arXiv subjects

Marco Tomamichel

Publications and source records attributed to Marco Tomamichel.

At least 37 records · Page 2Linked to original sources

arXiv2025

A Unified Approach to Quantum Contraction and Correlation Coefficients

The maximal correlation coefficient measures the linear correlation in a bipartite distribution and contraction coefficients measure how much information is lost under a noisy channel. Remarkably, Raginsky established a close relation between these two concepts by showing that the $\chi^2$ contraction coefficient equals the maximal correlation coefficient of the joint input/output distribution of the channel. In quantum theory, several generalizations of these concepts have been proposed, but none recover all the classical properties. Here we construct a framework in which the classical theory extends to the quantum setting. We introduce families of quantum maximal correlation coefficients and show that many impose limits on converting quantum states under local operations. We establish a family of quantum contraction coefficients are efficiently computable, yielding a generic efficient algorithm for mixing times of quantum channels with a full rank fixed point. Furthermore, we establish a quantum analogue of Raginsky's classical correspondence that relates these two families of quantities. To do this, we develop the operator-theoretic approach to Petz's family of non-commutative $L^{2}(p)$ spaces that extend the data processing inequality for variance to quantum theory.

quant-ph↗

arXiv2025

Quantum state-agnostic work extraction (almost) without dissipation

We investigate work extraction protocols designed to transfer the maximum possible energy to a battery using sequential access to $N$ copies of an unknown pure qubit state. The core challenge is designing interactions to optimally balance two competing goals: charging of the battery optimally using the qubit in hand, and acquiring more information by qubit to improve energy harvesting in subsequent rounds. Here, we leverage exploration-exploitation trade-off in reinforcement learning to develop adaptive strategies achieving energy dissipation that scales only poly-logarithmically in $N$. This represents an exponential improvement over current protocols based on full state tomography.

quant-ph↗

arXiv2025

Consecutive Measurement Tradeoffs in Quantum Cryptography

Measuring a quantum system can reveal information while disturbing its state. Consequently, the probabilities of obtaining desired outcomes in individual measurements constrain the probability of obtaining them in sequence. This interplay arises naturally in mistrustful quantum cryptography, where a dishonest participant may perform sequential measurements to extract more information than the protocol allows. We develop consecutive measurement theorems (CMTs) to quantify this constraint, relating the average success probability of a single measurement to that of an ordered pair of distinct measurements. For measurements on a common state, we derive the optimal lower bound on consecutive-measurement success as a function of single-measurement success and provide matching constructions over the entire parameter range. We also establish the first robust CMTs for measurements on different but nearby states, with closeness quantified by fidelity or trace distance. We use these results to strengthen sum-binding guarantees for relativistic bit commitment and soundness bounds for relativistic zero-knowledge proofs, and to obtain improved no-go results for quantum oblivious transfer and quantum private query. Our results establish CMTs as a versatile tool for translating consecutive-measurement constraints into concrete cryptographic bounds and suggest applications in other quantum-information settings.

quant-ph↗

arXiv2025

Quantum umlaut information

We study the quantum umlaut information, a correlation measure defined for bipartite quantum states $\rho_{AB}$ as a reversed variant of the quantum mutual information: $U(A;B)_\rho = \min_{\sigma_B} D(\rho_A\otimes \sigma_B\|\rho_{AB})$ in terms of the quantum relative entropy $D$. As in the classical case [Girardi et al., arXiv:2503.18910], this definition allows for a closed-form expression and has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given bipartite state is product or not. We generalise the umlaut information to quantum channels, where it also extends the notion of `oveloh information' [Nuradha et al., arXiv:2404.16101]. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. Inspired by recent results in entanglement theory, we then show as our main result that the regularised umlaut information constitutes a fundamental measure of the quality of classical information transmission over a quantum channel -- as opposed to the capacity, which quantifies the quantity of information that can be sent. This interpretation applies to coding assisted by activated non-signalling correlations, and the channel umlaut information is in general larger than the corresponding expression for unassisted communication as obtained by Dalai for the classical-quantum case [IEEE Trans. Inf. Theory 59, 8027 (2013)]. Combined with prior works on non-signalling--assisted zero-error channel capacities, our findings imply a dichotomy between the settings of zero-rate error exponents and zero-error communication. While our results are single-letter only for classical-quantum channels, we also give a single-letter bound for fully quantum channels in terms of the `geometric' version of umlaut information.

quant-ph↗

arXiv2025

Umlaut information

The sphere-packing bound quantifies the error exponent for noisy channel coding for rates above a critical value. Here, we study the zero-rate limit of the sphere-packing bound and show that it has an intriguing single-letter form, which we call the umlaut information of the channel, inspired by the lautum information introduced by Palomar and Verd\'u. Unlike the latter quantity, we show that the umlaut information is additive for parallel uses of channels. We show that it has a twofold operational interpretation: as the zero-rate error exponent of non-signalling-assisted coding on the one hand, and as the zero-rate error exponent of list decoding in the large list limit on the other.

cs.IT↗

arXiv2025

Some properties and applications of the new quantum $f$-divergences

Recently, a new definition for quantum $f$-divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum $f$-divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum $f$-divergences. We leverage these expressions to prove new properties of these $f$-divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new $f$-divergences, and prove inequalities between these divergences for various functions.

quant-ph↗

arXiv2024

String commitment from unstructured noise

Noisy channels are a foundational resource for constructing cryptographic primitives such as string commitment and oblivious transfer. The noisy channel model has been extended to unfair noisy channels, where adversaries can influence the parameters of a memoryless channel. In this work, we introduce the unstructured noisy channel model as a generalization of the unfair noisy channel model to allow the adversary to manipulate the channel arbitrarily subject to certain entropic constraints. We present a string commitment protocol with established security and derive its achievable commitment rate, demonstrating the feasibility of commitment against this stronger class of adversaries. Furthermore, we show that the entropic constraints in the unstructured noisy channel model can be derived from physical assumptions such as noisy quantum storage. Our work thus connects two distinct approaches to commitment, i.e., the noisy channel and physical limitations.

cs.IT↗

arXiv2024

Quantum conditional entropies from convex trace functionals

We study geometric properties of trace functionals that generalize those in [Zhang, Adv. Math. 365:107053 (2020)], arising from a novel family of conditional entropies with applications in quantum information. Building on new convexity results for these functionals, we establish data-processing inequalities and additivity properties for our entropies, demonstrating their operational significance. We further prove completeness under duality, chain rules, and various monotonicity properties for this family. Our proofs draw on tools from complex interpolation theory, multivariate Araki--Lieb and Lieb--Thirring inequalities, variational characterizations of trace functionals, and spectral pinching techniques.

quant-ph↗

arXiv2024

Sample Optimal and Memory Efficient Quantum State Tomography

Quantum state tomography is the fundamental physical task of learning a complete classical description of an unknown state of a quantum system given coherent access to many identical samples of it. The complexity of this task is commonly characterised by its sample-complexity: the minimal number of samples needed to reach a certain target precision of the description. While the sample complexity of quantum state tomography has been well studied, the memory complexity has not been investigated in depth. Indeed, the bottleneck in the implementation of na\"ive sample-optimal quantum state tomography is its massive quantum memory requirements. In this work, we propose and analyse a quantum state tomography algorithm which retains sample-optimality but is also memory-efficient. Our work is built on a form of unitary Schur sampling and only requires streaming access to the samples.

quant-ph↗

arXiv2024

Error exponent of activated non-signaling assisted classical-quantum channel coding

We provide a tight asymptotic characterization of the error exponent for classical-quantum channel coding assisted by activated non-signaling correlations. Namely, we find that the optimal exponent--also called reliability function--is equal to the well-known sphere packing bound, which can be written as a single-letter formula optimized over Petz-R\'enyi divergences. Remarkably, there is no critical rate and as such our characterization remains tight for arbitrarily low rates below the capacity. On the achievability side, we further extend our results to fully quantum channels. Our proofs rely on semi-definite program duality and a dual representation of the Petz-R\'enyi divergences via Young inequalities.

quant-ph↗

arXiv2024

Fundamental Limit on the Power of Entanglement Assistance in Quantum Communication

The optimal rate of reliable communication over a quantum channel can be enhanced by pre-shared entanglement. Whereas the enhancement may be unbounded in infinite-dimensional settings even when the input power is constrained, a long-standing conjecture asserts that the ratio between the entanglement-assisted and unassisted classical capacities is bounded in finite-dimensional settings [Bennett et al., IEEE Trans. Inf. Theory 48, 2637 (2002)]. In this work, we prove this conjecture by showing that their ratio is upper bounded by $o(d^2)$, where $d$ is the input dimension of the channel. An application to quantum communication with noisy encoders and decoders is given.

quant-ph↗

arXiv2024

Continuity of entropies via integral representations

We show that Frenkel's integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes-Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of 'filtered' relative entropy distances.

quant-ph↗

arXiv2024

Matrix Majorization in Large Samples with Varying Support Restrictions

We say that a matrix $P$ with non-negative entries majorizes another such matrix $Q$ if there is a stochastic matrix $T$ such that $Q=TP$. We study matrix majorization in large samples and in the catalytic regime in the case where the columns of the matrices need not have equal support, as has been assumed in earlier works. We focus on two cases: either there are no support restrictions (except for requiring a non-empty intersection for the supports) or the final column dominates the others. Using real-algebraic methods, we identify sufficient and almost necessary conditions for majorization in large samples or when using catalytic states under these support conditions. These conditions are given in terms of multivariate divergences that generalize the R\'enyi divergences. We notice that varying support conditions dramatically affect the relevant set of divergences. Our results find an application in the theory of catalytic state transformation in quantum thermodynamics.

math.ST↗

arXiv2024

Learning pure quantum states (almost) without regret

We initiate the study of sample-optimal quantum state tomography with minimal disturbance to the samples. Can we efficiently learn a precise description of a quantum state through sequential measurements of samples while at the same time making sure that the post-measurement state of the samples is only minimally perturbed? Defining regret as the cumulative disturbance of all samples, the challenge is to find a balance between the most informative sequence of measurements on the one hand and measurements incurring minimal regret on the other. Here we answer this question for qubit states by exhibiting a protocol that for pure states achieves maximal precision while incurring a regret that grows only polylogarithmically with the number of samples, a scaling that we show to be optimal.

quant-ph↗

arXiv2024

Quantum Channel Simulation in Fidelity is no more difficult than State Splitting

Characterizing the minimal communication needed for the quantum channel simulation is a fundamental task in the quantum information theory. In this paper, we show that, in fidelity, the quantum channel simulation can be directly achieved via quantum state splitting without using a technique known as the de~Finetti reduction, and thus provide a pair of tighter one-shot bounds. Using the bounds, we also recover the quantum reverse Shannon theorem in a much simpler way.

quant-ph↗

arXiv2024

On the composable security of weak coin flipping

Weak coin flipping is a cryptographic primitive in which two mutually distrustful parties generate a shared random bit to agree on a winner via remote communication. While a stand-alone secure weak coin flipping protocol can be constructed from noiseless quantum communication channels, its composability remains unexplored. In this work, we demonstrate that no weak coin flipping protocol can be abstracted as a simple black-box resource with composable security. Despite this, we also establish the overall stand-alone security of quantum weak coin flipping protocols under composition in sequential order.

quant-ph↗

arXiv2024

Linear bandits with polylogarithmic minimax regret

We study a noise model for linear stochastic bandits for which the subgaussian noise parameter vanishes linearly as we select actions on the unit sphere closer and closer to the unknown vector. We introduce an algorithm for this problem that exhibits a minimax regret scaling as $\log^3(T)$ in the time horizon $T$, in stark contrast the square root scaling of this regret for typical bandit algorithms. Our strategy, based on weighted least-squares estimation, achieves the eigenvalue relation $\lambda_{\min} ( V_t ) = \Omega (\sqrt{\lambda_{\max}(V_t ) })$ for the design matrix $V_t$ at each time step $t$ through geometrical arguments that are independent of the noise model and might be of independent interest. This allows us to tightly control the expected regret in each time step to be of the order $O(\frac1{t})$, leading to the logarithmic scaling of the cumulative regret.

cs.LG↗

arXiv2023

A fixed-point algorithm for matrix projections with applications in quantum information

We develop a fixed-point iterative algorithm that computes the matrix projection with respect to the Bures distance on the set of positive definite matrices that are invariant under some symmetry. We prove that the fixed-point iteration algorithm converges exponentially fast to the optimal solution in the number of iterations. Moreover, it numerically shows fast convergence compared to the off-the-shelf semidefinite program solvers. Our algorithm, for the specific case of Bures-Wasserstein barycenter, recovers the fixed-point iterative algorithm originally introduced in (\'Alvarez-Esteban et al., 2016). Our proof is concise and relies solely on matrix inequalities. Finally, we discuss several applications of our algorithm in quantum resource theories and quantum Shannon theory.

quant-ph↗