Stationary quantum Markov process for the Wigner function

A. Hayashi; M. Horibe; T. Hashimoto

REVIEW

Stationary quantum Markov process for the Wigner function

T. Hashimoto, M. Horibe, A. Hayashi

Quantum Physics · Submitted 28 Feb 2006 · 12 references

Pith's verdict: The U(1) phase extension is a genuine structural improvement over Cohendet et al., giving stationary, independent-particle rates, but the branching rule at α=0,π is load-bearing and informal because every trajectory hits the boundary in finite time.

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arXiv IDquant-ph/0602227 (version 4)

Pith IDPIJU437J, Pith's permanent record of this paper and its review

CitationA. Hayashi, M. Horibe, T. Hashimoto. "Stationary quantum Markov process for the Wigner function." arXiv:quant-ph/0602227 (2006). https://pith.science/paper/PIJU437J

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Editor's note

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Colleague,

The one thing to know: this paper actually delivers a genuine structural step. The U(1) phase extension is a real trick — it turns Cohendet et al.'s distribution-dependent, time-nonstationary rates into fixed rates with independent particles. That is new, not a reparameterization.

The central derivation holds up. In Sec. 2.4, the first Fourier moment of ϕ obeys the lattice Liouville equation once you set D/(2|\tilde h|) = 1/\hbar. The integration by parts in Eq. (40) works because ϕ(α) is even and vanishes at the endpoints. The N/4 phase factors cancel correctly. The polar decomposition in Eq. (21) presumably makes \tilde h nonnegative, so the weights in Eq. (41) are probabilities. The paper should state that explicitly; it's a minor clarity issue.

The real problem is the boundary. Eq. (28) gives d(cos α)/dt = D cos α, so every particle with cos α ≠ 0 hits α=0 or π in finite time. The branching rule in Sec. 2.5 is one sentence: replace a particle at 0 by two at ±π/3. No rule for π, no weights, no proof it preserves the first moment or yields a Markov process for all times. The Summary's admission that branching ratios are future work confirms this is a formal derivation, not a constructed process. This is load-bearing, but I think it can be patched with a proper boundary condition. It just isn't in the paper.

Also, the N=3 example has an algebraic error: for H = S_z with ω = e^{2πi/3}, the θ values should be 3/4 and 9/4, not both 3/2. Checkable.

Who should read this: people working on stochastic formulations of quantum mechanics or phase-space QMC. There's no numerical demo, so it's not a practical tool yet. But the idea is honest and the derivation is mostly clean. It deserves a serious referee. I'd send it to review and ask for a rigorous branching rule, a proof of existence for all times, and a corrected example.

Referee report

Significance

The proposal is conceptually interesting: if the boundary/branching issue were resolved, it would give a stochastic representation in which individual particles evolve independently, in contrast to earlier stochastic formulations of the Wigner-function equation that require knowledge of the full distribution (Guerra and Marra; Cohendet et al.). The derivation is self-contained and the matching condition (33) is a normalization rather than a fitted parameter, which is a genuine strength. However, because every deterministic trajectory reaches the singular boundary in finite time and the branching mechanism is only sketched, the existence of the advertised Markov process is not yet established. The algebraic errors in Eq. (32) and in the N=3 example are local but must be corrected before the construction can be accepted.

Major objections (3)

[2.5 and Summary] The construction of the Markov process is incomplete in a way that is load-bearing for the paper's central claim. The generator in Eq. (35) contains the singular drift term (D/tan α')∂/∂α, and Eq. (28) implies d(cos α)/dt = D cos α for the non-jumping part, so every trajectory reaches α=0 or α=π in finite time unless it is exactly at α=±π/2. The branching rule is introduced only by example ('a particle which reaches α=0 can be replaced by two particles at α=+π/3 and -π/3'); no branching law is specified for α=π, no weights or daughter-phase-space distribution are given, and no proof is supplied that the resulting piecewise process is Markov, conserves total probability, exists for all t, or preserves ρ(m,n)=∫dα e^{iα}φ(α,m,n). Since the Summary states that the branching/vanishing ratio is 'a problem for future studies,' Eq. (32) is at present a formal moment identity rather than a consequence of a fully constructed stochastic process.
[2.4, Eq. (32)] Equation (32) appears to contain a spurious factor 1/i: the preceding Eq. (31) has no such factor, and the target equation (22) has none. Read literally, Eq. (32) gives dρ/dt = [D/(2i|\tilde h|)]∑..., so the substitution D/(2|\tilde h|)=1/ħ from Eq. (33) would yield dρ/dt = (1/(iħ))∑..., not the Liouville equation (22). If the factor i is a typographical artifact it must be removed; as printed, this is an error in the central derivation.
[3.3, N=3 example] The N=3 example contains an inconsistency with the polar decomposition (21). With ω=e^{2πi/3}, the value \tilde H(0,1)=i/√3 requires ω^{θ(0,1)}=i, i.e. θ(0,1)=3/4 (mod 3), while \tilde H(0,-1)=-i/√3 requires θ(0,-1)=-3/4 (mod 3), i.e. 9/4 on the interval [0,3). The quoted value θ(0,1)=θ(0,-1)=3/2 gives ω^{3/2}=e^{iπ}=-1, which would force \tilde h to be negative or imaginary and is incompatible with interpreting \tilde h as the nonnegative modulus in Eq. (21). The example should be corrected because it is the only demonstration of the simulation procedure.

Minor objections (3)

[2.3, Eqs. (21) and (41)] State explicitly that \tilde h(m,n) is the nonnegative modulus in the polar decomposition, so that \tilde h(m-m',n-n')/|\tilde h| in Eq. (41) is a probability; also require |\tilde h|>0.
[3.2, Eq. (41)] The notation P(m,n;m',n') should be defined as the conditional probability of moving from (m',n') to (m,n), since the arrow relation is not otherwise stated.
[2.5] The phrase 'a finite time interval can be maintained between the branchings' is imprecise; the intended statement should be quantified, e.g., for each particle, between successive branchings, and how the branching choices control that interval.

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The paper

The paper claims

By attaching a U(1) phase angle to every lattice phase-space point, the paper constructs a stationary Markov process whose first Fourier moment exactly obeys the quantum Liouville equation for the Wigner function.

The reading

The paper constructs a stationary Markov process on an extended lattice phase space whose ensemble reproduces the quantum Liouville equation for the Wigner function. The key device is to attach an extra continuous phase angle $\alpha\in U(1)$ to every lattice point and define the Wigner function as the first Fourier moment $\rho(m,n)=\int d\alpha\,e^{i\alpha}\phi(\alpha,m,n)$ of a nonnegative distribution $\phi$. Each particle then evolves independently: with a rate set by the Hamiltonian's lattice coefficients it jumps to another phase point, and otherwise its phase drifts deterministically. Choosing the jump rate $D/(2|\tilde h|)=1/\hbar$ makes this moment satisfy exactly the quantum Liouville equation. This yields a simulation scheme for quantum dynamics whose cost per trial is linear in the number of particles rather than requiring the whole distribution at every step.

What carries the argument

The central object is the extended phase space $U(1)\times\mathbb{Z}_N\times\mathbb{Z}_N$ together with the identity $\rho(m,n)=\int d\alpha\,e^{i\alpha}\phi(\alpha,m,n)$. This identity turns a Wigner function, which can be negative, into the first Fourier moment of a genuine probability density, so sampling $\phi$ gives $\rho$. The Markov generator in Eq. (35) has two parts: a drift in the phase $\alpha$ with coefficient $D/\tan\alpha$ that produces the singular points, and jumps whose rates are the polar-decomposed Hamiltonian coefficients $\tilde h$. The condition $D/(2|\tilde h|)=1/\hbar$ fixes the physical time scale and is what turns the moment evolution into the quantum Liouville equation.

Extended reading notes

Core claim

The paper claims that the time evolution of a Wigner function on $\mathbb{Z}_N\times\mathbb{Z}_N$ with $N$ odd is exactly the first Fourier moment of a stationary Markov process on $U(1)\times\mathbb{Z}_N\times\mathbb{Z}_N$. Concretely, if $\phi(\alpha,m,n;t)$ follows the generator in Eq. (35) and $\rho(m,n)=\int d\alpha\,e^{i\alpha}\phi$, then $\rho$ satisfies the quantum Liouville equation when $D/(2|\tilde h|)=1/\hbar$. The transition kernel is built from the polar decomposition $\tilde H(m,n)=\tilde h(m,n)\omega^{\theta(m,n)}$ of the Hamiltonian's expansion coefficients, with jump probability $\tilde h/|\tilde h|$ and a deterministic phase rotation with generator singular at $\alpha=0,\pi$. The process is stationary and treats particles independently, removing the earlier need to know the whole distribution to move one particle. Branching at $\alpha=0,\pi$ and vanishing at $\alpha=\pm\pi/2$ are built into the prescription.

Load-bearing premise

The jump rule treats the polar-decomposition coefficients $\tilde h$ as nonnegative weights; if any are negative, Eq. (41) assigns a negative transition probability and the Markov process as stated does not exist.

Editorial extensions

If this is right

Each particle in the ensemble can be moved independently, so storage and per-step cost scale with the number of simulated particles rather than with the full distribution.
Because $\phi$ is nonnegative, the Wigner function is obtained as the sampled average of the phase $e^{i\alpha}$, so no sign cancellation is needed in the Monte Carlo estimate.
The same time-independent generator works for the whole evolution, eliminating the need to solve the Schrödinger equation or recompute transition rates at each step.
Branching at $\alpha=0,\pi$ and vanishing at $\alpha=\pm\pi/2$ are part of the prescription, and a finite time interval between branchings can be maintained.
The matching condition $D/(2|\tilde h|)=1/\hbar$ connects the simulation clock to Planck's constant, making the process a direct stochastic representation of quantum dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the construction is right, a natural next step is to test the same first-Fourier-moment trick on even-sized lattices or continuous phase space; the paper does not address those cases.
The nonnegativity of $\tilde h$ may restrict the class of Hamiltonians for which the jump rule is valid; a useful extension would be to characterize that class or find an alternative decomposition that restores nonnegativity.
The branching and vanishing points could be read as a stochastic image of the negative regions of the Wigner function, with particles appearing and disappearing where positivity fails; the paper leaves that physical picture open.
Because the paper leaves the branching rate as a problem for future study, an estimator of the expected number of particles over time would determine whether the method stays practical in high dimensions.

Original abstract

As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space Z_N x Z_N with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle. The process has branching and vanishing points, though a finite time interval can be maintained between the branchings. The procedure to perform a simulation using the process is presented.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,143 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.74 · human participants 0.02 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.94 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space Z_N x Z_N with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle. The process has branching and vanishing points, though a finite time interval can be maintained between the branchings. The procedure to perform a simulation using the process is presented.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space Z_N x Z_N with N odd.	contribution 0.99	0.92	0.43	asserted 1.00
s1	By introducing a phase factor extension to the phase space, each particle can be treated independently.	method 0.94	0.84	0.11	asserted 0.99
s2	This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle.	contribution 0.66	0.51	0.17	asserted 1.00
s3	The process has branching and vanishing points, though a finite time interval can be maintained between the branchings.	result 0.45	0.81	0.11	asserted 0.94
s4	The procedure to perform a simulation using the process is presented.	method 0.81	0.78	0.06	asserted 0.95

What kind of paper the classifier reports this is: theoretical result 0.69 · evidence from proof or derivation 0.74.

What was examined, and what was not

sources read

arXiv listing abstract v4

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (12)

[1]

R. P. Feynman, A. R . Hibbs, Quantum mechanics and path integra ls, McGraw-Hill, 1965

View on Pith 1965
[2]

D. Bohm, A suggested interpretation of the quantum theory in t erms of ”hidden” variables I, II, Phys. Rev., 85, 166, 180, 1952

View on Pith 1952
[3]

J. B. Anderson, A random walk simulation of the Schr¨ odinger equ ation: H + 3 , J. Chen. Phys., 63, 1499, 1975; Quantum chemistry by random walk. H 2P, H + 3 D3h1A′ 1, H 23Σ + u , H 41Σ + g , Be1S, J. Chen. Phys., 65, 4121, 1976; Quantum chemistry by random w alk: Higher accuracy, J. Chen. Phys., 73, 3897, 1980

View on Pith 1975
[4]

J. M. Thijssen, Computational Physics, Cambridge University Pr ess, 1999

View on Pith 1999
[5]

H. Gould, J. Tobochnik, An Introduction to Computer Simulation M ethods, Chapter 18.1, Addison-Wesley, 1996

View on Pith 1996
[6]

E. Nelson, Derivation of Schr¨ odinger equation from Newtonian m echanics, Phys. Rev., 150, 1076–1085, 1966

View on Pith 1966
[7]

M. Nagasawa, Schr¨ odinger Equation and Diﬀusion Theory, Birkh ¨ auser, 1993; and references therein

View on Pith 1993
[8]

T. Hashimoto, Comments on Nelson’s quantum stochastic proces s - tunneling time and domain structure -, in Quantum Information, p.91 – 102, World Scientiﬁc, 1 999; T. Hashimoto, T. To- momura, The estimation of tunneling time by the use of Nelson’s quant um stochastic process - towards a comparison with a neutron interference experiment -; in Quantum Info...

View on Pith 2000
[9]

F. Guerra, R. Marra, Discrete stochastic variational principles and quantum mechanics, Phys. Rev. D, 29, 1647–1655, 1984

View on Pith 1984
[10]

O. Cohendet, Ph. Combe, M. Sirugue, M. Sirugue-Collin, A stoch astic treatment of the dynamics of an integer spin, J. Phys. A, 21, 2875–2883, 1988

View on Pith 1988
[11]

E. Wigner, On the quantum correction for thermodynamic equilib rium, Phys. Rev., 40, 749–759, 1932

View on Pith 1932
[12]

W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001

View on Pith 2001

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

R. P. Feynman, A. R . Hibbs, Quantum mechanics and path integra ls, McGraw-Hill, 1965

View on Pith 1965

[2] [2]

D. Bohm, A suggested interpretation of the quantum theory in t erms of ”hidden” variables I, II, Phys. Rev., 85, 166, 180, 1952

View on Pith 1952

[3] [3]

J. B. Anderson, A random walk simulation of the Schr¨ odinger equ ation: H + 3 , J. Chen. Phys., 63, 1499, 1975; Quantum chemistry by random walk. H 2P, H + 3 D3h1A′ 1, H 23Σ + u , H 41Σ + g , Be1S, J. Chen. Phys., 65, 4121, 1976; Quantum chemistry by random w alk: Higher accuracy, J. Chen. Phys., 73, 3897, 1980

View on Pith 1975

[4] [4]

J. M. Thijssen, Computational Physics, Cambridge University Pr ess, 1999

View on Pith 1999

[5] [5]

H. Gould, J. Tobochnik, An Introduction to Computer Simulation M ethods, Chapter 18.1, Addison-Wesley, 1996

View on Pith 1996

[6] [6]

E. Nelson, Derivation of Schr¨ odinger equation from Newtonian m echanics, Phys. Rev., 150, 1076–1085, 1966

View on Pith 1966

[7] [7]

M. Nagasawa, Schr¨ odinger Equation and Diﬀusion Theory, Birkh ¨ auser, 1993; and references therein

View on Pith 1993

[8] [8]

T. Hashimoto, Comments on Nelson’s quantum stochastic proces s - tunneling time and domain structure -, in Quantum Information, p.91 – 102, World Scientiﬁc, 1 999; T. Hashimoto, T. To- momura, The estimation of tunneling time by the use of Nelson’s quant um stochastic process - towards a comparison with a neutron interference experiment -; in Quantum Info...

View on Pith 2000

[9] [9]

F. Guerra, R. Marra, Discrete stochastic variational principles and quantum mechanics, Phys. Rev. D, 29, 1647–1655, 1984

View on Pith 1984

[10] [10]

O. Cohendet, Ph. Combe, M. Sirugue, M. Sirugue-Collin, A stoch astic treatment of the dynamics of an integer spin, J. Phys. A, 21, 2875–2883, 1988

View on Pith 1988

[11] [11]

E. Wigner, On the quantum correction for thermodynamic equilib rium, Phys. Rev., 40, 749–759, 1932

View on Pith 1932

[12] [12]

W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001

View on Pith 2001