Operational quantum reconstruction, Feynman rules, composition algebras, and beyond

Feynman's formulation of quantum theory is remarkable in its combination of formal simplicity and computational power. However, as a formulation of the abstract structure of quantum theory, it is incomplete as it does not account for most of the fundamental mathematical structure of the standard von Neumann--Dirac formalism such as the unitary evolution of quantum states. In this paper, we show how to reconstruct the entirety of the finite-dimensional quantum formalism starting from Feynman's rules with the aid of a single physical postulate, the no-disturbance postulate. This postulate states that a particular class of measurements have no effect on the outcome probabilities of subsequent measurements performed. We also show how it is possible to derive both the amplitude rule for composite systems of distinguishable subsystems and Dirac's amplitude-action rule, each from a single elementary and natural assumption, by making use of the fact that these assumptions must be consistent with Feynman's rules.Feynman's formulation of quantum theory is remarkable in its combination of formal simplicity and computational power. However, as a formulation of the abstract structure of quantum theory, it is incomplete as it does not account for most of the fundamental mathematical structure of the standard von Neumann--Dirac formalism such as the unitary evolution of quantum states. In this paper, we show how to reconstruct the entirety of the finite-dimensional quantum formalism starting from Feynman's rules with the aid of a single physical postulate, the no-disturbance postulate. This postulate states that a particular class of measurements have no effect on the outcome probabilities of subsequent measurements performed. We also show how it is possible to derive both the amplitude rule for composite systems of distinguishable subsystems and Dirac's amplitude-action rule, each from a single elementary and natural assumption, by making use of the fact that these assumptions must be consistent with Feynman's rules.

arXiv:1403.3527Physical Review AMar 20, 201414 citations

Operator-language Feynman rules for driven-dissipative quantum systems: from mean field to non-Gaussian photon correlations

Peter Ehlers, Phi Hung Nguyen, Daniel Soh

We construct Feynman rules for Lindblad master equations in the operator language of quantum optics: a fixed algebra of elementary operator moves and closed-form propagators for perturbation theory about an exactly solvable, in most cases Gaussian, generator. The expansion separates into a kinematic part (the shift algebra on the Wick-ordered eigenoperators $O_{mn}=:(c^\dagger)^m c^n:$ , from which the vertex table for a specified interaction is assembled) and a dynamical part that enters only through the resolvent of the unperturbed generator, diagonal and additive for any stable, diagonalizable Gaussian $L_0$ , including thermal and squeezed reservoirs and correlated decay, and equally available for a two-level emitter, driven or undriven. In the driven Kerr cavity the loop expansion is the semiclassical expansion and the series is asymptotic. In a strongly driven two-level atom, dressed lines replace a series of finite radius by one accurate deep into saturation, one line per Mollow component. For Kerr rings and chains the cost is polynomial in the number of sites at fixed perturbative order. On a three-site ring the diagrams recover the non-Gaussian part of the three-photon correlation that the Gaussian cumulant closure lacks; carried to tenth order they give the smallest $g^{(3)}$ error of the methods compared for $U\leq0.25κ$ at a cost below that of a fourth-order closure, and third- and fourth-order closures are more accurate at moderate coupling. The same rules, extended to the counting-field-tilted generator, give the photon-counting cumulants of a disordered eight-site chain beyond direct diagonalization; quantum-jump trajectories at three couplings agree with them to within $1.6$ standard errors, within one at the strongest coupling, where they place the third cumulant six standard errors above the Gaussian value.We construct Feynman rules for Lindblad master equations in the operator language of quantum optics: a fixed algebra of elementary operator moves and closed-form propagators for perturbation theory about an exactly solvable, in most cases Gaussian, generator. The expansion separates into a kinematic part (the shift algebra on the Wick-ordered eigenoperators $O_{mn}=:(c^\dagger)^m c^n:$, from which the vertex table for a specified interaction is assembled) and a dynamical part that enters only through the resolvent of the unperturbed generator, diagonal and additive for any stable, diagonalizable Gaussian $L_0$, including thermal and squeezed reservoirs and correlated decay, and equally available for a two-level emitter, driven or undriven. In the driven Kerr cavity the loop expansion is the semiclassical expansion and the series is asymptotic. In a strongly driven two-level atom, dressed lines replace a series of finite radius by one accurate deep into saturation, one line per Mollow component. For Kerr rings and chains the cost is polynomial in the number of sites at fixed perturbative order. On a three-site ring the diagrams recover the non-Gaussian part of the three-photon correlation that the Gaussian cumulant closure lacks; carried to tenth order they give the smallest $g^{(3)}$ error of the methods compared for $U\leq0.25κ$ at a cost below that of a fourth-order closure, and third- and fourth-order closures are more accurate at moderate coupling. The same rules, extended to the counting-field-tilted generator, give the photon-counting cumulants of a disordered eight-site chain beyond direct diagonalization; quantum-jump trajectories at three couplings agree with them to within $1.6$ standard errors, within one at the strongest coupling, where they place the third cumulant six standard errors above the Gaussian value.

quant-pharXiv:2610.00667Oct 2, 20260 citations

Quantum Logic and Quantum Reconstruction

Allen Stairs

arXiv:1501.05492Foundations of PhysicsFeb 25, 20153 citations

Operational Quantum Logic: An Overview

Bob Coecke, David Moore, Alexander Wilce

arXiv:quant-ph/0008019Springer eBooksJan 1, 200052 citations

O_{mn}=:(c^\dagger)^m c^n:

, from which the vertex table for a specified interaction is assembled) and a dynamical part that enters only through the resolvent of the unperturbed generator, diagonal and additive for any stable, diagonalizable Gaussian

L_0

, including thermal and squeezed reservoirs and correlated decay, and equally available for a two-level emitter, driven or undriven. In the driven Kerr cavity the loop expansion is the semiclassical expansion and the series is asymptotic. In a strongly driven two-level atom, dressed lines replace a series of finite radius by one accurate deep into saturation, one line per Mollow component. For Kerr rings and chains the cost is polynomial in the number of sites at fixed perturbative order. On a three-site ring the diagrams recover the non-Gaussian part of the three-photon correlation that the Gaussian cumulant closure lacks; carried to tenth order they give the smallest

g^{(3)}

error of the methods compared for

U\leq0.25κ

at a cost below that of a fourth-order closure, and third- and fourth-order closures are more accurate at moderate coupling. The same rules, extended to the counting-field-tilted generator, give the photon-counting cumulants of a disordered eight-site chain beyond direct diagonalization; quantum-jump trajectories at three couplings agree with them to within

1.6

standard errors, within one at the strongest coupling, where they place the third cumulant six standard errors above the Gaussian value.