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arXiv:2511.12700 · PRX Quantum

Moments of quantum channel ensembles

Matthew Duschenes, Diego García-Martín, Zoë Holmes, M. Cerezo

quant-phcond-mat.stat-mechSubmitted Nov 16, 2025• Announced Nov 18, 2025• 1 citation
v1 · Nov 16, 2025v2 · Sep 9, 2026
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Abstract

Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate $t$-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders $t$, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel $t$-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest.

Journal reference
PRX Quantum 7, 033050 (2026)
DOI
10.1103/grdp-jph9
Comments
11+26 pages, 5+5 figures

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Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate ttt-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders ttt, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel ttt-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest.
The intersection of physics and information theory has advanced rapidly in the past few decades. Information-theoretic concepts have been employed to formulate physical laws, while quantum information processing, such as quantum computation and communication, has been developed both theoretically and experimentally. Simultaneously, understanding the universal or specific behaviors of quantum many-body systems has remained a central challenge since shortly after the inception of quantum mechanics. This is studied in part by statistical mechanics, which models complex many-body systems using simple ensembles. In this thesis, we investigate information-theoretic problems in quantum many-body dynamics that lie at the intersection of these avenues. To begin with, we investigate the extent of randomness, captured by unitary designs, in random quantum circuits. Quantum circuits serve as discrete models for random many-body quantum dynamics. Specifically, we study how the internal structure of random gates influences the generation of randomness in quantum circuits. We demonstrate that the spectral gap of structured random circuits can be larger than that of unstructured ones, implying that the former can converge to designs faster than the latter. In this analysis, free-fermionic techniques play a crucial role in obtaining exact results on the spectrum of second-moment operators. We then investigate the effect of quantum measurements on the quantum complexity in random circuits. Here, the quantum complexity of a state is defined as the minimum number of local gates required for its exact preparation in quantum circuits. We prove that in monitored random quantum circuits, the complexity undergoes a measurement-induced phase transition by leveraging techniques from percolation theory. Above the critical measurement rate, the complexity saturates quickly. Below the threshold, it grows at least linearly for an exponentially long time. Finally, we discuss quantum communication scenarios in noisy quantum dynamics, where we simulate the identity channel by combining a given noisy quantum dynamics and our quantum operational strategies. Specifically, we consider multi-time noisy processes and optimize our strategies to simulate near-noiseless channels, both of which are mathematically formulated using quantum combs. We introduce formal concepts of simulation and capacity for quantum combs, and we derive lower-bounds on the minimum simulation error in terms of semidefinite programming (SDP). We impose certain conditions on our strategies, such as the positive partial transpose (PPT) preserving constraint. We then analyze these SDPs for simple models, such as the depolarizing process, to determine their fundamental communication limits.
We develop systematic frameworks for characterizing the entanglement properties of two-qubit channels beyond unitary settings. We introduce averaged local-unitary invariants, referred to as moments, obtained from Haar integrals over input states or unitaries. These moments provide computable descriptions of how a quantum channel can create, preserve, or destroy bipartite entanglement. We first show that second-order moments yield criteria for non-entangling and entanglement-breaking channels, which allow us to detect entanglement-creating and entanglement-preserving channels. We then demonstrate that higher-order moments can capture additional information and distinguish channels beyond second-order moments alone. Finally, we show that combinations of moments associated with different channel families improve the discrimination of locally inequivalent two-qubit unitaries.
Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.
Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρρρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.
We develop systematic frameworks for characterizing the entanglement properties of two-qubit channels beyond unitary settings. We introduce averaged local-unitary invariants, referred to as moments, obtained from Haar integrals over input tates or unitaries. These moments provide computable descriptions of how a quantum channel can create, preserve, or destroy bipartite entanglement. We first show that second-order moments yield criteria for non-entangling and entanglementbreaking channels, which allow us to detect entanglement-creating and entanglement-preserving channels. We then demonstrate that higher-order moments can capture additional information and distinguish channels beyond second-order moments alone. Finally, we show that combinations of moments associated with different channel families improve the discrimination of locally inequivalent two-qubit unitaries.
The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these quantities using tools from statistical mechanics or via quantum information-based inequalities. By focusing on the second moment of one-dimensional unitary circuits where nearest-neighboring gates act on sets of qudits (with open and closed boundary conditions), we show that one can exactly compute the associated spectral gaps. Indeed, having access to their functional form allows us to prove several important results, such as the fact that the spectral gap for closed boundary condition is exactly the square of the gap for open boundaries, as well as improve on previously known bounds for approximate design convergence. Finally, we verify our theoretical results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random orthogonal and symplectic circuits.