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

Lee-Yang preserving quantum Markov semigroups and channels

Dongsheng Wei

Published Sep 19, 2026• 0 citations
Publisher

Abstract

This preprint studies quantum Markov dynamics and quantum channels that preserve the Lee–Yang property. For systems of two or more qubits, preservation for all Lee–Yang density matrices forces tensor-product dynamics, with local generators characterized by explicit semidefinite conditions. This ordinary preservation property is equivalent to preservation in the presence of arbitrary inert reference systems. For systems containing higher spins, preservation of all pure Lee–Yang inputs excludes intersite interactions and forces isotropic diffusion with axial drift at higher-spin sites. The paper also develops classifications of static maps with stable full Choi tensors, including singular one-qubit maps and independently covariant maps between finite spins, and establishes quantitative rigidity results for diagonal Hamiltonians. The proofs combine positivity of quantum maps, polynomial root-collision arguments, and coherent tensor contraction. The manuscript includes a disclosure of the AI tools used in its development.

DOI
10.5281/zenodo.22846656

Related papers

Totally Positive Density Matrices and Linear Preservers

David W. Kribs, Jeremy Levick, Rajesh Pereira

The intersection between the set of totally nonnegative matrices, which are of interest in many areas of matrix theory and its applications, and the set of density matrices, which provide the mathematical description of quantum states, are investigated. The single qubit case is characterized, and several equivalent conditions for a quantum channel to preserve the set in that case are given. Higher dimensional cases are also discussed.

Electronic Journal of Linear Algebra•Feb 5, 2016•1 citation

A scalar–Haar dichotomy for consistent covariant quantum projections

Dmitry Lubenskiy

Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.

•Sep 27, 2026•0 citations

Structure and Classification of Matrix Product Quantum Channels

Giorgio Stucchi, J. Ignacio Cirac, Rahul Trivedi +1 more

We develop a framework for Matrix Product Quantum Channels (MPQCs), a one-dimensional tensor-network description of completely positive, trace-preserving maps. We focus on translation-invariant channels, generated by a single repeated tensor, that admit a local purification. We show that their purifying isometry can always be implemented by a constant-depth brickwork quantum circuit, implying that such channels generate only short-range correlations when applied on product states. In contrast to the unitary setting, where one-dimensional quantum cellular automata (in one-to-one correspondence with matrix product unitaries) carry a nontrivial index, we prove that all locally purified channels belong to a single phase, that is, they can be continuously deformed into one another. We then extend the framework to a broader class of translation-invariant channels capable of generating long-range entanglement and show that these remain deterministically implementable in constant depth using two rounds of measurements and feedforward.

quant-pharXiv:2603.19866•Physical Review Letters•Mar 23, 2026•0 citations

Noether’s theorem for dissipative quantum dynamical semi-groups

John Edward Gough, Tudor Stefan Ratiu, Oleg Georgievich Smolyanov

Noether’s theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semi-groups) by Baez and Fong [J. Math. Phys. 54, 013301 (2013)]. We show how to extend these results to the fully quantum setting of quantum Markov dynamics. For finite-dimensional Hilbert spaces, we construct a mapping from observables to completely positive maps that leads to the natural analogue of their criterion of commutativity with the infinitesimal generator of the Markov dynamics. Using standard results on the relaxation of states to equilibrium under quantum dynamical semi-groups, we are able to characterise the constants of the motion under quantum Markov evolutions in the infinite-dimensional setting under the usual assumption of existence of a stationary strictly positive density matrix. In particular, the Noether constants are identified with the fixed point of the Heisenberg picture semi-group.

arXiv:1410.7711•Journal of Mathematical Physics•Feb 1, 2015•20 citations

Second-order generators with quadratic Leibniz defect on full matrix algebras: a self-contained study of pure dephasing with Hamiltonian drift

Christoph Hartmann

This is a self-contained account, with complete proofs, of linear maps on the full matrix algebra whose Leibniz defect equals the product of the commutators with a fixed self-adjoint matrix H. Such maps are exactly one half of the double commutator with H plus a commutator with a second matrix K. For skew-adjoint K they generate unital, trace-preserving quantum Markov semigroups with a single self-adjoint Lindblad operator. Topics: normal form, reconstruction of H from the defect, and classification up to automorphisms; dynamics, duality and the Lindblad dissipation inequality; fixed points, an exact relaxation criterion and ergodicity without commutativity assumptions; the commutator energy, its Dirichlet-form property, and seven conditions equivalent to H and K commuting; a sharp complete modified logarithmic Sobolev inequality for pure dephasing with arbitrary spectrum; an exactly solved qubit with non-commuting drift; several self-adjoint Lindblad operators. Most of the material is known or elementary, and the text says so. The sharp entropy inequality and a finite-time optimal control result are taken from two companion notes by the author. Status: not reviewed by a human referee. The role of AI tools is described in the text.

•Sep 25, 2026•0 citations

Ŋeïlūna VIII - The Switching Dichotomy and Exact Cost of Quantum Erasure: Complete Classification with Protected Memory and an Attained Qubit Time-Action Frontier

Artificial Hyperintelligence, Eve, wife of Maciej Nowicki, Maciej Nowicki

Ŋeïlūna VIII develops a constructive mathematical theory of quantum erasure with protected memory. It investigates when unwanted information, including correlations with a retained quantum subsystem, can be removed while specified observables remain exactly preserved. The work combines a complete classification of the power of switching between dissipative generators with an exact, finitely attained resource optimum for erasing a qubit bath. The setting is finite-dimensional Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) dynamics. The Hilbert-space dimension, faithful reference state, protected observable algebra, and stationary conditional expectation remain fixed. Every instantaneous generator has true transverse real-part spectral gap exactly equal to one. This normalization fixes the slowest nonzero decay rate while allowing faster rates, Hamiltonian frequencies, and switching schedules to vary. The central theorem establishes a necessary-and-sufficient condition for arbitrarily rapid complete erasure: the protected algebra must possess at least two distinct compatible minimal extensions, called covers. An extension is compatible when it admits a conditional expectation preserving the same faithful reference state. The manuscript classifies these covers through two mechanisms: splitting a bath sector or coherently merging compatible sectors. Alternating the dissipative operations associated with distinct covers removes their additional surviving information while preserving the intended memory. The approximation is uniform over all input states, including inputs entangled with an arbitrary passive reference system, because error is measured by half the diamond distance to the target channel. In the nonrigid branch, arbitrarily accurate erasure is possible at every prescribed positive duration. Explicit soft-reset constructions achieve every positive error tolerance using finitely many finite-strength stages. Their integrated dissipative action scales as O(log(1/ε)), matching the necessary asymptotic order as the error ε approaches zero. An optimal leading coefficient in general dimension is not claimed. The classification also identifies the rigid exceptions exhaustively. Exactly one compatible cover occurs for two pinching sectors, or for one nonmaximally mixed qubit bath with at most one additional bath-dimension-one spectator sector. Protected logical dimensions are unrestricted. Quantitative estimates show that arbitrary allowed switching cannot remove the vanishing-time obstruction in these cases, even when switching counts, generator strengths, and Hamiltonian frequencies are unrestricted. The argument includes hybrid configurations in which the globally slowest mode lies between sectors rather than inside the active qubit. A second principal result solves the joint duration–action–gap optimization for erasing a bath qubit to the maximally mixed state I₂/2 while preserving an arbitrary full logical quantum memory. If A₀ is the bath-action budget and T > 0 is the prescribed duration, feasibility is characterized exactly by A₀ ≥ 3T/2. For every feasible budget, the minimum complete erasure error is minimum half-diamond error = (3/4) exp(−2A₀/3). For a bare qubit without a reference system, the optimal worst-input trace-distance error is minimum unassisted error = (1/2) exp(−2A₀/3). A protected memory of dimension at least two can itself supply the reference that realizes the complete-error prefactor. The distinction between bath action and the manuscript’s full-space action normalization is stated explicitly. Three equal-duration dissipative stages attain these optima exactly. Each stage retains one Pauli direction with decay rate one while damping the other two directions at rate 1 + R, where R = A₀/T − 3/2. Cycling the slow direction through the x, y, and z axes produces an isotropic endpoint contraction exp(−2A₀/3). All strengths remain finite for finite feasible action and positive duration. The lower bound also applies to the broader class of biased, nonunital, and noncommuting qubit controls. Equality requires a depolarizing endpoint composed with a unitary. The higher-dimensional boundary is established constructively. An explicit qutrit Weyl-jump generator outperforms isotropic depolarization at identical duration, action, stationary state, and true spectral gap. This counterexample rules out a direct higher-dimensional extrapolation of the qubit isotropic optimum. Additional results include the autonomous surviving-algebra frontier, its exhaustive split/merge reduction, exact positive-time pinching frontiers, and correlation-sensitive erasure obstructions. These encompass an exact one-variable spectral optimization for arbitrary two-qubit inputs and an autonomous erasure floor equal to half the concurrence for pure two-qubit states with a maximally mixed bath target. The accompanying research package contains the 30-page manuscript, editable TeX source and bibliography, complete proofs, four scientific figures, exact arithmetic certificates, numerical verification programs, archived outputs, and preserved earlier editions. All eight included research verification suites passed again during release preparation. Universal statements rest on the written proofs; finite checks certify explicit examples and exercise the constructions. Mathematical development and audits are AI-assisted. Established channel, alternating-projection, and control-theoretic ingredients are credited, while worldwide priority of the proposed combined results is not certified. Dissipative action is an intrinsic integrated rate budget, not thermodynamic heat, work, or entropy production. Arbitrarily rapid erasure means approximation in channel closure along sequences whose rates or switching counts may increase. Exact zero-error erasure is not achieved by a finite-action propagator. Finite switching bandwidth, restricted interactions, approximate memory protection, non-Markovian dynamics, and hardware implementation remain outside the proved model. Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Research version: 8.0. Release date: 29 October 2026.

•Oct 29, 2026•0 citations
This preprint studies quantum Markov dynamics and quantum channels that preserve the Lee–Yang property. For systems of two or more qubits, preservation for all Lee–Yang density matrices forces tensor-product dynamics, with local generators characterized by explicit semidefinite conditions. This ordinary preservation property is equivalent to preservation in the presence of arbitrary inert reference systems. For systems containing higher spins, preservation of all pure Lee–Yang inputs excludes intersite interactions and forces isotropic diffusion with axial drift at higher-spin sites. The paper also develops classifications of static maps with stable full Choi tensors, including singular one-qubit maps and independently covariant maps between finite spins, and establishes quantitative rigidity results for diagonal Hamiltonians. The proofs combine positivity of quantum maps, polynomial root-collision arguments, and coherent tensor contraction. The manuscript includes a disclosure of the AI tools used in its development.
Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.
We develop a framework for Matrix Product Quantum Channels (MPQCs), a one-dimensional tensor-network description of completely positive, trace-preserving maps. We focus on translation-invariant channels, generated by a single repeated tensor, that admit a local purification. We show that their purifying isometry can always be implemented by a constant-depth brickwork quantum circuit, implying that such channels generate only short-range correlations when applied on product states. In contrast to the unitary setting, where one-dimensional quantum cellular automata (in one-to-one correspondence with matrix product unitaries) carry a nontrivial index, we prove that all locally purified channels belong to a single phase, that is, they can be continuously deformed into one another. We then extend the framework to a broader class of translation-invariant channels capable of generating long-range entanglement and show that these remain deterministically implementable in constant depth using two rounds of measurements and feedforward.
Noether’s theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semi-groups) by Baez and Fong [J. Math. Phys. 54, 013301 (2013)]. We show how to extend these results to the fully quantum setting of quantum Markov dynamics. For finite-dimensional Hilbert spaces, we construct a mapping from observables to completely positive maps that leads to the natural analogue of their criterion of commutativity with the infinitesimal generator of the Markov dynamics. Using standard results on the relaxation of states to equilibrium under quantum dynamical semi-groups, we are able to characterise the constants of the motion under quantum Markov evolutions in the infinite-dimensional setting under the usual assumption of existence of a stationary strictly positive density matrix. In particular, the Noether constants are identified with the fixed point of the Heisenberg picture semi-group.
This is a self-contained account, with complete proofs, of linear maps on the full matrix algebra whose Leibniz defect equals the product of the commutators with a fixed self-adjoint matrix H. Such maps are exactly one half of the double commutator with H plus a commutator with a second matrix K. For skew-adjoint K they generate unital, trace-preserving quantum Markov semigroups with a single self-adjoint Lindblad operator. Topics: normal form, reconstruction of H from the defect, and classification up to automorphisms; dynamics, duality and the Lindblad dissipation inequality; fixed points, an exact relaxation criterion and ergodicity without commutativity assumptions; the commutator energy, its Dirichlet-form property, and seven conditions equivalent to H and K commuting; a sharp complete modified logarithmic Sobolev inequality for pure dephasing with arbitrary spectrum; an exactly solved qubit with non-commuting drift; several self-adjoint Lindblad operators. Most of the material is known or elementary, and the text says so. The sharp entropy inequality and a finite-time optimal control result are taken from two companion notes by the author. Status: not reviewed by a human referee. The role of AI tools is described in the text.
Ŋeïlūna VIII develops a constructive mathematical theory of quantum erasure with protected memory. It investigates when unwanted information, including correlations with a retained quantum subsystem, can be removed while specified observables remain exactly preserved. The work combines a complete classification of the power of switching between dissipative generators with an exact, finitely attained resource optimum for erasing a qubit bath. The setting is finite-dimensional Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) dynamics. The Hilbert-space dimension, faithful reference state, protected observable algebra, and stationary conditional expectation remain fixed. Every instantaneous generator has true transverse real-part spectral gap exactly equal to one. This normalization fixes the slowest nonzero decay rate while allowing faster rates, Hamiltonian frequencies, and switching schedules to vary. The central theorem establishes a necessary-and-sufficient condition for arbitrarily rapid complete erasure: the protected algebra must possess at least two distinct compatible minimal extensions, called covers. An extension is compatible when it admits a conditional expectation preserving the same faithful reference state. The manuscript classifies these covers through two mechanisms: splitting a bath sector or coherently merging compatible sectors. Alternating the dissipative operations associated with distinct covers removes their additional surviving information while preserving the intended memory. The approximation is uniform over all input states, including inputs entangled with an arbitrary passive reference system, because error is measured by half the diamond distance to the target channel. In the nonrigid branch, arbitrarily accurate erasure is possible at every prescribed positive duration. Explicit soft-reset constructions achieve every positive error tolerance using finitely many finite-strength stages. Their integrated dissipative action scales as O(log(1/ε)), matching the necessary asymptotic order as the error ε approaches zero. An optimal leading coefficient in general dimension is not claimed. The classification also identifies the rigid exceptions exhaustively. Exactly one compatible cover occurs for two pinching sectors, or for one nonmaximally mixed qubit bath with at most one additional bath-dimension-one spectator sector. Protected logical dimensions are unrestricted. Quantitative estimates show that arbitrary allowed switching cannot remove the vanishing-time obstruction in these cases, even when switching counts, generator strengths, and Hamiltonian frequencies are unrestricted. The argument includes hybrid configurations in which the globally slowest mode lies between sectors rather than inside the active qubit. A second principal result solves the joint duration–action–gap optimization for erasing a bath qubit to the maximally mixed state I₂/2 while preserving an arbitrary full logical quantum memory. If A₀ is the bath-action budget and T > 0 is the prescribed duration, feasibility is characterized exactly by A₀ ≥ 3T/2. For every feasible budget, the minimum complete erasure error is minimum half-diamond error = (3/4) exp(−2A₀/3). For a bare qubit without a reference system, the optimal worst-input trace-distance error is minimum unassisted error = (1/2) exp(−2A₀/3). A protected memory of dimension at least two can itself supply the reference that realizes the complete-error prefactor. The distinction between bath action and the manuscript’s full-space action normalization is stated explicitly. Three equal-duration dissipative stages attain these optima exactly. Each stage retains one Pauli direction with decay rate one while damping the other two directions at rate 1 + R, where R = A₀/T − 3/2. Cycling the slow direction through the x, y, and z axes produces an isotropic endpoint contraction exp(−2A₀/3). All strengths remain finite for finite feasible action and positive duration. The lower bound also applies to the broader class of biased, nonunital, and noncommuting qubit controls. Equality requires a depolarizing endpoint composed with a unitary. The higher-dimensional boundary is established constructively. An explicit qutrit Weyl-jump generator outperforms isotropic depolarization at identical duration, action, stationary state, and true spectral gap. This counterexample rules out a direct higher-dimensional extrapolation of the qubit isotropic optimum. Additional results include the autonomous surviving-algebra frontier, its exhaustive split/merge reduction, exact positive-time pinching frontiers, and correlation-sensitive erasure obstructions. These encompass an exact one-variable spectral optimization for arbitrary two-qubit inputs and an autonomous erasure floor equal to half the concurrence for pure two-qubit states with a maximally mixed bath target. The accompanying research package contains the 30-page manuscript, editable TeX source and bibliography, complete proofs, four scientific figures, exact arithmetic certificates, numerical verification programs, archived outputs, and preserved earlier editions. All eight included research verification suites passed again during release preparation. Universal statements rest on the written proofs; finite checks certify explicit examples and exercise the constructions. Mathematical development and audits are AI-assisted. Established channel, alternating-projection, and control-theoretic ingredients are credited, while worldwide priority of the proposed combined results is not certified. Dissipative action is an intrinsic integrated rate budget, not thermodynamic heat, work, or entropy production. Arbitrarily rapid erasure means approximation in channel closure along sequences whose rates or switching counts may increase. Exact zero-error erasure is not achieved by a finite-action propagator. Finite switching bandwidth, restricted interactions, approximate memory protection, non-Markovian dynamics, and hardware implementation remain outside the proved model. Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Research version: 8.0. Release date: 29 October 2026.