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REVIEW 2 major objections 4 minor 95 references

Quantum phase sensing with states out of thermal equilibrium

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Pith's one-line read The paper proves that the optimal phase-sensing precision of a quantum probe combined with a thermal bath is governed exactly by the probe's athermality, and gives closed-form bounds in finite dimensions and in linear optics.

desk verdict Strong paper with a real proof gap: the optical result is solid, but the finite-dimensional formula's proof needs a fix for incommensurate spectra. read the letter →

arxiv 2507.06030 v1 pith:YAGF7QAJ submitted 2025-07-08 quant-ph

Benjamin Yadin This is my paper · ORCID
classification quant-ph
keywords quantumFisherinformationphasesensingathermalitythermaloperationsinterferometrylatentcoherencenonclassicalitywitnessspeedlimit
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

Thermal backgrounds are normally treated as a nuisance in interferometry: the second input port is assumed to be vacuum. This paper shows that when the background is genuinely thermal, the athermality of the probe state is the resource that sets the phase-sensing precision. For any finite-dimensional system, the optimal quantum Fisher information under energy-conserving interactions with a thermal bath is an exact step-function integral of a "Fisher difference" over Boltzmann-weighted populations; in linear optics it reduces to a latent coherence $C_r(\rho)=\sum_{n\ge 1} f(p_n, r p_{n-1})\,n$ achieved by one balanced beam splitter. The same technique bounds the speed of system–bath evolution. A sympathetic reader should take away that brightness and mean photon number are replaced, at nonzero temperature, by a rigorously defined notion of athermality, one that can even witness optical nonclassicality.

What carries the argument

The carrying object is the Fisher difference $f(a,b)=(a-b)^2/(a+b)$, used in a blockwise optimization over conserved total energy. Because an energy-conserving unitary is block-diagonal in total energy, the maximal-QFI lemma from Ref. [40] applies independently to each block; optimizing the bath then leads to idealized baths whose degeneracies scale exponentially, $D(E+\epsilon)=D(E)e^{\beta\epsilon}$, which turns exact sums into the step-function integral of Eq. (2). In the optical setting the same machinery reduces to the latent coherence $C_r$, a convex, thermal-operation-monotone functional of the photon-number distribution that pairs adjacent Fock coefficients through $f(p_n, r p_{n-1})$; a single balanced beam splitter suffices to attain it, and the associated optimal measurement is a reweighted quadrature observable that reduces to homodyne detection at zero temperature.

What would settle it

Take the qubit case with a finite ladder bath of $K$ levels: the derivation in Appendix B shows the QFI saturates $f(p_0e^{-\beta\epsilon},p_1)\epsilon^2$ only as $K\to\infty$ through the probability factor $P_B(E<E_{\rm max}-\epsilon)$. If a finite or physically constrained bath were demonstrated to exceed this bound, or if the best achievable QFI in an engineered finite bath fell systematically below it for finite excitation probabilities, the claim that Eq. (3) is the optimal precision would be falsified.

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Extended reading notes

Core claim

The central claim is that the optimal phase sensitivity obtainable by mixing a phase-invariant probe state with a thermal bath, under only global unitarity and energy conservation, is exactly a function of the probe's athermality. For a finite-dimensional system with Hamiltonian $H_S=\sum_i \epsilon_i |\epsilon_i\rangle\langle\epsilon_i|$ and state $\rho_S=\sum_i p_i |\epsilon_i\rangle\langle\epsilon_i|$, the maximal QFI is $F_T(\rho_S,H_S)=\frac{1}{2}\int_0^{Z_S} dx\, f(\chi^\downarrow_q(x),\chi^\uparrow_q(x))(\chi^\downarrow_\epsilon(x)-\chi^\uparrow_\epsilon(x))^2$, where $f(a,b)=(a-b)^2/(a+b)$ and the step functions $\chi^\downarrow,\chi^\uparrow$ encode Boltzmann-weighted probabilities and energies ordered against the partition function $Z_S$. This vanishes if and only if $\rho_S$ is thermal, and is a monotone under thermal operations. For a single optical mode at frequency $\omega$ mixed with thermal modes through linear optics, the result is $(\hbar\omega)^2(\bar n_r+1)C_r(\rho_S)$ with $C_r(\rho_S)=\sum_{n\ge 1} f(p_n, r p_{n-1})\,n$, $r=e^{-\beta\hbar\omega}$, attained by one balanced beam splitter; at high temperature $C_1(\rho)=\tfrac{1}{2}F(\rho,x)$, making the same quantity a witness of $P$-function nonclassicality via $C_r(\rho_{\rm cl})\le (1-r)\langle N\rangle + r$. The same derivation gives the maximal interaction speed, $F^{\rm int}_T(\rho_S,H_S)=\frac{1}{2}\int_0^{Z_S} dx\, f(\chi^\downarrow_q(x),\chi^\uparrow_q(x))$, so that athermality alone bounds how fast a system and bath can evolve together.

Load-bearing premise

The bounds are suprema over idealized thermal baths whose degeneracies grow exponentially with energy, $D(E+\epsilon)=D(E)e^{\beta\epsilon}$, a property no finite physical bath can exactly satisfy, so for real finite baths the formulas are approached only in the limit of large, finely tuned reservoirs.

Editorial extensions

If this is right

  • If $F_T$ is the true optimum, then no energy-conserving unitary with any thermal bath at temperature $T$ can yield phase precision beyond Eq. (2); any reported sensitivity above it would indicate a missing resource such as coherence or an external phase reference.
  • The optical bound is tight for a single thermal mode and a balanced beam splitter, so a standard Mach–Zehnder with a thermal second port and a reweighted homodyne measurement is already optimal among all linear-optics networks.
  • The monotonicity of $F_T$ under thermal operations means partial thermalization can only degrade sensing precision, so probe preparation and the sensing interaction should be kept separate from any thermalizing contact.
  • At any nonzero background temperature, the latent coherence gives a nonclassicality witness: a measured $C_r$ above $(1-r)\langle N\rangle + r$ certifies $P$-function nonclassicality without requiring an external phase reference.
  • The athermality speed limit bounds system–bath evolution speed purely from the initial state's temperature mismatch, giving a thermodynamic speed limit for open-system dynamics.

Reading between the lines

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

  • Because the optimum requires a bath with exponentially growing degeneracies, realistic finite reservoirs will leave a gap; computing finite-bath corrections (e.g., truncating the energy ladder) could turn Eqs. (2), (5), and (10) into practically useful upper bounds and test how close engineered baths can come.
  • The latent coherence $C_r$ may be directly measurable in microwave or circuit-QED interferometers, where $\bar n_r$ is sizeable, using the reweighted quadrature strategy; this would extend interferometric phase sensing to regimes where vacuum-input assumptions fail.
  • The same formalism could be re-run for multiple copies or multi-mode resource states, where mode correlations (like those in filtered laser light) may modify the trade-off between phase sensing and illumination precision.
  • The paper leaves open whether the athermality speed limit is tight for practical open-system examples; studying specific master equations could settle that, and would connect the result to concrete clock and thermalization processes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies phase estimation when a finite-dimensional diagonal probe state is coupled to a thermal bath by an energy-conserving unitary, and defines F_T as the supremum of the quantum Fisher information (QFI) over all baths and all such unitaries. The central result, Eq. (2), expresses F_T as an integral of step functions built from the system energies and beta-ordered Boltzmann-weighted probabilities. For linear optics, Eq. (5) states that the optimal QFI is (hbar omega)^2 (nbar_r + 1) C_r(rho), with C_r(rho) = sum_{n>=1} f(p_n, r p_{n-1}) n, achieved by a single balanced beam splitter with one thermal ancilla mode. The paper also derives a speed-limit analogue Eq. (10), proves monotonicity of F_T under thermal operations, and relates C_r to nonclassicality, to quantum illumination, and to standard optical coherence.

Significance. If the finite-dimensional proof is completed, Eq. (2) is a rare closed-form solution to an optimization over arbitrary thermal baths and all energy-conserving unitaries, and it gives a precise operational meaning to athermality as a resource for phase sensing. The optical result Eq. (5) is self-contained, directly derived, and much stronger: it identifies an explicit optimal protocol (one balanced beam splitter) and a computable, convex, monotone quantity C_r that also serves as a nonclassicality witness and satisfies a clean complementarity relation with quantum illumination (Theorem 5). The paper is carefully structured, with explicit block decompositions, use of the external Lemma 1 from Ref. [40], and several cross-checked temperature limits. These strengths are substantial and make the manuscript potentially valuable for quantum thermodynamics and quantum metrology.

major comments (2)
  1. [Appendix C, Lemma 2; Eq. (C7); Eq. (B14)] The proof of Lemma 2 only treats bath energy shifts that are integer multiples of the qubit gap xi: Eq. (C7) sets epsilon/xi = l and computes ln[D'(E+epsilon)/D'(E)]. If a system gap epsilon_i is not an integer multiple of xi, then E+epsilon_i is not an energy of B' for any lattice energy E, so D'(E+epsilon_i)=0 and the combined degeneracy ratio is 0 rather than approximately e^{beta epsilon_i}. The sentence in Appendix C that xi is 'chosen small enough that all gaps are well approximated' does not repair this, because [U,H_S+H_B]=0 is an exact constraint: the term |epsilon_i>_S tensor |E-epsilon_i>_B in Eq. (B9) is simply absent when E-epsilon_i is not a bath energy. Consequently the rescaling argument leading from Eq. (B13) to Eq. (B14), and hence Eq. (2) and Eq. (10), is not justified for a generic non-commensurate system spectrum. A complete proof needs either a different bath construction whose spectrum is closed under addition of every system gap (for example, a multi-species qubit bath with independent gaps) with degeneracy slope approaching beta in the relevant energy window, or an explicit limiting argument showing that the supremum equals the step-function integral; alternatively, Theorem 1 must be restricted to commensurate spectra. As written, Eq. (2) is proven for qubits and for spectra commensurate with a single gap, but not for the claimed fully general finite-dimensional case.
  2. [Main text after Eq. (2); Appendix B, §B5, property (1)] The stated characterization that F_T vanishes precisely when p_i e^{beta epsilon_i} = p_j e^{beta epsilon_j} for every pair, equivalently rho_S = gamma_S, is false when H_S has degenerate energy levels. For example, take d=2 with epsilon_0=epsilon_1=0 and p_0 != p_1; then rho_S != gamma_S but H_S=0, so the phase encoding is trivial and F_T=0, and Eq. (2) indeed gives 0 because chi^down_epsilon is identically zero. The proof in Appendix B, around Eq. (B23), tries to handle epsilon_i=epsilon_j by taking a doubly degenerate bath level, but on the corresponding block H_S is proportional to the identity, so Lemma 1 yields zero QFI. The theorem and the resource-theoretic interpretation should either assume a nondegenerate spectrum or reformulate the zero set of F_T in terms of energy shells (equality of coarse-grained Gibbs occupations).
minor comments (4)
  1. [Appendix E and H] The equality case of the bound Eq. (8) is said to be characterized by tr(rho a rho a^dagger)=0. As written, this trace vanishes for every number-diagonal state, so it cannot characterize the states with support on only even or odd photon numbers. The intended condition appears to be tr(rho a^dagger rho a)=0, i.e., sum_n p_n p_{n+1}(n+1)=0, which is equivalent to p_n p_{n-1}=0 for all n.
  2. [Throughout] There are several typographical errors that should be corrected: 'intially', 'maxmise', 'annihiliation', 'indispensible', 'enocde', 'miminum', and 'ths work' in the introduction, preliminaries, and conclusion.
  3. [Introduction, Eq. (2)] The Fisher difference f(a,b) is defined in the main text with f(a,0)=a, but the convention f(0,0)=0 is only stated later in Appendix A; it would help to state this convention at first use so that the step-function integrand is unambiguous.
  4. [Appendix E] The proof of Theorem 3 implicitly sets hbar omega = 1 in the appendices, while the main-text Eq. (5) restores the factor (hbar omega)^2. A brief note to this effect would prevent confusion when comparing the appendices with the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central QFI formulas are derived from an external maximal-QFI lemma and explicit block decompositions, not fitted to their own outputs.

full rationale

The main result Eq. (2) is derived in Appendix B by decomposing the joint system-bath state into energy blocks (Eq. B9), applying Lemma 1 from the external reference [40] blockwise, and then optimizing bath degeneracies via Lemma 2; the step-function integral emerges as a continuum limit rather than being inserted to reproduce a target formula. No parameter is fitted to the claimed prediction; the inputs are the probe state, Hamiltonian, and temperature, and the output is an analytic expression. The optical result Eq. (5) is obtained by reducing to a single balanced beam splitter using published structural results ([41,42,45], some with author overlap) and then explicitly evaluating the QFI through Eqs. (E7)-(E10); the self-citation concerns a cosine-sine decomposition of passive linear unitaries, not the value of the latent coherence Cr. Monotonicity claims follow from standard QFI monotonicity and the optimization definition, not from assuming the conclusion. The only substantive caveat, the large-bath approximation in Lemma 2 regarding lattice/incommensurate spectra, is a mathematical attainability and correctness concern about the supremum, not a circular reliance on the result, so it does not raise the circularity score.

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

The result depends on standard quantum metrology and thermal-operations assumptions, not on fitted parameters. The two main physical postulates are the phase-invariance of the probe (no external phase reference) and the free choice of an arbitrarily large, energy-conserving bath. The mathematical workhorse is the external maximal-QFI lemma of Ref. [40]; the optical reduction uses the standard cosine-sine decomposition of passive linear unitaries. No new physical entities are introduced: latent coherence C_r and the Fisher difference f are defined quantities, not additional degrees of freedom.

assumptions (6)
  • domain assumption No external phase reference: the probe state can be taken diagonal in the energy eigenbasis (phase-invariant), so ρS = Σ p_i |ε_i><ε_i|.
    Used throughout the paper as the starting point for both finite-dimensional and optical results; without it, energy-basis coherence could contribute extra phase sensitivity and the latent coherence C_r would not be a complete quantifier. The assumption is stated in the Introduction and Preliminaries.
  • standard math Energy-conserving joint unitaries: [U, HS + HB] = 0, so U is block-diagonal in total energy.
    This defines the allowed thermal operations and is used to decompose the optimization into independent energy blocks in Appendix B.
  • standard math Lemma 1 from Ref. [40]: for a fixed spectrum, the maximal QFI under any unitary is 1/2 Σ f(p_i,p_{d-1-i})(ε_{d-1-i}-ε_i)^2.
    This external result is the engine of the finite-dimensional derivation; the present paper does not prove it.
  • domain assumption An ideal bath with degeneracies D(E+ε)=D(E)e^{β ε} on the relevant energy shell is admissible as a supremum, and finite baths can approximate it arbitrarily well (Lemma 2, Appendix C).
    The assumption that the bath can be chosen arbitrarily large and finely grained is part of the thermal operations framework; the tightness of Eq. (2) as a supremum relies on Lemma 2.
  • standard math For linear optics, any passive linear unitary on the signal and thermal ancilla modes reduces to a single beam splitter with one mode, using the cosine-sine decomposition and rotational invariance of γ_B^{⊗k} (Refs. [41,42,45]).
    Used in Appendix E to reduce the optical optimization to a single beam splitter; this is a standard result cited from prior work.
  • standard math For the nonclassicality witness, the bound F(σ_cl,H) ≤ 4⟨H^2 - :H^2:⟩_σ for classical states from Ref. [43].
    Used in Appendix G to derive Eq. (7).

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Cite this review

Pith. "Pith review of Quantum phase sensing with states out of thermal equilibrium." pith.science (2026). https://pith.science/paper/YAGF7QAJ

@misc{pith2026250706030,
  author       = {Pith},
  title        = {Pith review of: Quantum phase sensing with states out of thermal equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YAGF7QAJ}},
  note         = {Machine review of arXiv:2507.06030}
}
read the original abstract

Interferometry can be viewed generally as the measurement of a relative phase between two subsystems. I consider the problem of interfering a quantum resource state with a thermal bath, drawing a precise connection between the athermality of the resource and the resulting phase sensitivity. This is done by finding the fundamental sensing precision limit under the minimal conditions of global unitarity and energy conservation. The results here apply both to general finite-dimensional systems and to linear quantum optics. The same techniques further upper-bound the speed at which a system and bath can jointly evolve under an energy-conserving interaction.

Figures

Figures reproduced from arXiv: 2507.06030 by the authors.

Figure 1
Figure 1. FIG. 1. The main settings in this work: a) A phase-invariant [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the step functions used in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of step functions in the low-temperature limit. Here, [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The maximum QFI F (qubit) T for a qubit with energy gap ϵ combined with a thermal bath at temperature T, in units of ϵ 2 as a function of the excitation probability p1, for three different temperatures [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The large bath [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Latent coherence relative to different background temperatures, indexed by the Boltzmann ratio [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. A single resource state [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

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    Convexity: For any pair of parameterised states ρ(θ), σ(θ) and probability p, F [pρ(θ)+(1 −p)σ(θ)] ≤ pF [ρ(θ)]+ (1 − p)F [σ(θ)]. In the unitary case, this implies F (pρ + [1 − p]σ, H) ≤ pF (ρ, H) + (1 − p)F (σ, H)

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    Monotonicity: For any quantum channel (trace-preserving completely positive map) Λ, F [Λ(ρ{θ})] ≤ F[ρ(θ)]. A special statement for the unitary case holds fortranslationally invariant channels [36], i.e., those commuting with the unitary evolution such that Λ( e−iθH ρeiθH ) = e...

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    Then F (ρ, H) = P k F (ρk, Hk)

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    Properties (1-4) are easy to see

    f (x, y) is jointly convex, and therefore satisfies f (px + [1 − p]x′, py+ [1 − p]y′) ≤ pf (x, y) + (1 − p)f (x′, y′) ∀ p ∈ [0, 1]. Properties (1-4) are easy to see. (5) follows immediately from the derivative ∂xf (x, y) = (x − y)(x + 3y) (x + y)2 . (A4) Finally, for (6) we co...

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    Main result Here, we prove the main result on the maximal QFI for phase sensing in combination with a thermal bath: Theorem 1. For any d-dimensional system with Hamiltonian HS = Pd−1 i=0 |ϵi⟩ ⟨ϵi| in the state ρS =Pd−1 i=0 pi |ϵi⟩ ⟨ϵi|, the maximal phase-sensing QFI relative t...

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    Low-temperature limit First assume that ρS is full-rank, so that all pi > 0. As T → 0, q↓ and ϵ↓ are in the same order and have the same pattern of repetitions. For both χ↓ ϵ and χ↓ q, almost all the x-range is taken up by the lowest energy ϵ0 – see Fig. 3 – 11 hence lim T →0 ...

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    In addition, the step lengths in χ↓ ϵ and χ↓ q are all unity, so lim T →∞ FT (ρS, HS) = (d−1)/2X i=0 f (p↓ i , p↑ i )(ϵ↓ i − ϵ↑ i )2

    High-temperature limit As T → ∞, all Boltzmann factors tend to unity, e−βϵi → 1, so q = p↓. In addition, the step lengths in χ↓ ϵ and χ↓ q are all unity, so lim T →∞ FT (ρS, HS) = (d−1)/2X i=0 f (p↓ i , p↑ i )(ϵ↓ i − ϵ↑ i )2. (B17)

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    (3) in which a bath with uniform degeneracies D(kϵ) = 1 ∀ k = 0, 1, 2,

    Qubit case In the case d = 2, we present a separate, simpler derivation of Eq. (3) in which a bath with uniform degeneracies D(kϵ) = 1 ∀ k = 0, 1, 2, . . .is optimal. Given HS = ϵ |ϵ⟩ ⟨ϵ|, we have q|E = (p0 [D(E) times], p1eβϵ [D(E − ϵ) times] )↓, ϵ|E = (0 [D(E) times], ϵ[D(E ...

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    Properties of FT Here, we characterise FT as a formal quantifier of athermality due to the following properties:

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    FT (ρS, HS) = 0 if and only if ρS = γS

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    For property (1), if ρS = γS, then γS ⊗ γB ∝ e−β(HS +HB ) is invariant under all energy-conserving unitaries, hence FT vanishes

    For any thermal operation Λ, we have FT (Λ[ρS], HS) ≤ FT (ρS, HS). For property (1), if ρS = γS, then γS ⊗ γB ∝ e−β(HS +HB ) is invariant under all energy-conserving unitaries, hence FT vanishes. Conversely, if ρS ̸= γS, then there exists a pair i ̸= j such that ϵi ≥ ϵj and pi...

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    X m gm(m + 1) #

    It is easy to construct such a B′. For example, we take n identical noninteracting qubits, each with energy gap ξ, chosen small enough that all gaps in the system’s energy spectrum are well approximated within the gaps of HB′. This is done to effectively treat the bath as a co...

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    F ock state For a single nonvanishing pn, only two terms remain in the sum: one pairing pn = 1 with each of pn−1 = 0 and pn+1 = 0, so Cr(|n⟩ ⟨n|) = f (1, 0)n + f (0, r)(n + 1) = (1 + r)n + r. (F1)

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    Since pn/pn−1 = r′, we have Cr(γ′) = X n f (r′pn−1, rpn−1)n = X n f (r′, r)pn−1n = f (r′, r) ⟨N + 1⟩γ′ = (r′ − r)2 (r′ + r)(1 − r′)

    Thermal state We take ρ = γ′, thermal at temperature T ′ ̸= T . Since pn/pn−1 = r′, we have Cr(γ′) = X n f (r′pn−1, rpn−1)n = X n f (r′, r)pn−1n = f (r′, r) ⟨N + 1⟩γ′ = (r′ − r)2 (r′ + r)(1 − r′) . (F2) 20

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    We claim the following integral representation: Cr(ρλ) = r(λ + 1) − 3λ + 4λ r − 4 λ2 r2 Z 1 0 du u λ r −1eλ(u−1)

    Poisson statistics Let pn = e−λλn n! (which can be obtained from a phase-averaged coherent state |α⟩ with |α|2 = λ). We claim the following integral representation: Cr(ρλ) = r(λ + 1) − 3λ + 4λ r − 4 λ2 r2 Z 1 0 du u λ r −1eλ(u−1). (F3) The large-amplitude limit at fixed temper...

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    From the series expansion of (1 − r)−1, one can show that ⟨(N + 1)(k)⟩γ = k!(¯nr + 1)k and ⟨N(k)⟩γ = k!¯nk r , thus F (ρS ⊗ γB, Gk) = 4k!(¯nr + 1)k X n f (pn, rkpn−k)n(k)

    (x − k + 1). From the series expansion of (1 − r)−1, one can show that ⟨(N + 1)(k)⟩γ = k!(¯nr + 1)k and ⟨N(k)⟩γ = k!¯nk r , thus F (ρS ⊗ γB, Gk) = 4k!(¯nr + 1)k X n f (pn, rkpn−k)n(k). (G7) 23 Aside from k = 1, it is not clear if this quantity is a monotone under Gaussian ther...

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    [34], the coherence C of a bosonic field is defined as the maximal mean photon number in any single spatial mode, restricted to some frequency band

    First-order coherence In Ref. [34], the coherence C of a bosonic field is defined as the maximal mean photon number in any single spatial mode, restricted to some frequency band. This can be motivated by a connection to the peak value of the power spectrum, the Fourier transfo...

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    For two orthogonal modes, this reduces to ⟨NANB⟩; for A = B, it gives N 2 A − ⟨NA⟩

    Second-order coherence Second-order optical coherence describes correlations between the intensities of different modes (e.g., different spatial modes or with a time delay), considering expectation values of the form a†b†ba . For two orthogonal modes, this reduces to ⟨NANB⟩; f...

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