Experimental detection of microscopic environments using thermodynamic observables

Ivan Henao; Nadav Katz; Raam Uzdin

REVIEW 4 major objections 4 minor 49 references

Experimental detection of microscopic environments using thermodynamic observables

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Thermodynamic inequalities can act as black-box detectors: a negative passive-observable test reveals that a quantum system has been touched by a hidden environment.

desk verdict A credible first experimental demonstration of thermodynamic heat-leak detection on IBM hardware, but the 'unambiguous' claim outruns the error bars because calibration and initial-state systematics are not propagated. read the letter →

arxiv 1908.08968 v3 pith:3HR3EV6A submitted 2019-08-23 quant-ph

Ivan Henao , Raam Uzdin , Nadav Katz This is my paper · ORCID

classification quant-ph

keywords thermodynamicinequalitiesnon-unitaldynamicsheatleakdetectionpassiveobservablesglobalpassivityquantumerrordiagnosticsshadowestimationthermodynamics

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

The paper claims that a violation of a thermodynamic inequality can act as a black-box detector for a hidden environment: if the change in the mean value of a suitably passive observable is negative, the system's evolution cannot be unital, so it must have leaked heat. For initial states that are products of thermal states, the relevant observables are built from $B=-\ln\rho_s$, and experiments with four-qubit and three-qubit systems coupled to a single-qubit environment show that negative values appear only when the environment is switched on. The same data show that shifted and deformed passive observables can detect leaks that the standard Clausius inequality and simple powers of $B$ miss. The paper also proves that evaluating these tests requires a number of measurements that scales polynomially with system size, in contrast to tomography-based diagnostics.

What carries the argument

The load-bearing object is the family of passive observables $F_{\alpha,\delta}=(B-\delta I)^\alpha$, where $B=-\ln\rho_s$ and $\delta$ is a shift chosen so that the eigenvalues of $F$ remain a non-decreasing function of the eigenvalues of $B$. Passivity guarantees $\Delta\langle F_{\alpha,\delta}\rangle\ge0$ under any unital transformation, so a negative value signals non-unitality; the expansion $\Delta\langle(B-\delta I)^\alpha\rangle=\sum_{k=0}^\alpha \binom{\alpha}{k}(-1)^k\delta^k\Delta\langle B^{\alpha-k}\rangle$ shows why the shift works, since odd-$k$ terms carry negative signs and can outweigh positive even-$k$ contributions. A second mechanism, passivity deformation, replaces $B$ by $B_{\rm def}=\sum_i \beta_i^{({\rm def})}H_i$ with effective temperatures that keep the observable passive, which reweights the expansion and enables detection when unshifted observables fail. On the resource side, the paper uses shadow estimation of local $k$-qubit operators $\tilde{H}_{k,\{i_m\}}$ to show that $\Delta\langle F_{\alpha,\delta}\rangle$ can be evaluated with $N\sim O\!\left(\frac{\log(n/\mu)}{\varepsilon^2}(n+\delta)^{2\alpha}\right)$ measurements, polynomial in the number of qubits.

What would settle it

Prepare a product thermal state on a processor, apply only unital channels (random mixtures of unitaries or depolarizing noise) with the ancilla environment decoupled, and check that every $\Delta\langle F_{\alpha,\delta}\rangle$ stays nonnegative; then deliberately perturb the readout-correction matrix $M$ (for example, swap the detection probabilities of one qubit) and see whether a negative test appears. If a calibration error alone can produce a violation, the 'hidden-environment' reading of the inequality is not self-contained.

Watch

Extended reading notes

Core claim

The central claim is that $\Delta\langle F\rangle=\mathrm{Tr}[F(\rho_s'-\rho_s)]<0$ for a passive observable $F$ is a certificate of non-unital dynamics, which the authors call a heat leak. Given an initial product of thermal states $\rho_s=\otimes_i e^{-\beta_i H_i}/\mathrm{Tr}(e^{-\beta_i H_i})$, the observable $B=-\ln\rho_s$ satisfies $\Delta\langle B\rangle\ge0$ for every unital evolution; the same holds for every $F$ that commutes with $B$ and has eigenvalues given by a non-decreasing function of the eigenvalues of $B$. The experiments implement engineered environment couplings on small superconducting processors: with the environment coupled, the test $\Delta\langle F_{3,\delta}\rangle$ with $F_{3,\delta}=(B-\delta I)^3$ becomes negative in the interval $1.8\lesssim\delta\lesssim2.6$, while $\Delta\langle B\rangle$ and $\Delta\langle F_{2,\delta}\rangle$ do not detect it; in a second experiment, a deformed observable $B_{\rm def}=\beta_0(H_1+H_3)$ makes $\Delta\langle F^{({\rm def})}_{5,\delta}\rangle$ negative for $0.8\lesssim\delta\lesssim2.1$. A theoretical analysis shows that all these tests can be estimated from mean values of local observables, with a measurement count scaling polynomially in the number of qubits.

Load-bearing premise

The whole detection logic rests on the initial state being exactly the product of thermal states in Eq. (1) after readout correction by the calibrated matrix $M$; if that inversion is biased or the state carries residual correlations, a negative test value could appear without any hidden environment, or a real leak could be missed.

Editorial extensions

If this is right

Heat-leak detection becomes a black-box test: only the initial and final mean values of passive observables are needed, with no trajectory information and no model of the noise channel.
Because the measurement count grows polynomially with the number of qubits for fixed observable power and shift range, the tests remain feasible where full state or process tomography is not.
Shifted and deformed passive observables detect leaks that the standard Clausius inequality $\Delta\langle B\rangle\ge0$ and unshifted powers of $B$ miss, so observable choice expands the detectable class.
The tests are self-contained: they need no comparison with a classical simulation of the ideal circuit, which becomes intractable for large devices.
In a constructed example, global passivity detects an environment that leaves resource-theory free-energy constraints unviolated, so the method covers cases where other thermodynamic frameworks fail.

Reading between the lines

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

The same violation-as-diagnostic principle should transfer to other platforms—bosonic modes, trapped ions, or photonic circuits—where $B=-\ln\rho_s$ can be defined and passive observables measured, although the paper only demonstrates qubit processors.
Because the paper does not propagate uncertainty in the readout-correction matrix $M$ into the reported confidence intervals, a natural extension is a sensitivity analysis quantifying how much calibration bias can fake or mask a heat leak.
Sweeping $\alpha$ and $\delta$ yields a curve of test values that may act as a coarse fingerprint of the non-unital part of a noise channel, going beyond binary detection toward classification.
The same thermodynamic logic could be applied to detect dephasing by choosing passive observables in rotated bases, since pure dephasing is unital and will not trigger the current tests; the paper lists dephasing as an open problem.

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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

Colleague,

The one-liner: this is the first experiment I know of that actually runs these passive-observable "heat leak" tests on superconducting hardware, and it shows the idea can work. But the paper oversells the word "unambiguous": the reported confidence intervals ignore exactly the systematics that could produce a false violation.

What's genuinely new: the Melbourne and Essex experiments, the passivity-deformed observables (setting one beta to zero), and the supplemental scaling bound using shadow tomography. The authors also do the right thing in the controls: they run the environment decoupled, they simulate the ideal circuit, and they check that the measured initial state is close to a product thermal state. They are honest in the discussion that the tests miss some non-unital errors and don't touch dephasing. The polynomial measurement cost is correctly stated as conditional on fixed alpha and delta ranges.

The first soft spot is a real theoretical error: the text says every unital evolution is a convex mixture of unitaries. That's false for d>2. The inequality itself probably survives because for any unital channel the dephased final state is majorized by the initial product state, and monotone functions of B then give Delta >=0. But the proof as written is wrong and should be fixed.

The bigger issue is experimental. In Melbourne, B is built from the initial populations recovered from M^{-1}, and the same populations are used to evaluate Delta. The error bars in the supplemental are shot-noise only. They don't propagate uncertainty in the readout matrix, the 10-batch spread in the initial populations, or the 0.005 L2 distance from the nearest product thermal state. If M^{-1} is slightly biased, the function -ln p can change a lot for low-population states, and a unital or even unitary evolution on a slightly non-thermal state could produce a negative Delta in the reported window. The decoupled control is not a clean isolation of the engineered environment: it removes the CNOTs, so it also removes their gate errors. So the central claim, that a negative Delta unambiguously flags a heat leak, lacks a demonstrated robustness bound.

There is also some post-hoc observable selection: they scan alpha and delta and report the region that works, without a multiple-comparison correction. That's a minor issue given the decoupled control and simulation, but worth noting.

Who should read this: people building quantum-device validation tools, and quantum thermodynamics experimentalists. It deserves a serious referee; the experiments are real and the concept is useful. My recommendation: revise, fix the unital statement, propagate the calibration and initial-state uncertainty, and then it's publishable.

Referee Report

4 major / 4 minor

Summary. This paper proposes to detect a hidden environment coupled to a quantum system by looking for violations of thermodynamic inequalities. For a product thermal initial state ρ_s = ⊗ e^{-β_i H_i}/Z_i, passive observables F that are diagonal in the energy basis with non-decreasing eigenvalues are predicted to satisfy Δ⟨F⟩ = Tr[F(ρ'_s − ρ_s)] ≥ 0 for unital evolutions. A negative Δ⟨F⟩ is therefore interpreted as a heat leak, i.e., a non-unital component. The authors implement tests with F_{α,δ} = (B − δI)^α and with a deformed observable B_def on IBM quantum processors: a four-qubit system plus one-qubit environment (Melbourne) and a three-qubit system plus one-qubit environment (Essex). They report detection windows (Melbourne 1.8 ≲ δ ≲ 2.6 for F_{3,δ}; Essex 0.8 ≲ δ ≲ 2.1 for a deformed F^{(def)}_{5,δ}) that are negative by more than three standard deviations when the environment is coupled, and non-negative in the decoupled-environment controls. A supplemental analysis uses classical shadows to argue that evaluating these tests requires only O(poly(n)) measurements for fixed α and δ.

Significance. If the claims hold, the work offers a scalable, tomography-free diagnostic for non-unital errors in quantum devices, with the unusual feature that the test is independent of the details of the noise model. Strengths of the manuscript include genuine hardware experiments on two IBM processors, decoupled-environment control runs, and a scaling argument that correctly reduces the estimation of (B − δI)^α to local observables whose shadow norm is independent of n. The central inequality itself is more robust than the paper's stated justification: it holds for all unital maps via the doubly stochastic action on diagonals, not only for mixtures of unitaries, so the detection principle is theoretically sound. However, the experimental evidence for unambiguous detection is not yet airtight because the readout-calibration uncertainty and the non-unital gate-error component of the environment CNOTs are not quantified. With a corrected theoretical statement and the requested robustness analysis, this would be a worthwhile contribution to quantum device characterization.

major comments (4)

[Main text, paragraph after Eq. (2)] The statement 'For any unital evolution, the final state ρ′s can be written as ρ′s = ∑i qi U(i)s(ρs)U(i)†s' is false for dimensions d > 2, and the following 'i.e.' incorrectly equates 'non unital' with 'not representable as a mixture of unitaries'. In the present four-qubit setting (d = 16), this is a substantive mathematical error. The inequality Δ⟨F⟩ ≥ 0 does, however, hold for every unital map when ρs is diagonal in the eigenbasis of F, because the diagonal of the output state is a doubly stochastic image of the initial diagonal and F has non-decreasing eigenvalues. I recommend replacing the mixture-of-unitaries justification with this majorization argument and defining non-unitality in the standard sense; the detection logic (violation ⇒ non-unital) then remains valid.
[Supplemental 'Detector noise and characterization of initial states' and 'Statistical error'; main text around Eq. (4)] The error bars do not include the uncertainty in the readout calibration matrix M or the statistical error in the estimate of {p_i^s} from the ten batches, even though B and F are built from those same {p_i^s}. Equations (S12)–(S15) treat F_{α,δ} as a fixed operator and use only the shot noise of the initial and final populations. A biased M^{-1} (from crosstalk or drift) or the reported 0.005 L2 distance between the recovered populations and the nearest product thermal state could reorder the eigenvalues of B and make Δ⟨F_{3,δ}⟩ negative for a unitary or unital evolution. Please provide a quantitative bound: propagate the uncertainty in M (for example, by bootstrapping the calibration data) and verify that the detection intervals 1.8 ≲ δ ≲ 2.6 (Melbourne) and 0.8 ≲ δ ≲ 2.1 (Essex) survive that propagation.
[Main text, Figs. 2 and 3 and the discussion of the decoupled controls] The decoupled-environment control removes the environment CNOTs, so it does not control for the additional gate errors introduced by those CNOTs in the coupled circuit. The observed negative Δ⟨F⟩ could therefore be caused by the non-unital component of the CNOT gate errors themselves rather than by the engineered thermal coupling. This does not invalidate the method—any actual non-unital component is a heat leak—but it weakens the specific physical claim that the source is the intended environment. An additional control that varies the coupling strength (the number of environment CNOTs or the environment temperature) and shows a systematic dependence of the detection interval would substantially strengthen the demonstration.
[Supplemental, section on sensitivity of heat leak tests, after Eq. (S7)] The derivation gives Δ⟨F⟩ = −Σ_j Δf_j arξ_j, so a negative Δ⟨F⟩ requires a positive arξ_j. Two sentences later the text states that Δ⟨F⟩ < 0 'only if arξ_j < 0', which is inconsistent with the displayed formula. Since the proof of Theorem 1 is consistent with the formula, this appears to be a sign typo, but it should be corrected because it obscures the logic of the sensitivity claim.

minor comments (4)

[Supplemental, 'Statistical error'] In the sentence about the Essex processor, the text says 'n = nMel = 4 batches'; the symbol should be nEss (or a similar distinct label) to avoid confusion with the Melbourne batch count.
[Abstract and main text] There are several typographical errors: 'enviroment' in the abstract, 'in constrast' in the introduction, and 'not a applied' in the supplemental material. A careful proofreading pass is needed.
[Main text, paragraph after Fig. 2] The sentence 'for 0 ≤ α ≤ 4 no detection occurs with the observables B^α' should be qualified as 'no unambiguous detection within three standard deviations', since the supplemental data show that the mean value of Δ⟨B^α⟩ becomes negative for α ≳ 3.4 even though the confidence interval still contains positive values.
[Main text, Eq. (6)] The expansion in Eq. (6) is written with a sum over k from 0 to α, and the sign discussion is helpful; it would be even clearer to state explicitly that the bound on δ from the passivity condition must be imposed when α is even, as is later done for F_{2,δ}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the thermodynamic tests are based on externally checkable inequalities and measured populations, not on fitted predictions.

full rationale

The derivation chain is self-contained. Equations (2) and (3) are stated as mathematical constraints for unital dynamics on product-thermal initial states, with Eq. (2) attributed to the Clausius inequality [34] and Eq. (3) to the passive-observable theorem of [27]; neither is inferred from the experimental data, and the cited theorem is parameter-free and does not contain the detection claim. The experiment measures initial and final populations in independent circuits, constructs B_i = -ln p_i^s from the measured initial populations, and evaluates Eq. (4) directly; the sign of Delta<F> is not fitted to produce detection. A violation of the inequality implies non-unital dynamics by the theorem, not by redefinition: 'heat leak' is introduced as a description of non-unital behavior, and the inequality is the evidence. The polynomial scaling result is derived in the Supplemental from the external shadow-tomography bound [29] plus error propagation; it is anchored in the paper's own appendix only in the sense that the derivation is included there. Self-citations [27,28] are present but the load-bearing result is a published theorem with stated assumptions that do not include the target experiment, so they do not constitute circular evidence. The unpropagated readout-inversion uncertainty and the 0.005 L2 distance to a product thermal state are experimental robustness concerns, not circularity of the derivation. Overall: no significant circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, conserved quantities, or dimensions are introduced. The passivity deformation reweights existing Hamiltonian terms and is a construction, not a new entity. The observable B is built from measured initial populations, so the inverse-temperature entries above are the parameters the central claim depends on.

free parameters (6)

Measured inverse temperatures of Melbourne system qubits = beta_i from ground populations p0 = 0.557, 0.611, 0.586, 0.612
Used to construct B and all passive observables; measured from the actual initial state rather than the programmed state.
Measured inverse temperature of Melbourne environment qubit = p0^e = 0.782
Defines the engineered thermal environment used in the circuit.
Measured inverse temperatures of Essex system and environment = p0 = 0.944, 0.652, 0.652, and p0^e = 0.806
Used to construct B and the deformed observable for the Essex experiment.
Passivity deformation setting beta_1 to zero in Essex = beta_1 = 0
Chosen by hand to make the deformed observable detect the heat leak; not derived from a first-principles criterion.
Essex rotation angles = {0.3*pi, 0.4*pi, 0.4*pi, 0.15*pi}
Set the thermal populations used in the coherent-state mixing preparation.
Test order alpha and shift windows = alpha=3, delta in [1.8,2.6]; alpha=5, delta in [0.8,2.1]
Selected after scanning because lower-order and nondeformed tests failed to give unambiguous detection; no multiple-comparison correction is applied.

assumptions (5)

standard math Majorization decomposition: rho' = sum_i q_i U_i rho U_i^dagger iff rho majorizes rho'
Invoked via [33] to justify that unital evolutions are mixtures of unitaries before Eq. (2).
ad hoc to paper All standard unital evolutions are convex mixtures of unitaries
The paper's wording identifies unital with random-unitary; this is false for multiqubit systems under the standard definition and is not flagged.
domain assumption Initial state is exactly of the form Eq. (1), a tensor product of thermal states
Passivity tests require this form; only L2 closeness 0.005 to a product thermal state is checked.
domain assumption Readout noise is a fixed invertible linear map M, and M^{-1} yields true populations
The measurement matrix is calibrated per device; calibration uncertainty is not propagated into reported confidence intervals.
standard math Shadow tomography bound Eq. (S1) from [29] and the error-propagation steps are valid
This is the basis for the claimed polynomial measurement cost Eq. (S6).

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Cite this review

Pith. "Pith review of Experimental detection of microscopic environments using thermodynamic observables." pith.science (2026). https://pith.science/paper/3HR3EV6A

@misc{pith2026190808968,
  author       = {Pith},
  title        = {Pith review of: Experimental detection of microscopic environments using thermodynamic observables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3HR3EV6A}},
  note         = {Machine review of arXiv:1908.08968}
}

read the original abstract

Modern thermodynamic theories can be used to study highly complex quantum dynamics. Here, we experimentally demonstrate that the violation of thermodynamic constraints allows to detect the coupling of a quantum system to a hidden environment. By using the IBM quantum superconducting processors, we perform thermodynamic tests to detect a qubit environment interacting with a system composed of up to four qubits. The experiments are complemented by theoretical findings that show efficient scalability of the tests with respect to system size. Hence, they may be useful to detect an open system dynamics in situations where other methods (e.g. quantum state tomography) are practically infeasible.

Figures

Figures reproduced from arXiv: 1908.08968 by the authors.

**Figure 3.** FIG. 3. Heat leak tests performed in the five-qubit Essex [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Estimate the initial and ﬁnal populations{pi} and {p′ i}

work page

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price to pay

The eigenvalues ofF are obtained by applying a non-decreasing functionf to the eigenvalues ofB. The properties 1 and 2 imply that ∆ ⟨ F ⟩ = Tr[F( ∑ i qiU (i) s ρsU (i)† s −ρs)]≥ 0. (3) Here, we consider the set of passive observables{Fα,δ}≡ {(B− δI)α}, where α is a positive integer and I is the identity operator. Denoting the eigenvalues ofB asBi, withBi≤...

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[3] [3]

small-size

Evaluate the quantitiesξj ≡ ∑j i=1p′↓ i −∑j i=1p↓ i. If ξj > 0 for some 1≤j≤ 2n, there is a heat leak. If ξj≤ 0 for 1≤ j≤ 2n, there can be a heat leak that is undetectable using this method. This kind of heat leak corresponds to a transformation such that ρ′ is not majorized byρ, yet the majorization relation ρ≻ D(ρ′) (which is equivalent toξj≤ 0 for 1≤j≤...

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[4] [4]

Sort them in non-increasing order

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Esposito, U

M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009)

2009

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α-free energies

This is read- ily deduced by substituting the explicit expressions Ry(±θ)|0⟩ = cos (θ 2 ) |0⟩± sin (θ 2 ) |1⟩, (S18) into Eq. (S16). A product of an arbitrary numberN of thermal qubit statescanalsobeexpressedasamixtureanalogoustoEq. (S16). Let ρi(θi) = cos2(θi 2 ) |0⟩i⟨0| +sin2(θi 2 ) |1⟩i⟨1| be the state of theith qubit, and let|ψ(θ)⟩i≡ Ry(θ)|0⟩i. By wri...

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Breuer and F

H.-P. Breuer and F. Petruccione,Open quantum systems (Oxford university press, 2002)

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Kosloﬀ, Entropy 15(6), 2100 (2013)

R. Kosloﬀ, Entropy 15(6), 2100 (2013)

work page 2013