Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1209.5233

Spectral moments and entropy rigidity of quantum channels: a three-mode photonic witness

Laizhen Luo, Lin Zhang

Published Sep 24, 2012• 0 citations
arXivHTMLPDF

Abstract

Majorization and von Neumann entropy provide related but inequivalent descriptions of spectral disorder in quantum optical states. We investigate the channels preserving these descriptions and propose a three-mode photonic test of their difference. For channels with equal input and output dimensions, we show that constant output purity and third spectral moment on the unitary orbit of one simple-spectrum state determine the majorization-preserving channel forms, without assuming unitality.For qubits, output purity alone is sufficient. Preservation of entropy equality is substantially more restrictive: in dimension $d\geqslant3$, even a local condition near the maximally mixed state permits only replacement and unitary channels. Qubits admit an additional depolarizing family, but this exception disappears under extension by an idle auxiliary qubit. We construct exactly entropy-matched qutrit inputs whose output entropies split at cubic order under nontrivial depolarizing noise. A proposed single-photon implementation uses three optical modes, randomized Weyl transformations and population or spectral-moment measurements.Explicit expressions are also obtained for preparation mismatch,readout uncertainty and input-independent loss. The results connect quadratic and cubic spectral information with experimentally interpretable tests of quantum-channel structure.

DOI
10.48550/arxiv.1209.5233

Related papers

Information-Theoretic Bounds on Entropy Generation in Quantum Channels

Tibor Svoboda, Milos Bartos, Igor Novotny

Landauer's principle sets a fundamental lower bound on the entropy generated when information is erased, but the tightness of that bound for realistic, noisy quantum channels used in physical information-processing devices remains only partially characterized. We derive and numerically test information-theoretic lower bounds on entropy generation for three canonical quantum channel models, the amplitude-damping channel, the depolarizing channel, and the dephasing channel, expressing each bound in terms of the channel's quantum capacity and its deviation from unitarity, and compare these bounds against directly computed von Neumann entropy production across 1,200 randomly sampled input states per channel type at 22 noise-strength settings spanning damping/depolarizing/dephasing parameters from 0.02 to 0.90. Across all three channel families, measured entropy production exceeded the derived lower bound in every one of the 3,600 trials, with a median excess (bound tightness gap) of 0.087 bits and an interquartile range of 0.041 to 0.156 bits, confirming the bounds are valid but revealing systematic looseness that varies by channel type. The amplitude-damping channel showed the tightest bound performance, with a median gap of 0.052 bits at low-to-moderate damping strength (gamma < 0.4), while the depolarizing channel showed the loosest bound, with a median gap of 0.134 bits across the full noise range, a 2.6-fold difference attributable to the depolarizing channel's more uniform mixing of the Bloch sphere. Gap magnitude correlated strongly with channel noise strength (Spearman rho = 0.81, p < 0.001 pooled across channel types), growing roughly threefold from the lowest to highest noise settings tested, indicating the bound is tightest in the near-unitary, low-decoherence regime relevant to near-term quantum error correction. A correction term incorporating the channel's output purity reduced the median gap to 0.029 bits (a 67% reduction) without requiring knowledge of the specific input state, at the cost of a more complex closed-form expression requiring numerical evaluation of the channel's Choi matrix eigenvalues. These results quantify, for the first time across three major channel families on a common footing, how much headroom exists between an easily computable information-theoretic bound and actual thermodynamic entropy cost, information directly relevant to estimating the true energetic overhead of near-term noisy quantum information processing.

•Dec 13, 2022•0 citations

Relating Entropies of Quantum Channels

Dariusz Kurzyk, Łukasz Pawela, Zbigniew Puchała

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

arXiv:2106.09629•Entropy•Aug 10, 2021•3 citations

Characterizing quantum channels from local-unitary invariants

Salwa Shaglel, Satoya Imai

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.

quant-pharXiv:2606.03722•Jun 3, 2026•0 citations

Entropic characterization of quantum operations

Wojciech Roga, M. Fannes, Karol Życzkowski

We investigate decoherence induced by a quantum channel in terms of minimal output entropy and of map entropy. The latter is the von Neumann entropy of the Jamiolkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 x Ψ2 holds.

arXiv:1101.4105•Jan 21, 2011•1 citation

ENTROPIC CHARACTERIZATION OF QUANTUM OPERATIONS

Wojciech Roga, Karol Życzkowski, M. Fannes

We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1acting on a finite dimensional system, there exists a class of channels Φ2sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1⊗ Ψ2holds.

International Journal of Quantum Information•Jun 1, 2011•27 citations

Moments of Quantum Channel Ensembles

Matthew Duschenes, Diego García-Martín, Zoë Holmes +1 more

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 noise), 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 in moment calculations.

quant-pharXiv:2511.12700•PRX Quantum•Nov 18, 2025•1 citation
Majorization and von Neumann entropy provide related but inequivalent descriptions of spectral disorder in quantum optical states. We investigate the channels preserving these descriptions and propose a three-mode photonic test of their difference. For channels with equal input and output dimensions, we show that constant output purity and third spectral moment on the unitary orbit of one simple-spectrum state determine the majorization-preserving channel forms, without assuming unitality.For qubits, output purity alone is sufficient. Preservation of entropy equality is substantially more restrictive: in dimension d⩾3d\geqslant3d⩾3, even a local condition near the maximally mixed state permits only replacement and unitary channels. Qubits admit an additional depolarizing family, but this exception disappears under extension by an idle auxiliary qubit. We construct exactly entropy-matched qutrit inputs whose output entropies split at cubic order under nontrivial depolarizing noise. A proposed single-photon implementation uses three optical modes, randomized Weyl transformations and population or spectral-moment measurements.Explicit expressions are also obtained for preparation mismatch,readout uncertainty and input-independent loss. The results connect quadratic and cubic spectral information with experimentally interpretable tests of quantum-channel structure.
Landauer's principle sets a fundamental lower bound on the entropy generated when information is erased, but the tightness of that bound for realistic, noisy quantum channels used in physical information-processing devices remains only partially characterized. We derive and numerically test information-theoretic lower bounds on entropy generation for three canonical quantum channel models, the amplitude-damping channel, the depolarizing channel, and the dephasing channel, expressing each bound in terms of the channel's quantum capacity and its deviation from unitarity, and compare these bounds against directly computed von Neumann entropy production across 1,200 randomly sampled input states per channel type at 22 noise-strength settings spanning damping/depolarizing/dephasing parameters from 0.02 to 0.90. Across all three channel families, measured entropy production exceeded the derived lower bound in every one of the 3,600 trials, with a median excess (bound tightness gap) of 0.087 bits and an interquartile range of 0.041 to 0.156 bits, confirming the bounds are valid but revealing systematic looseness that varies by channel type. The amplitude-damping channel showed the tightest bound performance, with a median gap of 0.052 bits at low-to-moderate damping strength (gamma < 0.4), while the depolarizing channel showed the loosest bound, with a median gap of 0.134 bits across the full noise range, a 2.6-fold difference attributable to the depolarizing channel's more uniform mixing of the Bloch sphere. Gap magnitude correlated strongly with channel noise strength (Spearman rho = 0.81, p < 0.001 pooled across channel types), growing roughly threefold from the lowest to highest noise settings tested, indicating the bound is tightest in the near-unitary, low-decoherence regime relevant to near-term quantum error correction. A correction term incorporating the channel's output purity reduced the median gap to 0.029 bits (a 67% reduction) without requiring knowledge of the specific input state, at the cost of a more complex closed-form expression requiring numerical evaluation of the channel's Choi matrix eigenvalues. These results quantify, for the first time across three major channel families on a common footing, how much headroom exists between an easily computable information-theoretic bound and actual thermodynamic entropy cost, information directly relevant to estimating the true energetic overhead of near-term noisy quantum information processing.
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.
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.
We investigate decoherence induced by a quantum channel in terms of minimal output entropy and of map entropy. The latter is the von Neumann entropy of the Jamiolkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 x Ψ2 holds.
We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1acting on a finite dimensional system, there exists a class of channels Φ2sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1⊗ Ψ2holds.
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 noise), 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 in moment calculations.