Mixed-state geometric phases of coherent and squeezed spin states

Chih-Chun Chien; Hao Guo; Jia-Chen Tang; Xin Wang; Xu-Yang Hou

REVIEW 2 major objections 3 minor 78 references

Mixed-state geometric phases of coherent and squeezed spin states

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read At finite temperature, both the Uhlmann phase and the interferometric geometric phase of a $j=3/2$ coherent spin state and a $j=1$ one-axis squeezed spin state jump abruptly, while the $j=1$ two-axis squeezed spin state avoids such…

desk verdict Clean exact IGP results for CSS and one-axis squeezing, but the two-axis no-transition claim rests on a wrong trace in Eq. (51); corrected trace is real and shows finite-temperature jumps. read the letter →

arxiv 2502.07268 v4 pith:WTY27TPK submitted 2025-02-11 quant-ph cond-mat.mes-hall

Xin Wang , Jia-Chen Tang , Xu-Yang Hou , Hao Guo , Chih-Chun Chien This is my paper · ORCID

classification quant-phcond-mat.mes-hall PACS 03.65.Vz

keywords Uhlmannphaseinterferometricgeometricspincoherentstatessqueezedfinite-temperaturetopologicaltransitionone-axistwistingtwo-axiscounter-twistingmixed-statephases

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether mixed-state geometric phases can signal finite-temperature phase transitions in spin systems, and answers with exact and numerical calculations for three canonical state families. For a spin $j=3/2$ coherent state and a $j=1$ one-axis squeezed state, the Uhlmann phase and the interferometric geometric phase both change discontinuously at specific temperatures, with the Uhlmann phase jumping by quantized values. The same quantities vary smoothly for a $j=1$ two-axis squeezed state, so the presence or absence of a transition depends on which spin-state family is used. The reason to care is that these phases are measurable or simulable, so they can expose topological or geometric structure of finite-temperature quantum states.

What carries the argument

The two calculational engines are the Uhlmann phase and the interferometric geometric phase. The Uhlmann phase is the holonomy of the Uhlmann bundle, obtained from parallel transport of a purification $W=\sqrt{\rho}\,U$ of the density matrix, with $\theta_U(C)=\arg\mathrm{Tr}\left[\rho(0)\,\mathcal{P}e^{-\oint_C A_U}\right]$. The IGP is the phase $\theta_G=\arg\mathrm{Tr}[\rho(0)U(t)]$ accumulated under a unitary $U(t)$ that satisfies the parallel-transport condition $\langle n(t)|\dot U U^\dagger|n(t)\rangle=0$ for every eigenstate $|n(t)\rangle$ of $\rho(t)$. The paper evaluates both phases for the Gibbs families above, using explicit identities for $D^\dagger dD$, $S^\dagger dS$, and $K^\dagger dK$, and the difference in behavior is tied to the Hamiltonian structures the unitaries generate.

What would settle it

Compute the Uhlmann phase for the $j=3/2$ coherent spin state along $\theta=\pi/2$ by directly integrating the parallel-transport equation for $\sqrt{\rho}$ as a function of temperature; if no discontinuous jump in $\theta_U$ occurs near $T\approx0.321\,\omega_0$, $0.376\,\omega_0$, and $0.493\,\omega_0$, the claimed topological transitions are artifacts of the calculation. Alternatively, in an NMR or trapped-ion experiment, prepare the thermal family and measure the IGP for the one-axis squeezed state at fixed $\Theta_f$ while sweeping $T$; a failure to find a $\pi$-jump near $\Theta_f^c=4\arccos\left(-\tfrac12\operatorname{sech}(\omega_0/T)\right)$ would falsify the central prediction.

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Extended reading notes

Core claim

The central discovery is that temperature alone, without changing any Hamiltonian parameter, can switch a mixed-state geometric phase between $0$ and $\pi$ in certain spin-state families. Working within thermal Gibbs families $\rho(\lambda)= e^{-\beta H(\lambda)}/Z$ generated by the coherent displacement $D(\zeta)=e^{\zeta J_+-\bar\zeta J_-}$, the one-axis squeezing $S(\Theta)=e^{-i\Theta J_x^2/2}$, and the two-axis squeezing $K(z)=e^{zJ_+^2-\bar z J_-^2}$, the authors find that the $j=3/2$ coherent spin state along the equator has Uhlmann-phase jumps at $T_1^c\approx0.321\,\omega_0$, $T_2^c\approx0.376\,\omega_0$, and $T_3^c\approx0.493\,\omega_0$, while the IGP jumps at a temperature that depends on the final polar angle, with $T_c=0.408\,\omega_0$ for $\theta_f=3\pi/4$. The $j=1$ one-axis squeezed state exhibits a Uhlmann-phase jump at $T_c\approx0.68\,\omega_0$ and IGP jumps whose critical squeezing parameter obeys $\Theta_f^c=4\arccos\left(-\tfrac12\operatorname{sech}(\omega_0/T_c)\right)$. The $j=1$ two-axis squeezed state, by contrast, has both phases continuous in temperature, indicating no finite-temperature geometric transition.

Load-bearing premise

The calculations take the state at every parameter value to be the thermal equilibrium Gibbs state of the instantaneous Hamiltonian, $\rho(\lambda)= e^{-\beta H(\lambda)}/Z$, so if an actual experiment or simulation does not remain in this thermal family, the computed Uhlmann and IGP values will not describe what is measured.

Editorial extensions

If this is right

If the calculations are right, a $j=3/2$ coherent spin state cooled through $T\approx0.321\,\omega_0$ along the equator would show a measurable switch in Uhlmann phase from $\pi$ to $0$, followed by further jumps at $T\approx0.376\,\omega_0$ and $T\approx0.493\,\omega_0$.
For the $j=1$ one-axis squeezed state, the Uhlmann phase drops from $\pi$ to $0$ at $T\approx0.68\,\omega_0$, while the IGP jumps by $\pi$ when the squeezing parameter reaches $\Theta_f^c=4\arccos\left(-\tfrac12\operatorname{sech}(\omega_0/T)\right)$, giving two independent finite-temperature geometric signatures.
The two-axis squeezed state can be warmed through the same temperature range without any jump, so the presence or absence of a transition is controlled by the squeezing protocol itself, not just by temperature.
In the zero-temperature limit the Uhlmann phase for the coherent spin state reduces to the Berry phase of the ground state, recovering the Uhlmann-Berry correspondence for these states.
Because a spin-$j$ state can be assembled from $2j$ qubits, the predicted jumps are concrete targets for ancilla-based Uhlmann-phase measurement protocols on quantum simulators.

Reading between the lines

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

Going beyond the paper, the same machinery suggests a diagnostic rule: unitary families whose generator couples only states with $\Delta m=0$ or $\pm2$, as in two-axis squeezing, may suppress temperature-induced Uhlmann jumps, while families with $\Delta m=\pm1$ coupling, as in coherent displacement, allow them.
The paper treats each state as the Gibbs state of the instantaneous Hamiltonian; if a physical experiment slowly sweeps the squeezing parameter in a closed cycle instead, the realized density matrix will generally lag behind this thermal family, and the predicted jumps would broaden or shift. Testing this with a Lindblad simulation would isolate how much of the transition is an equilibrium propert
Because the IGP jump for the coherent spin state occurs only when the final polar angle lies in $(\pi/2,3\pi/2)$, the phenomenon is path-dependent; a natural next question is whether the critical temperature itself depends on the chosen loop geometry in a way that could be engineered in interferometric experiments.
The absence of transitions for two-axis squeezing, if confirmed, connects naturally to the fact that its parameter manifold is two-dimensional while the one-axis case is one-dimensional; this suggests a topological origin for the difference that the paper does not fully unpack.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me get straight to the point. The paper has a set of exact results that look right—the j=3/2 CSS IGP and the j=1 one-axis SSS IGP are clean and new—but the central contrast, that the two-axis SSS shows no finite-temperature transitions, comes from a wrong trace evaluation in Eq. (51). The stress-test note is correct: for the chosen path φ=π/2 and j=1, z=-i tan(θ/2), and K(z) in the |±1> block is exp(-2i t σ_x) with t=tan(θ/2). So K_{11}=cos(2t), K_{1,-1}=-i sin(2t). Tracing against ρ(0)=diag(e^{βω_0},1,e^{-βω_0})/(2 cosh(βω_0)+1) gives a real number: [2 cosh(βω_0) cos(2t)+1]/[2 cosh(βω_0)+1]. The printed Eq. (51) has an imaginary part and an extra constant that cannot come from this trace. The real numerator changes sign for cos(2t) ∈ (-0.5,0), so the IGP jumps from 0 to π at a critical temperature. Taking t=0.95 gives a jump near βω_0≈1.0, inside the plotted range [0,3π/4]. That directly contradicts Fig. 6's top panel and the paper's no-transition conclusion.

What's good: Sections III and IV.A are mostly solid. The CSS Uhlmann phase is a sign-flipped version of the authors' own spin-j result from Ref. [68], which they state openly; the j=3/2 jumps are then an evaluation of that formalism, so they're not completely new but they're correctly derived. The CSS IGP formula in Eq. (33) and the one-axis SSS IGP formula in Eq. (39) with the transition condition Eq. (40) are genuinely new and checkable. The paper is clearly written and does a fair job of distinguishing Uhlmann phase from IGP. The one-axis Uhlmann phase is numerical and lacks Trotter convergence data; that's a minor but real request for revision.

The two-axis Uhlmann phase is also numerical, and I don't see a comparable error there, but the IGP error is enough to sink the paper's headline. The thermal-family assumption is fine; conjugating a Gibbs state by a unitary is a legitimate construction within the model, so I don't weight that concern heavily.

Verdict: reject as is, but this is fixable. Replace Eq. (51), recompute Fig. 6, and the two-axis conclusion likely changes from 'smooth' to 'jumps for some range'. A serious referee should see the corrected version, but the current manuscript shouldn't be published. I'd send it to review rather than desk reject, because the exact formulas are easy to check and the error is specific enough to fix in one round.

Referee Report

2 major / 3 minor

Summary. The paper analyzes two mixed-state geometric phases, the Uhlmann phase and the interferometric geometric phase (IGP), for coherent spin states (CSSs) and one- and two-axis spin squeezed states (SSSs). It presents exact formulas for the j=3/2 CSS and j=1 one-axis SSS and reports numerical results for the j=1 two-axis SSS. The central claimed finding is that the CSS and one-axis SSS exhibit finite-temperature phase transitions in both phases, whereas the two-axis SSS varies smoothly with temperature and therefore shows no temperature-induced transitions. The paper also discusses possible experimental and quantum-simulation implementations.

Significance. If the central contrast were correct, the paper would provide clean exact examples of mixed-state geometric phases that behave differently under temperature variation, with possible implications for metrology and finite-temperature topology. The CSS Uhlmann result is explicitly a sign-modified special case of the authors' earlier spin-j formalism (Ref. [68]), and the one-axis IGP formula is simple and appears correctly derived from the stated trace. These parts are useful, but the central claim about the two-axis SSS rests on Eq. (51), which is demonstrably incorrect; the corrected trace is real and exhibits finite-temperature jumps for the plotted parameter range. The claimed smoothness of the two-axis IGP and the corresponding conclusion are therefore unsupported.

major comments (2)

[Sec. IV.B.2, Eq. (51)] The printed formula for Tr[rho(0)K] is incorrect. Along the chosen meridian phi=pi/2, with alpha=2 tan(theta/2), the j=1 matrix K has K11=K33=cos alpha, K13=-i sin alpha, K31=i sin alpha, and K22=1, while rho(0) is diagonal in the Jz basis. The off-diagonal elements of K therefore do not contribute to the trace, and the exact result is Tr[rho(0)K] = [2 cosh(beta omega0) cos alpha + 1] / [2 cosh(beta omega0) + 1], which is real. The imaginary term -2i sin(...) in Eq. (51) cannot appear. The real numerator changes sign when cosh(beta omega0) = -1/(2 cos alpha) for cos alpha in (-1/2,0); for example, alpha = 1.9 rad gives a sign change at beta omega0 ~ 0.98, so the IGP jumps from 0 to pi at finite temperature for a theta_f value inside the plotted range [0, 3pi/4]. This directly contradicts the top panel of Fig. 6 and the paper's conclusion that the two-axis SSS IGP varies smoothly with temperature.
[Sec. VI and Abstract] The conclusion that the j=1 two-axis SSS has no temperature-induced transitions is the paper's central message, but it is based on the erroneous Eq. (51). Since the corrected evaluation produces finite-temperature jumps for some squeezing parameters, the abstract, the concluding paragraph, and the discussion of a qualitative difference between two-axis squeezing and the other cases need to be revised. The two-axis IGP result cannot be repaired locally without changing the main claim of the paper.

minor comments (3)

[Sec. III.A, Eq. (25)] The denominator in the expression for the Uhlmann connection is written with lambda_n + lambda_n, which should be lambda_n + lambda_m; this appears to be a typographical error that does not affect the subsequent result, but it should be corrected.
[Sec. IV.B.1, Eq. (41)] The line 'where \hat H = e^{-\beta \omega_0 \hat J_z}' should read '\hat H = \omega_0 \hat J_z'; as printed, the expression is dimensionally inconsistent and the density matrix notation is confusing.
[Sec. IV.B, Eqs. (45) and (47)] The notation sin2(2), sin(4), and sin(8) is ambiguous: it is unclear whether these are powers of sine, products, or sine functions of arguments measured in radians; the numerical coefficients in Eq. (47) should be checked and the notation clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the results are direct evaluations of the Uhlmann and IGP definitions from stated thermal families, with no fitted parameters; the main concern is an apparent algebraic error in Eq. (51), which is a correctness issue rather than circularity.

full rationale

The paper's central quantities are obtained by substituting explicitly stated unitary families into the defining formulas for the Uhlmann connection and the IGP trace: Eqs. (24), (10), and (41) define the thermal density matrices, Eq. (16) defines the Uhlmann connection, and Eqs. (22), (37), and (51) define the IGP. No free parameters are fitted, and no transition criterion is imposed from outside the calculation. The CSS Uhlmann connection is derived in Eqs. (25)-(28) from the Uhlmann formula, and the statement that it differs by a sign from Ref. [68] is a reuse/cross-check of a parameter-free prior result by the same group, not a logical reduction of the present derivation to its own input. The IGP results for the CSS and one-axis SSS are closed-form evaluations of the defining trace; the two-axis IGP is likewise written as the trace of rho(0)K(z), although the printed evaluation in Eq. (51) appears to contain an algebraic error because the trace should be real while Eq. (51) includes an imaginary sine term. That issue, if confirmed, would invalidate the claim of smooth two-axis IGP behavior, but it is a mathematical error and not a circular derivation. The high-temperature Uhlmann limits are supported by independent arguments and a cited prior result, and the T-to-zero Berry correspondence is proven in Appendix A. There is no fitted parameter renamed as a prediction, no uniqueness theorem imported from a self-citation to force a choice, and no ansatz smuggled in through citation. Therefore the paper has no significant circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No fitted physical constants are introduced; the Trotter discretization step is unstated. The derivations rely on standard geometric-phase formalisms and on the Gibbs-family assumption, and no new particles, fields, or conserved quantities are proposed.

free parameters (2)

Trotter-Suzuki path-discretization step = not reported
The j=1 one-axis SSS Uhlmann phase (Fig. 3) is computed numerically as a path-ordered exponential; the discretization step and convergence threshold are not given, so the stated Tc about 0.68 omega0 depends on an unstated numerical parameter.
two-axis IGP theta range cutoff = 3pi/4
Chosen to avoid rapid oscillations near theta = pi; it restricts the plotted IGP results in Fig. 6 and is a hand-set numerical cutoff.

assumptions (4)

standard math Uhlmann parallel-transport condition and connection formula in Eq. (16) correctly define the Uhlmann phase.
Standard formalism from Refs. [17,63]; not re-derived in this paper.
standard math The IGP is arg Tr[rho(0)U(t)] under the strengthened parallel-transport condition, and this condition is always satisfiable.
Standard IGP framework from Ref. [27].
domain assumption The density matrix at every point on the parameter loop is the Gibbs state of the instantaneous Hamiltonian: rho(lambda) = exp(-beta H(lambda))/Z.
Used for CSS in Eq. (24), for one-axis SSS in Eq. (10), and for two-axis SSS in Eq. (41). The computed phases describe an equilibrium family, not an arbitrary non-equilibrium evolution.
ad hoc to paper The chosen paths (equator for CSS Uhlmann, longitude for CSS IGP, Theta in [0,4pi] for one-axis SSS, equator and meridian for two-axis SSS) are representative and physically relevant.
The presence or absence of jumps is path-dependent; the paper only checks these specific loops and does not prove robustness to other paths.

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Cite this review

Pith. "Pith review of Mixed-state geometric phases of coherent and squeezed spin states." pith.science (2026). https://pith.science/paper/WTY27TPK

@misc{pith2026250207268,
  author       = {Pith},
  title        = {Pith review of: Mixed-state geometric phases of coherent and squeezed spin states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WTY27TPK}},
  note         = {Machine review of arXiv:2502.07268}
}

abstract

Two mixed-state geometric phases, known as the Uhlmann phase and interferometric geometric phase (IGP), of spin coherent states (CSSs) and spin squeezed states (SSSs) are analyzed. Exact solutions and numerical results of selected examples are presented. For the $j = 3/2$ CSS, the Uhlmann phase exhibits finite-temperature topological phase transitions characterized by abrupt jumps. The IGP for the same state similarly shows discontinuous jumps as the temperature varies. In the case of the $j = 1$ one-axis SSS, both Uhlmann phase and IGP display discrete finite-temperature jumps. By contrast, the $j = 1$ two-axis SSS shows no such transitions because the Uhlmann phase and IGP both vary smoothly with temperature. We also briefly discuss potential realizations and simulations related to these phenomena in spin systems.

Figures

Figures reproduced from arXiv: 2502.07268 by the authors.

**Figure 1.** Uhlmann phase θU versus temperature for the CSS with j = 3 2 along θ = π 2 . Jz =   − 3 2 0 0 0 0 − 1 2 0 0 0 0 1 2 0 0 0 0 3 2   . (29) We also assume that the evolution starts from φ = 0, so the initial density matrix is given by ρ(0) = 1 Z e iπ 2 Jy e −βω0Jz e − iπ 2 Jy according to Eq. (24). A straightforward evaluation shows that the Uhlmann phase generated in this process is θU = arg Tr 1 Z e βω0Jx… view at source ↗

**Figure 2.** (Top panel) Contour plot of the IGP θG versus temperature T and θf for j = 3 2 . The geometric phase transitions when θf ∈ ( π 2 , 3π 2 ) are indicated by the boundaries separating the two values of θG. (Bottom panel) θG versus T for the case with θf = 3π 4 . a two-level Hermitian quantum system, a finite change in temperature in general does not induce any jump of the IGP [38, 39]. For a three-level system, a care… view at source ↗

**Figure 3.** Uhlmann phase θU as a function of temperature T for the j = 1 one-axis SSS. A π-valued jump is observed at Tc ≈ 0.68ω0. It is in general challenging to find a nontrivial evolution path along which AU is proportional to a constant matrix. The difficulty thus hinders the discovery of an analytical expression of the Uhlmann phase. Nevertheless, numerical results can be obtained by using the Trotter-Suzuki approximati… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The IGP θG as a function of T and Θf for the j = 1 one-axis SSS, where Θf ∈ [0, 4π]. θG exhibits discontinuities with respect to both Θf and T . At finite temperatures, θG always experiences a π-jump when crossing Θc f = 4 arccos − sech ω0 T /2 , indicating a geom…

**Figure 5.** Figure 5: Uhlmann phase θU of the j = 1 two-axis SSS as a function of temperature T on semi-log scale. The smooth curve indicates the absence of a transition. However, the main difference lies in the complexity of K(z) = ezJ2 +−zJ¯ 2 − , which can be difficult to disentangle. As…

**Figure 6.** Figure 6: (Top panel) Contour plot of the IGP θG of the j = 1 two-axis SSS as a function of T and θf , where θf ∈ [0, 3π 4 ]. θG is continuous with respect to both θf and T (apart from a jump at T = 0). (Bottom panels) θG vs T at θf = π/2, where θG exhibits no jumps (left), and …

Discussion (0). Continue with ORCID to comment.

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angle” between them undergoes an abrupt transition from “parallel

Uhlmann Phase For the SSS shown in Eq. (5), the parameter space is one-dimensional. Therefore, we need to determine the value of Θ to deﬁne a cyclic process for density ma- trices, which is essential for generating the Uhlmann phase. The density matrix of a system in thermal equi- librium governed by H(Θ) show in Eq. (10) is given by 7 ρ(Θ) = 1 Z e−βH (Θ)...

work page

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(21), the oper- ator ˜S automatically satisﬁes the strengthened parallel- transport condition of the IGP given by Eq

The interferometric geometric phase As shown in the discussion around Eq. (21), the oper- ator ˜S automatically satisﬁes the strengthened parallel- transport condition of the IGP given by Eq. (20). Ac- cordingly, the IGP accumulated during this evolution is given by θG(t) = arg Tr[ρ(0) ˜S(t)] = arg ( ∑ m λ mνmeiφ m ) , (37) where λ m is the m-th eigenvalu...

work page

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(11) under two-axis squeez- ing, the parameter space becomes two-dimensional

Uhlmann Phase For the SSS given by Eq. (11) under two-axis squeez- ing, the parameter space becomes two-dimensional. The Uhlmann process can be generated by a cyclic path ei- ther on the complex plane or equivalently on the sphere parametrized by ( θ, φ ) with z = e −iφ tan ( θ 2 ) . The den- sity matrix of a system in thermal equilibrium governed by the ...

work page

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Of course, such a path also respects the condition (19)

The IGP For the two-axis SSS, we ﬁnd that the strengthened parallel transport condition (20) of the IGP holds for a nontrivial process which starts from the pole at θ = 0 and proceeds along the meridian with φ = π 2 under the transformation K(θ, φ ). Of course, such a path also respects the condition (19). The density matrix ρ(t) of this unitary evolution...

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