arXiv subjects

Marco Tomamichel

Publications and source records attributed to Marco Tomamichel.

At least 37 records · Page 2Linked to original sources

arXiv2026

A strong converse for stabilizer codes over Pauli channels via the blowing-up lemma

We prove a strong converse for quantum communication over Pauli channels within the class of stabilizer codes. If a code whose code space is a full joint eigenspace of a stabilizer group transmits above the coherent information of its own input state, its entanglement fidelity decays exponentially in the block length; the encoder may be any isometry onto that space and the decoder any channel. For memoryless channels this determines the $\varepsilon$-quantum capacity of the class for every $\varepsilon < 1$, so that tolerating a constant error buys no rate; for antidegradable channels, such as the depolarizing channel with error probability $p \in [1/4, 3/4]$, that capacity is zero, while for $p \in [1/4,1/2)$ partial-transposition bounds provably cannot certify a strong converse. The proof uses neither additivity assumptions nor semidefinite relaxations: optimal decoding succeeds precisely on an event in a product probability space, so the blowing-up lemma of Ahlswede, Gács and Körner applies, and the side information it produces is charged against the coherent information. The argument also constrains near-deterministic decoding for codes of any kind, and we isolate the encoder-side statement that would extend it to all of them.

quant-ph↗

arXiv2026

Hockey stick $f$-divergences

In this paper we give a systematic and unified treatment and extensions of various results on a new notion of quantum $f$-divergences defined from quantum hockey stick divergences, the theory of which has been developed recently in \cite{BHT_fdiv,HircheTomamichel_integral,LiuHircheCheng2025}. In particular, we consider non-normalized states and hockey stick $f$-divergences defined from more general notions of quantum hockey stick divergences, as well as a somewhat more general form of the integral representation defined in terms of an additional real parameter. We also consider the extension of the theory to general von Neumann algebras, and extend various results from \cite{HircheTomamichel_integral,LiuHircheCheng2025} to this setting. Our main results here are the representation of the hockey stick $f$-divergences in terms of Neyman-Pearson error probabilities, which was given in the finite-dimensional case in \cite{LiuHircheCheng2025}, an extension of Jen\v cová's result \cite{Jencova2023} on the detection of reversibility of a quantum channel on a pair of states in terms of the hockey stick divergences, and an extension of a result in \cite{HircheTomamichel_integral} showing that the regularized hockey stick Rényi $α$-divergences coincide with the Petz-type Rényi divergences for $α\in(0,1)$ and with the sandwiched Rényi divergences for $α>1$. Moreover, we give some partial results on the characterization of when different notions of quantum $f$-divergences give the same value on a pair of quantum states.

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arXiv2026

Partially smoothed information measures

Smooth entropies are a tool for quantifying resource trade-offs in (quantum) information theory and cryptography. In typical bi- and multi-partite problems, however, some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states. We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed. This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting. In particular, we immediately get asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting. For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well.

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arXiv2026

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.

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arXiv2026

Additivity of quantum relative entropies as a single-copy criterion

The fundamental goal of information theory is to characterize complex operational tasks using efficiently computable information quantities, Shannon's capacity formula being the prime example of this. However, many tasks in quantum information can only be characterized by regularized entropic measures that are often not known to be computable and for which efficient approximations are scarce. It is thus of fundamental importance to understand when regularization is not needed, opening the door to an efficiently computable characterization based on additive quantities. Here, we demonstrate that for a large class of problems, the question of whether regularization is needed or not can be determined at the single-copy level. Specifically, we demonstrate that regularization of the Umegaki relative entropy, along with related quantities such as the Petz and sandwiched relative entropies, is not needed if and only if a single-copy optimizer satisfies a certain property. These problems include hypothesis testing with arbitrarily varying hypotheses as well as quantum resource theories used to derive fundamental bounds for entanglement and magic state distillation. We derive the Stein, Chernoff, and Hoeffding exponents for these problems and establish necessary and sufficient conditions for their additivity, while also presenting partial results for the strong converse exponent.

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arXiv2026

Reliability of asymptotic work extraction

Extracting work from quantum states is a fundamental task in quantum thermodynamics. Previous studies have primarily focused on determining the best achievable rate of work extraction, and remarkably, this characterization appeared to remain unchanged regardless of the choice of allowed processes: whether one considers the operationally motivated class of energy-conserving thermal operations, or the axiomatic class of Gibbs-preserving operations, the optimal extractable work is given by the Helmholtz free energy. Here, we challenge this perspective, showing that a more refined analysis of the asymptotic performance of work extraction reveals significant differences in the performance for the two different classes of free operations. Precisely, we focus on the trade-off between the extraction rate and its reliability, characterized by the optimal asymptotic speed at which the extraction error can be suppressed. We establish that the reliability of Gibbs-preserving operations and of thermal operations are respectively characterized by the Petz and the sandwiched Rényi relative entropies, demonstrating that the former in general strictly outperforms the latter, and providing new interpretations of several information-theoretic divergences. Our analysis reveals that operational constraints such as energy conservation impose stronger limitations on the achievable precision of quantum tasks than can be inferred from their asymptotic rates, thereby questioning the use of Gibbs-preserving operations as a mathematically convenient substitute for the physically realizable thermal processes.

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arXiv2026

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 naïve 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.

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arXiv2026

Learning Pure Quantum States in Any Dimension (Almost) Without Regret

We extend quantum state tomography with minimal cumulative disturbance, first investigated in [arXiv:2406.18370], to arbitrary finite-dimensional pure states. A learner sequentially receives fresh copies of an unknown pure state, chooses a rank-one projector for each copy using the previous outcomes, and performs the corresponding two-outcome projective measurement. The goal is to learn the state while keeping the chosen projectors close to the unknown state in order to minimize disturbance. The qubit solution relies on the special geometry of the Bloch sphere and does not extend directly to qudits, where pure states form a curved manifold. We show that this obstruction can be overcome by working locally on the pure-state manifold. The algorithm proceeds in epochs. In each epoch, it fixes a current estimate, measures pairs of nearby rank-one projectors obtained by moving in opposite tangent directions, and takes differences of the corresponding outcomes. This gives an exact linear observation of the tangent component of the error. The resulting local linear models are combined with a robust variance-adaptive estimator and a hot-start regularization that transfers precision across epochs. For every unknown pure state in dimension $d$, after $T$ measured copies, our protocol achieves cumulative regret $\mathcal{O}(d^3\log^2 T)$, and at each intermediate time $t\leq T$ its current estimate has online infidelity $\mathcal{O}(d^3\log(T)/t)$. Hence, pure-state tomography with essentially no cumulative disturbance is not a peculiarity of qubits but a geometric phenomenon that persists for qudits.

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arXiv2026

Tight Contraction Rates for Primitive Channels under Quantum $f$-Divergences

Data-processing inequalities capture the phenomenon that two probability distributions can only become less distinguishable under any common post-processing. For more fine-grained inequalities, one turns to strong data-processing inequality (SDPI) constants, which give the strongest inequalities for a given channel and reference state for a fixed measure of distinguishability. These quantities have been used to quantify the rate at which time-homogeneous Markov chains contract towards a fixed point both in the classical and quantum setting. In this work, we establish that quantum $f$-divergences satisfy a local reverse Pinsker inequality, which implies the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative $χ^2$-divergence. Using quantum-detailed balance, we establish a sufficient condition for these bounds to be tight. Finally, we apply these results to Petz, Matsumoto, and Hirche-Tomamichel $f$-divergences, establishing new and strengthening previously known results.

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arXiv2026

Conditions for Large-Sample Majorization of Pairs of Flat States in Terms of $α$-z Relative Entropies

We offer the first operational interpretation of the $α$-z relative entropies, a measure of distinguishability between two quantum states introduced by Jakšić et al. and Audenaert and Datta. We show that these relative entropies appear when formulating conditions for large-sample or catalytic relative majorization of pairs of flat states and certain generalizations of them. Indeed, we show that such transformations exist if and only if all the $α$-z relative entropies for $α$<1 of the two pairs are ordered. In this setting, the $α$ and z parameters are truly independent from each other. These results also yield an expression for the optimal rate of converting one flat state pair into another. Our methods use real-algebraic techniques involving preordered semirings and certain monotone homomorphisms and derivations on them.

quant-ph↗

arXiv2026

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 $χ^2$ 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.

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arXiv2026

Projections with Respect to Bures Distance and Fidelity: Closed-Forms and Applications

We derive simple and unified closed-form expressions for projections with respect to fidelity (equivalently, the Bures and purified distances) onto several sets of interest. These include projections of bipartite positive semidefinite (PSD) matrices onto the set of PSD matrices with a given marginal, and projections of ensembles of PSD matrices onto the set of PSD decompositions of a given matrix, with important special cases corresponding to projections onto the set of quantum channels (via the Choi isomorphism) and onto the set of measurements. We introduce prior-channel decompositions of completely positive (CP) maps, which uniquely decompose any CP map into a prior PSD matrix and a quantum channel. This decomposition generalizes the Choi-Jamiolkowski isomorphism by establishing a bijective correspondence between arbitrary bipartite PSD matrices and channel-state pairs, and we show that it arises naturally from the fidelity projections developed here. As applications, we show that the pretty good measurement - associated with a weighted ensemble - is the fidelity projection of the ensemble onto the set of measurements, and that the Petz recovery map - associated with a reference state and forward channel - is the projection of a CP map (constructed from the channel-state pair) onto the set of reverse quantum channels, thereby recasting the well-known identification of the Petz map with quantum Bayes' rule in information-geometric terms. Our results also provide an information-geometric underpinning of the Leifer-Spekkens quantum state over time formalism [Leifer and Spekkens, Phys. Rev. A 88, 052130 (2013)].

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arXiv2025

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 (Álvarez-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.

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arXiv2025

Quantum umlaut information

We study the quantum umlaut information, a correlation measure defined for bipartite quantum states $ρ_{AB}$ as a reversed variant of the quantum mutual information: $U(A;B)_ρ= \min_{σ_B} D(ρ_A\otimes σ_B\|ρ_{AB})$ 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.

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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 Verdú. 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

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↗

arXiv2025

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 Rényi 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↗

arXiv2025

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.

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