arXiv subjects

Marco Tomamichel

Publications and source records attributed to Marco Tomamichel.

At least 19 recordsLinked to original sources

arXiv2026

A complete characterisation of conditional entropies

Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of Rényi entropies, parameterized by a real number $α$. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterisation of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterisation of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We show that the most general conditional entropy is a generalized convex combination of conditional entropies defined as exponential averages of Rényi entropies of the conditioned distribution, parameterized by a real number and a probability measure on the positive reals.

cs.IT↗

arXiv2026

Monotonicity of the Rényi channel capacity under non-signaling assisted channel simulation

We give an alternative variational proof that non-signaling correlations shared between the sender and receiver cannot increase the Rényi capacity of a classical channel. This monotonicity follows from the data-processing inequality for channel divergences established by Oufkir, Yao and Berta. Our approach works directly with the information-radius representation of the capacity. The constraints defining non-signaling boxes are linear in the box, while the Rényi capacity depends on the channel nonlinearly. We use a Legendre transform to express the capacity as an extremum over a family of affine functionals in which the channel appears linearly. Combining this method with separate arguments for the Shannon capacity and endpoint orders, we recover monotonicity at every order from zero to infinity. We then derive consequences for standard coding exponent functions: non-signaling simulation cannot increase the random-coding or sphere-packing exponent below capacity, or decrease the strong-converse exponent above capacity.

quant-ph↗

arXiv2026

On the Power of Adaptivity in Testing Quantum States in Fidelity

We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state $ρ$ and the description of a state $σ$, the goal is to test whether $ρ=σ$, or whether $ρ$ and $σ$ are far in a given distance measure. In equivalence testing, $σ$ is also unknown and only accessible via samples. Independence testing decides whether $ρ_{AC}=ρ_A\otimesρ_C$, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap $\varepsilon$ in trace distance: in the single-copy measurement setting with $d$-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses $Θ(d^{3/2}/\varepsilon^2)$ samples and is optimal in general, even without adaptivity. In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state $σ$ of rank $r$ does not benefit from adaptivity and requires $\widetildeΘ(r^{3/2}/\varepsilon)$ samples. For equivalence testing and independence testing, we provide adaptive algorithms using $\widetilde{O}(\min\{d^{3/2}/\varepsilon^2,d^{9/4}/\varepsilon\})$ and $\widetilde{O}(\min\{(d_Ad_C)^{3/2}/\varepsilon^2,d_A^{9/4}d_C^{3/4}/\varepsilon\})$ samples, for $d_A\geq d_C$, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in $\ell_2$-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that $\widetildeΩ(1/\varepsilon^2)$ samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.

quant-ph↗

arXiv2026

Quantum information processing under maximal distance measures

The measures of distance frequently used in quantum information, such as trace distance or fidelity-based purified distance, are the smallest extensions of the corresponding distance measures in classical information theory. However, this way of extending classical distances is not unique, and a whole spectrum of other extensions can be defined. Here we study the question of how quantum information processing tasks are affected by a change of the error metrics to the maximal extensions of the classical distance measures, imposing more demanding error requirements on operational tasks. We find that the performance of some tasks, such as channel coding or quantum resource distillation, is not affected whatsoever by this change. However, for other tasks such as channel simulation and quantum resource dilution, we show that the rates are now given not by functions of the standard (Umegaki) quantum relative entropy, but ones based on the Belavkin-Staszewski relative entropy. Our results connect the Belavkin-Staszewski relative entropy with operational tasks and reveal that known reversibility results in quantum information, including the reverse Shannon theorem and the asymptotic reversibility of quantum resource transformations, no longer hold under maximal distance measures.

quant-ph↗

arXiv2026

Fast and Sure-ious Quantum Process Tomography

Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity $\varepsilon$ from a number of channel uses proportional to $1/\varepsilon$. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.

quant-ph↗

arXiv2026

Strong Converses for Quantum Channel Capacities from Blowing-Up Lemmata

A quantum channel can transmit quantum states, classical messages, or classical messages that remain secret from its environment. We prove exponential strong converses for all three tasks over arbitrary finite-dimensional memoryless quantum channels, at their respective regularized capacities and allowing arbitrary entangled inputs across channel uses and joint decoding. Above quantum or classical capacity, entanglement fidelity or decoding success probability, respectively, decays exponentially with the number of channel uses. Above private capacity, fidelity to an ideal shared secret key decays exponentially; in particular, vanishing secrecy error forces decoding success to vanish. The proofs combine one-shot quantum blowing-up lemmas, which convert codes with low success into reliable codes with a controlled loss in code size, with a low-degree polynomial approximation of the channel's Stinespring image projector. The loss is governed by the projective tensor norm of the approximation across the receiver--environment bipartition. For memoryless channels, the polynomial construction gives exponential accuracy while keeping the rate loss arbitrarily small, turning the corresponding weak converses into strong converses.

quant-ph↗

arXiv2026

Parallel classical simulation of noisy shallow circuits: no quantum advantage in 1D

We consider quantum circuits consisting of $d$ layers of nearest-neighbor two-qubit gates acting on $n$ qubits arranged on a line, where every qubit is independently depolarized with a constant probability before each layer. We describe a randomized parallel algorithm which samples from the output distribution of any such circuit to within total variation error $δ$, with parallel runtime $2^{O(d)}\log\log(n/δ)$ and $n 2^{O(d)}$ elementary real-arithmetic operations. Without noise, the same approach gives an exact sampler with parallel runtime $O(\log n)$. For constant depth, we further show that the input/output behavior of the noisy quantum circuit is reproduced up to a constant error by a randomized $\mathsf{AC}^0$-circuit, that is, a Boolean circuit of polynomial size and constant depth with unbounded fan-in AND and OR gates and NOT gates. Consequently, every relation problem solved by a noisy constant-depth quantum circuit in one dimension is also solved, with essentially the same success probability, by a randomized $\mathsf{AC}^0$-circuit. This rules out, for noisy circuits in one dimension, the unconditional quantum advantage established for noisy shallow circuits in two and three dimensions, where polynomial-size classical circuits over the same gate set require depth $Ω(\log n/\log\log n)$. Our algorithm exploits the fact that depolarizing noise cuts a one-dimensional circuit into independent pieces of logarithmic width, each of which can be sampled exactly in parallel.

quant-ph↗

arXiv2026

Approximate majorization and high-order capacity of quantum depolarizing channels

A central challenge in quantum information theory is to determine whether entanglement provides an advantage for information processing when coherent operations across multiple channel uses are allowed. For depolarizing channels, King's proof of the additivity of the minimum output entropy rules out such an advantage for classical communication at leading asymptotic order, while leaving open the possibility of an advantage for higher orders. In this work, we conjecture a majorization relation for the outputs of depolarizing channels and prove an approximate version of this conjecture. Our proof combines an exact ordering of shifted log-determinants with an anti-concentration bound for the spectrum of the product-state output, refining King's analysis based on Schatten norms. As a direct application, we obtain a tight third-order expansion for the classical capacity of quantum depolarizing channels. Consequently, entangled codewords and collective decoding can improve the message size over the product-coding strategy by at most $O(\log \log n)$ bits, sharpening our understanding of the role of entanglement in the finite-blocklength regime.

quant-ph↗

arXiv2026

The exact strong converse exponent for private communication over quantum channels

We determine the exact strong converse exponent for secret-key transmission and generation over general finite-dimensional quantum wiretap channels, measuring reliability and secrecy jointly by squared fidelity to an ideal secret key. We introduce a novel Renyi private information whose regularization characterizes this exponent. An exact integral representation in terms of ordinary private information gives a uniform continuity bound, showing that the regularized quantity converges to private capacity as the Renyi order approaches one. This establishes for any wiretap channel an exponential fidelity decay at every rate above private capacity.

quant-ph↗

arXiv2026

Making the most of leftovers: Improved privacy amplification for quantum key distribution

The amount of secret key that can be obtained from a quantum key distribution run depends on both the physically observed error rates and the mathematical bounds used to certify security. For finite datasets, conservative bounds force users to discard a substantial fraction of the potentially available key. Here we further refine and extend the privacy amplification bounds achievable through the recent leftover hash lemma of Regula and Tomamichel [arXiv:2603.04493] and incorporate them into the security analysis of quantum key distribution based on entropic uncertainty relations. This improves on state-of-the-art key rates in finite-block regimes, certifying more secret key from the same experimental data without changes to the protocol, and outperforming techniques based on entropy accumulation. The results illustrate how sharper mathematical estimates can directly increase the usable output of a quantum communication system.

quant-ph↗

arXiv2026

The strong converse exponent of composable randomness extraction against quantum side information

We find a tight characterization of the strong converse exponent for randomness extraction against quantum side information. In contrast to previous tight bounds, we employ a composable error criterion given by the fidelity (or purified distance) to a uniform distribution in product with the marginal state. Our characterization is formulated in terms of the club-sandwiched conditional entropy, recently introduced by Rubboli, Goodarzi, and Tomamichel. It extends the corresponding result for classical side information, whose strong converse exponent was first established by Berta and Yao and later recovered by Li, Li, and Yu within this entropic framework. This provides the first precise operational interpretation of this family of conditional entropies in the quantum setting.

quant-ph↗

arXiv2026

Sharp Data-Processing Region for Quantum Conditional Rényi Entropies

We study the three-parameter family of quantum conditional Rényi entropies $H_{α,z}^λ$ introduced in \textit{Commun. Math. Phys.} 407, 180 (2026). We prove that the previously established data-processing region is sharp. We then characterize the club-sandwiched family through a classical coarse-graining principle: within the full data-processing region, universal monotonicity under deterministic coarse-graining of the classical conditioned register holds if and only if $z = α$. Finally, we establish rigidity of entropic duality by showing that any universal duality relation for finite-dimensional tripartite pure states must satisfy the known parameter relations. Thus, the data-processing region, the club-sandwiched subfamily, and the duality transformation are each uniquely characterized by their respective universal properties.

quant-ph↗

arXiv2026

Second-Order Expansion of Privacy Amplification Under f-Divergence Criteria

We derive the second-order asymptotics of randomness extraction from memoryless sources with side information under security criteria based on a broad class of Csiszàr f-divergences, treating both a fixed reference side-information marginal and optimization over that marginal. The conditional varentropy decomposes into fluctuations of the conditional entropy across different values of the side information and the average variance of the conditional surprisal for each value. Without marginal optimization, these contributions yield a Gaussian-mixture second-order profile. With marginal optimization, they combine into the total conditional varentropy, yielding a single Gaussian profile. As corollaries, we obtain second-order expansions for Rényi-entropy criteria of all orders $α\in (0,1)$ and recover the known expansion for total variation distance.

cs.IT↗

arXiv2026

No information transmission through quantum channels above capacity

We show that the capacity of a quantum channel demarcates a phase transition: while reliable transmission below capacity is always possible, any attempt to transmit information above it fails catastrophically. Specifically, we prove exponential strong converse theorems for unassisted quantum and classical communication over arbitrary finite-dimensional memoryless quantum channels. At rates beyond the respective capacity, the entanglement-generation fidelity and the success probability for classical communication decay exponentially with the number of channel uses. This rules out transmission above capacity even when one tolerates arbitrarily large errors. Our proof follows the classical Arimoto strategy, augmented by a crucial new ingredient: integral representations of Rényi information measures that lead to asymptotic continuity bounds for Rényi capacities.

quant-ph↗

arXiv2026

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↗

arXiv2026

Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification

We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.

quant-ph↗

arXiv2026

Strong Converse Exponent of Quantum State Merging

We determine the strong converse exponent for the entanglement cost of quantum state merging, showing that it is characterized by the optimized $α$-$z$ conditional Rényi entropies with $z=α/2\in[1/2,1]$. This contrasts with the sandwiched conditional Rényi entropies that typically govern strong converse exponents in quantum information theory. As a consequence, we derive the strong converse exponent of the partially smoothed conditional min-entropy in purified distance. This exponent is governed by club-sandwiched conditional entropies, whereas global smoothing leads to a sandwiched expression.

quant-ph↗

arXiv2026

Weak Log-Majorization for Negative Lim-Pálfia Power Means

For $0 \le α\le 1$ and $-1 \le r < 0$, let $\#_{r,α}$ denote the weighted Kubo-Ando power mean of order $r$ and weight $α$. Ando proved that, for every unitarily invariant norm, $\| A \ \#_{r,α}\ B \| \le \| A \|\ \#_{r,α}\ \| B \|$. We show that this inequality remains valid also for the Schatten $q$-quasi-norms with $0 < q < 1$. More generally, our main result resolves a problem recently posed by Hiai and Lim in the stronger form of weak log-majorization for multivariable Lim-Pálfia power means of negative order. It yields analogous Schatten quasi-norm inequalities for every finite family of positive definite matrices.

math.FA↗