem. We also study the regime of large $n$ and obtain the expression of the success probability that is non-vanishing. Finally, we address the case in which the observer is blind to the reference and the anomalous states. This scenario requires a universal protocol for which we prove that in the asymptotic limit, the success probability corresponds to the average of the known state scenario.

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