Untwining multiple parameters at the exclusive zero-coincidence points with quantum control

In this paper we address a special case of"sloppy"quantum estimation procedures which happens in the presence of intertwined parameters. A collection of parameters are said to be intertwined when their imprinting on the quantum probe that mediates the estimation procedure, is performed by a set of linearly dependent generators. Under this circumstance the individual values of the parameters can not be recovered unless one tampers with the encoding process itself. An example is presented by studying the estimation of the relative time-delays that accumulate along two parallel optical transmission lines. In this case we show that the parameters can be effectively untwined by inserting a sequence of balanced beam splitters (and eventually adding an extra phase shift on one of the lines) that couples the two lines at regular intervals in a setup that remind us a generalized Hong-Ou-Mandel (GHOM) interferometer. For the case of two time delays we prove that, when the employed probe is the frequency-correlated biphoton state, the untwining occurs in correspondence of exclusive zero-coincidence (EZC) point. Furthermore we show the statistical independence of two time delays and the optimality of the quantum Fisher information at the EZC point. Finally we prove the compatibility of this scheme by checking the weak commutativity condition associated with the symmetric logarithmic derivative operators.


Introduction
Quantum metrology [1][2][3] targets the problem of estimating physical quantities as precise as possible by exploiting the advantages offered by quantum coherence and (unconventional) quantum measurements, so as to obtain a higher estimation precision than the classical estimation scheme. Compared to the extensively investigated quantum estimation of a single parameter, the quantum estimation of multiple parameters is more important for many practical applications such as the magnetometry [4], gyroscopy [5], or quantum network [6]. However, the much challenges the quantum multi-parameter estimation task has, since it is extremely difficult to simultaneously and optimally recover a collection of (say) k independent (unknown) parameters from the same experimental setup [7][8][9][10][11][12][13][14]. The most general multi-parameter estimation task can be casted in a black-box scenario [15] where the collection of to-be-estimated quantities expressed by the real vector τ = {τ 1 , . . . , τ k } are initially imprinted into the state of a quantum probing system via an encoding map Λ τ that depends parametrically upon τ . The value of τ is hence recovered through measurements performed on a statistically significant set of M copies of the imprinted state of the probe. Under these conditions the ultimate precision of multiple parameters is gauged by the quantum Cramér Rao bound (QCRB) [16][17][18] that establishes a lower bound for mean square error matrix of the problem via the inverse of the quantum Fisher information matrix (QFIM). It is well known however that in many cases of physical interests the multi-parameter QCRB precision threshold cannot be attained (not even in the asymptotic limit of large M) due to the compatibility problems pertaining the non-commutativity of quantum operations [10,11,13,[19][20][21]. An even more drastic limitation occurs when one faces what we may call an intertwined multi-parameter scenario, i.e. when the black-box map Λ τ admits the same generator for some of the k components of the vector τ or, more generally, when a subset of such parameters have linearly dependent generators. When this happens the QFIM turns out to be singular, the QCRB to be meaningless and the individual components of τ cannot be individually discerned and estimated [22]. Examples of this behavior are well known in computation biology and chemistry where they are typically identified as sloppy models [23][24][25]. It is clear that in these cases neither a careful optimization of the input state of the probeρ, nor a careful choice of the measurement procedure will enable us to recover the values of τ .
In the present manuscript we present an example of this phenomenon considering the estimation of multiple time delays τ 1 , τ 2 , . . . , τ k on a transmission line, with respect to a second reference line, which could be generated for example by length differences in the paths. It has been shown that the estimation precision with respect to the time delay can approach the attosecond scale (i.e. the length-difference reaches to the nanometer scale) [26]. However, by sending light on this couple of lines and measuring it at the end we can only estimate the total time delay (or the mean delay for unit of length in the transmission line), but the information on the spatial distributions of such delays is totally inaccessible to us, because it hasn't been codified in a discernible way in the first place. One motivation of our study is exactly to explore a way to simultaneously estimate (or untwine) these intertwined parameters with the individual highest precision, i.e. to achieve the simultaneous optimal estimation of multiple intertwined parameters. The only possibility we find at our disposal is to interfere directly with the encoding process Λ τ , e.g. by means of quantum control acting between the encoding of the different parameters. Clearly a possible solution is to open the transmission line at regular time interval to read each individual delay. While effective, this strategy has however the major drawback that it requires us to effectively destroy the transmission line. In contrast to this active and invasive technique we show that the passive solution exists that can do the same job without signal leakage. Concretely, the solution allowing us to effectively untwining the imprinting of the various parameters, is to alternate the encoding of the individual delays with a unitary control of the probe, so that each parameter will be encoded in a different way and can be discerned from the others by a measurement at the end of the lines. The simplest form of unitary control for light traveling on a couple of transmission lines is a 50:50 beam splitter (BS). If we insert a BS between each encoding of the time-delay, and add an achromatic phase [27], to further generalize the unitary control we are led automatically to the setting of the generalized Hong-Ou-Mandel (GHOM) interferometer, which was discussed in [28]. The GHOM interferometer is an extension of the 'HOM effect' [29] that exhibits what was dubbed exclusive zero-coincidence (EZC) points, i.e. a one-to-one correspondence between the contemporary absence of all the time delays and zero values of the coincidence counts at the output of the device when the input state of the device is a frequency-correlated biphoton state. Therefore, we take the GHOM setting as a representative scheme of untwining multiple interwined parameters by introducing quantum control specified as the 50:50 BS and achromatic phase. We then compute the QFI for a two-photon symmetric input state at the EZC point, explicitly obtaining an invertible matrix, and proving therefore the success of the untwining of the time delays. We observe also that at the EZC point the QFIM looses the off-diagonal components (hence ensuring the statistical independence of the estimation of τ 1 and τ 2 ), while attaining the maximum values of the diagonal entries (therefore providing the maximal estimation accuracy for both the parameters).
The manuscript is organized as follows: in section 2 we give a brief review of the theory of quantum parameter estimation, formalize the intertwined multi-parameter scenario, and discuss the compatibility issues that may arise when the parameters are not intertwined. In section 3 we present out a case study based on the estimation of multiple time delays along an optical transmission line. Here after recalling some basic facts about the GHOM interferometer we show how for the case of two delays, this setup can be used to effectively untwine such parameters. We also show that at the EZC point the time-delay parameters are also statistically independent and that the QCRB can be saturated. In the last part of section 3, we also discuss the singularities of the different QFIMs in the cases of three or four time delays. Some conclusions and the possible applications are presented in section 4.

Quantum multi-parameter estimation
In this section we review some basic mathematical tools that, on one hand, allow us to identify those cases where the multi-parameter estimation is formally impossible (intertwined configurations), and on the other hand permit to clarify the compatibility issues that affect the efficiency of those where the estimation is possible.

Intertwined vs not-intertwined configurations with quantum control
Consider a multi-parameter quantum estimation scenario in which a collection of k unknown parameters represented by the vector τ = {τ 1 , . . . , τ k } are imprinted on a probing quantum system via an assigned mappingρ (ρ being the input state of the probe). A characterization of the precision attainable in the process is provided by the k × k, positive semidefinite, QFIM H τ of the model [16-18, 30, 31] whose elements can be expressed via the spectral decompositionρ τ = l p l |l⟩⟨l| of the probe state as an identity which for the pure stateρ τ = |Ψ τ ⟩⟨Ψ τ | simplifies into (in the above expressions ∂ ℓ represents the partial derivative with respect to τ ℓ , while Re[•] means extracting the real part). We can now distinguish two different scenarios depending on the invertibility of the QFIM. If H τ is invertible (i.e. if Det[H τ ] > 0) for at least one special choice of the input stateρ, then the recovering of the values of τ is possible (at least in principle) with an ultimate estimation precision that is gauged by the QCRB [16-18, 30, 31], i.e.

Cov[τ
the identity between the second and third term being achieved when H τ is diagonal, ensuring the statistical independence of the estimation of the individual parameters of the model. Equation (5) proves the consistency of equation (4) with the bound one would obtain in the special case where one attempts to recover the ith component of τ in the scenario where the remaining k − 1 components of such vector are known. If on the contrary H τ is not invertible (i.e. if Det[H τ ] = 0) for all possible choices of the input state of the probe, we are in presence of an intertwined multi-parameter estimation scenario in which the mapping (1) is characterized by linearly dependent generators for each of the k components τ 1 ,…, τ k . A paradigmatic example of this sort occurs e.g. when Λ τ is induced by a sequence of unitary transformationŝ U τ1 ,Û τ2 ,…,Û τ k generated by the same Hamiltonian generatorĤ i.e.
with τ := k j=1 τ j (an explicit instance of this kind of model is given in the next section). Clearly under these special circumstances the probe stateρ τ , irrespectively from the choice of its input configurationρ, will only carry information on the linear combination τ making the reconstruction of the individual components of τ impossible (a fact signalled by the loss of meaning of (4) whose right-hand-side is now effectively divergent). In other words, in these cases neither a careful optimization of the initial probe stateρ, nor a careful choice of the measurement procedure will enable us to untwine τ . In the face of this difficulty, we can resort to control-enhanced quantum parameter estimation procedures, i.e. interfering with the probe evolution between the encoding of the different parameters. References [32,33] already demonstrated that with the help of quantum controls acting the adjacent two systems in a sequence structure, the optimized quantum dynamics is capable to achieve the multi-parameter simultaneous optimal estimation. Accordingly it is significant to explore the possibility of untwining the multiple parameters by alternating the encoding of individual parameters with quantum controls, as shown in figure 1 (b). In this case equation (7) is renewed asΛ whereÛ c is the introduced quantum control andŨ τ plays a same role asÛ τ in equation (7). It is necessary to stress that while certainlyŨ τ ̸ =Ûτ , the two schemes are still produced by the same local encoding stepsÛ τ k 's: they only differ in the way such local steps are allowed to operate on the system. In the following section 3, we take the GHOM interferometry as an explicit example to clarify the feasibility of untwining multiple delays.

Compatibility and asymptotic achievability of the QCRB
A second main issue associated with multi-parameter estimation procedures is that the imperfect knowledge of one of the parameters tends to deteriorate the precision of estimating the others. This implies that for k ⩾ 2 the inequality (4) is typically not reachable even in the asymptotic limit of large M values. Sufficient requirements for this to happen have been identified [12,13,20,21] when certain commutativity conditions hold true for the symmetric logarithmic derivative (SLD) operator which allows one to express the variation ofρ τ with the jth component of the vector τ in terms of the following Lyapunov matrix equation ∂ jρτ = (L jρτ +ρ τLj )/2. Specifically we can say that the inequality (4) saturates for sufficiently large M if does happen that the following condition holds true an identity that for the pure stateρ τ = |ϕ τ ⟩⟨ϕ τ | simplifies into using the fact that in this case (9) rewrites as [14] For a pure encoded state |Ψ τ ⟩ the weak commutativity condition (11) implies also the strong commutativity, that is, there exists a set of SLD operators such that [L ′ i ,L ′ j ] = 0 [12]. When the encoded state is pure the SLD operators are not univocally specified, and if the weak commutativity holds, this freedom allows us to find a set of commuting SLD operators. The common bases of these operators can be used to design the projective measurement to be realized on |Ψ τ ⟩ having the Classical Fisher Information Matrix reaching the QFIM. The QCRB can then be saturated asymptotically in M with a maximum likelihood estimator for example.
Besides as discussed in section 2.1, the behavior of the QFIM H τ can be changed from non-invertible to invertible such that the intertwined parameters can be untwined by optimizing the original quantum dynamics with quantum controls. Accordingly, another problem is whether these parameters can be recovered simultaneously and optimally, i.e. investigating the compatibility of the scheme depicted by figure 1(b) by checking the weak commutativity condition (10) associated with the SLD operators (12). In the following section 3, we take the GHOM interferometry as an explicit example to investigate the asymptotic achievability of the associated QCRB as a result of the mentioned compatibility.

A case study
In this section we focus on an explicit example of an intertwined multi-parameter model based on the quantum optical setting. As shown in panel (a) of figure 1 it consists into two parallel multimode optical lines connecting a sender to a remote receiver while experiencing k relative temporal delays τ 1 ,…, τ k that are spatially distributed along the path, which results into a global delay of the emergent signals represented exactly by the quantity τ of equation (7). As discussed in section 2.1 under these circumstances there is no way that the two parties will be able to recover (even approximatively) the individual values of the τ 1 ,…, τ k . Of course a solution of the problem would be to grant access to each of the individual portions of the line where the delays are introduced: it is clear however that this choice requires a complete redesign of the setup which will have a huge impact on the blue print of the original model (one communication line connects two remote parties). What we are looking for instead is a much less invasive modification for the scheme. For this purpose we adopt the GHOM interferometric setup introduced in [28] which only accounts for introducing sequences of balanced BSs connecting the upper and lower lines of the scheme, plus possibly a collection of achromatic phases [27]-see figure 2. In this GHOM setting quantum controls are specified as the balanced beamsplitters plus some achromatic phases so as to achieve the control-enhanced multi-delays estimation. As a matter of fact, the simplest form of unitary control for light traveling on a couple of transmission lines is exactly a 50:50 BS. As we shall see explicitly for the case of k = 2, this choice is successful as the new QFIM of the scheme associated with a two-photon input state, is explicitly non-singular, hence allowing us to untwine two time delays. Interestingly enough the proposed setup also permits to fulfill the weak commutativity condition (11), hence ensuring the possibility of asymptotically reaching the associated QCRB (4), and (for the special case of τ 1 = τ 2 = 0) the saturation of the second inequality in (5) with maximum values for diagonal QFI components.

The GHOM interferometer
Here we briefly review the main feature of the GHOM interferometric setup introduced in [28] as a method for pin-pointing the contemporary zero values of multiple independent time-delay parameters expressed by the vector τ = {τ 1 , . . . , τ k }. Figure 2 gives the schematic diagram of the setting, which includes k cascaded phase-shift modules that induce two opposite phase shifts φ ℓ (ω)/2 and −φ ℓ (ω)/2 (ℓ = 1, 2, . . . , k) along the upper and lower arms respectively. The k uses of phase-shift modules are interleaved with k balanced BSs. This kind of allocation of phases is usually known as the symmetric phase-shift [34][35][36][37]. For ℓ > 1 the phase shift φ ℓ (ω) is constituted by a time-delay element τ ℓ (the ℓth parameter we need to estimate), and by a frequency-independent achromatic wave-plate θ ℓ [27] that instead represents the control parameter of the setup, i.e.
Yet the first phase-shift module only contains a time delay element, i.e. φ 1 (ω) = ωτ 1 , which corresponds to a conventional HOM interferometer [29]. The initial state of the interferometer is assumed to be a symmetric, frequency-correlated biphoton pure state where |∅⟩ stands for the Fock vacuum state, whereâ † 1 (ω) andâ † 2 (ω) are Bosonic creation operators that describe a photon of frequency ω that enters the device along the upper and lower input port respectively, and where finally Ψ s (ω, ω ′ ) is the biphoton joint spectral amplitude (JSA) following the normalization conditioń dω´dω ′ |Ψ s (ω, ω ′ )| 2 = 1, and possessing the exchanging symmetry Ψ s (ω, ω ′ ) = Ψ s (ω ′ , ω). The output state that emerges from the interferometer can now be expressed as the sum of two orthogonal terms, i.e.  1, 2, . . . , k) along the upper and lower arms respectively. The initial probe state is produced from the frequency-correlated biphoton source and two single-photon detectors T1, T2 are used for measuring. This GHOM setting can be used for untwining the delays and allows for the recovering of the individual delays, in which the balanced beamsplitters plus some possible achromatic phases play the role of quantum controls so as to achieve the control-enhanced multi-delays estimation.
with the (not necessarily normalized) vectors |Φ τ ⟩ and |Υ τ ⟩ representing respectively events in which the two photons of the model emerges either on the same ports (biphoton bunching) or in distinct port (biphoton anti-bunching)-see appendix A for details. Accordingly the probability of the coincidence counts where each detector at the output of the device captures only one photon, corresponds to The presence of an EZC point in the model emerges by observing that for special values of k there exists an optimal choiceθ := {θ 2 , . . . ,θ k } of the achromatic wave-plates vector θ := {θ 2 , . . . , θ k } such that a one-to-one correspondence relation can be established between the zero point of the coincidence counts and the contemporary absence of all the time delays [28,38], i.e.
In particular in [38] it has been shown that this happens for k = 2 withθ 2 = π/2, for k = 4 with θ 3 = arccos(cotθ 2 cotθ 4 ) under the assumption that sinθ 2 sinθ 4 ̸ = 0, while notably no solutions exist for k = 3. Equation (17) shows that under EZC conditions (i.e. for θ =θ) the GHOM provides a method to simultaneously pin-point the zero values of all the components of the τ vector: this naturally suggests that the same setting could be used to improve the efficiency of the estimation of the delays. In the next section we check this fact by focusing on the simplest (yet not trivial) case k = 2. In this scenario we shall see that indeed by setting θ 2 =θ 2 not only the QFIM measured on the output signal of the GHOM setup can be made invertible, but also that both the inequalities of equation (5) are locally achieved for τ = 0. We stress however that the optimal measurements that would lead to the saturation of the QCRB would not be the simple photon-detections associated with the EZC condition (17) but, as explained at the end of section 2, projective measures derived from the SLD operators (12).

Joint time-delay estimation via a GHOM interferometer
First of all in section 3.2.1 we show that the QFI matrix (3) of the output state (15) of a k = 2 GHOM interferometer is non-singular, by explicitly computing its entries and plotting its determinant. As anticipated at the beginning of the section, this ensures the possibility of recovering both τ 1 and τ 2 is at variance with what's happening in the original setting of panel (a) of figure 1 which only allows for the estimation of τ = τ 1 + τ 2 . We also show that enforcing the EZC condition (17), at this special point one gets the saturation of the second inequality of equation (5) with maximum values of the diagonal QFIM elements. In section 3.2.2 we prove that there exists a measurement at the end of the two lines reaching the QFIM, by proving the validity of the weak commutativity condition (12) for the case of k = 2 time delays, this is sufficient to the pure encoded state [12]. Finally in section 3.2.3, we numerically investigate the GHOM settings with three or four time delays. The similar results have been obtained in the case of four time delays, i.e. the GHOM interferometer of k = 4, under the EZC condition θ =θ (specifically we have numerically tested the model using θ 2 = π/3, θ 3 = arccos 1/ √ 3 , θ 4 = π/4), gives a nonsingular QFIM. Notably instead for k = 3 time delays we have not been able to find a configuration of θ such that the QFIM is nonsingular, a fact that mimics the observation of [38] that no EZC point can be found under this condition. In panel (a) we report the determinant of the QFI matrix for two time-delay parameters, equation (19), computed for the symmetric biphoton pure state having JSA expressed by equation (21), with the achromatic phase θ2 = π/2 (θ2 is involved in the phase-shift module φ2(ω) of the GHOM setting as shown in figure 2 and equation (13)). Panel (b) represents the same quantity but in absence of the achromatic phase, i.e. θ2 = 0. In both cases the determinant is non-null for the plotted region of parameters. Red ellipses mark the regions where the determinants assume their maximal values. Here τ 1 and τ 2 are always rescaled by the inverse of the width Ω2 of the biphoton JSA function (see equation (21)), and Ω1 = Ω2/3, ω0 = 5Ω2, Ω2 = 1 are set for the simulation.

Invertibility of the QFIM
We start observing that the bunching and anti-bunching components of the output state (15) |Ψ τ ⟩ remain orthogonal even under differentiation, i.e.
for all i, j. Inserting hence equation (15)

into equation (3) and invoking the decomposition (A.1) of appendix A we can write
where in the second identity we used (18), and where we defined To get further insight on the structure of the matrix H τ we now take a paradigmatic two-mode Gaussian function to be the JSA Ψ s (ω, ω ′ ) of the input signal, i.e.
which locally allocates to each photon an average frequency ω 0 and a spread ∆ω = Ω 2 2 + 4Ω 2 1 /2. Inserting equation (21) into equation (19) all the integrals present in such expressions can be analytically solved allowing for close, yet cumbersome, formulas which we report in appendix B. The resulting function allows for an explicit evaluation of the determinant of the QFIM that turns out to be always strictly positive as shown in figures 3 and 4. Here Det[H τ ] is plotted as a function of τ 1 and τ 2 for θ 2 = θ 2 = π/2 (special choice of θ 2 that enable the EZC condition (17)), and for θ 2 = 0 (no achromatic phases). Notice that for both these values of θ 2 the Det[H τ ] is not null meaning that to untwin τ 1 and τ 2 there is not need to enforce the  (19) is plotted, computed for the symmetric biphoton pure state having JSA expressed by equation (21). Panel (a) and (b) show the determinant as a function of τ 1 respectively for θ2 = π/2 and θ2 = 0, while panel (c) and (d) show the determinant as a function of τ 2 again respectively for θ2 = π/2 and θ2 = 0. In the plotted region of parameters the determinant is non-null and the QFI matrix is invertible. Here τ 1 and τ 2 are always rescaled by the inverse of the width Ω2 of the biphoton JSA function (see equation (21)), and Ω1 = Ω2/3, ω0 = 5Ω2, Ω2 = 1 are set for the simulation.
EZC condition (the mere presence of the BSs suffice for this scope). Yet, setting the achromatic phase θ 2 = π/2 the estimation model seems to yield some extra advantages that appears as we analyze in details the various elements of the QFIM. Plots of all these expressions as a function of the delay parameters τ 1 and τ 2 are presented in figures 5 and 6. In particular the first show that when θ 2 = π/2, both [H τ ] 11 and [H τ ] 22 simultaneously exhibits a global maximum in correspondence of the EZC point τ 1 = τ 2 = 0, It actually turns out that irrespective on the specific choice of the input state given in (21) the off-diagonal term [H τ ] 12 θ2=π/2 is null on the principal axis of the (τ 1 , τ 2 ) space, i.e.
Indeed from equation (19) the off-diagonal component of the QFI matrix can be expressed as with from which it is easy to verify (23). In particular this holds at the EZC point implying that under such condition the QFIM of the problem is diagonal so that the gap between the second and the third terms of equation (5) saturates, and the two time-delay parameters are statistically independent. Thus it can be seen that the employment of θ 2 = π/2 is not only necessary for enforcing the EZC condition, but also pivotal for achieving the statistical independence between two to-be-estimated time-delay parameters.

Weak commutativity condition
Here we show that at the output of the GHOM interferometer the condition (11) is met for all values of τ . Indeed exploiting the identities (18), equation (11) simplifies as To verify that such an identity is indeed valid we now notice that invoking equations (A.4)-(A.6) one can show that ⟨∂ 1 Φ τ |∂ 2 Φ τ ⟩ is a real quantity, i.e.
and that due to the fact that the integrand is explicitly anti-symmetric for the exchange of ω with ω ′ . We can hence conclude that in present case the bound (4) saturates at least asymptotically in the limit of large M. Notice also that this result is valid not just when we met the EZC point condition (17), but for all choices of τ 1 , τ 2 and θ 2 as long as the JSA function |Ψ s (ω, ω ′ )| 2 is symmetric.

GHOM settings of k = 3 and k = 4
In this section, we focus on the GHOM estimation scheme with a higher k, for instance k = 3 and 4, and try to explore whether the invertibility of the QFIM still keeps under the EZC condition. Firstly to analyze the singularity of the QFIM with respect to three time-delay parameters on the point τ 1 = τ 2 = τ 3 = 0, we study the functional dependence of its determinant upon the full (τ 2 , τ 3 ), (τ 1 , τ 3 ) and (τ 1 , τ 2 ) planes with the given value τ 1 = 0, τ 2 = 0 and τ 3 = 0 respectively. The simulation results are depicted in figure 7, in which achromatic phase shifts θ 2 and θ 3 are set to be zeros for simplifying the calculation (this is acceptable since whatever the values of θ 2 , θ 3 , the EZC condition of equation (17) does not hold in the case of k = 3 [28]).  ) and (τ3, τ4) planes, in which θ2 = π/3, θ4 = π/4, θ3 = arccos 1/ √ 3 and the corresponding remaining parameters are set to be zeros. Here τ 1, τ 2, τ 3 and τ 4 are rescaled by the inverse of the width Ω2 of the biphoton JSA function (see equation (21)), and Ω1 = Ω2/3, ω0 = 5Ω2, Ω2 = 1 are set for the simulation. Figure 7 shows that the determination of the QFIM for three time-delay parameters is always zero on the point τ 1 = τ 2 = τ 3 = 0, which is opposite to the case of k = 2 shown in figure 3. In other words, the QFIM is non-invertible or singular for the case of k = 3. If we further generalize the above consideration into the GHOM scheme with four time delays, the QFIM will be nonsingular similar to the case of k = 2. To analyze the singularity of the QFIM with respect to four time-delay parameters on the point τ 1 = τ 2 = τ 3 = τ 4 = 0, we respectively study the functional dependence of its determinant upon the full (τ 1 , τ 2 ), (τ 1 , τ 3 ), (τ 1 , τ 4 ), (τ 2 , τ 3 ), (τ 2 , τ 4 ) and (τ 3 , τ 4 ) planes with the remaining parameters are set to be zeros. The simulation results are exhibited in figure 8 where the corresponding achromatic phase shifts θ 2 = π/3, θ 3 = arccos 1 √ 3 , θ 4 = π/4 (this configuration makes the EZC condition of equation (17) hold in the case of k = 4 [28]).

Conclusions
In the present work we have presented a method that untwines multiple parameters from an intertwined multi-parameter scenario and achieves the multi-delays simultaneous optimal estimation by introducing some necessary quantum controls. As an explicit example, we have showed the untwining of two time-delay parameters in a GHOM interferometer, and proved that the resulting estimation scheme is in many ways optimal around the EZC point. Specifically, at the EZC point every time-delay parameter can be estimated with the individual highest precision (the diagonal elements of the QFIM reach the individual maxima) and the statistical independence between them can be achieved (the off-diagonal elements of the QFIM are zeros). Furthermore, the multi-parameter QCRB can be saturated at least in the asymptotic limit of infinite copies (the weak commutation condition can be satisfied). Finally, we prove that the similar results have been obtained in the case of k = 4 time delays with a GHOM interferometer under the EZC condition θ =θ (specifically we have numerically tested the model using θ 2 = π/3, θ 3 = arccos 1/ √ 3 , θ 4 = π/4). Notably instead for k = 3 time delays we have not been able to find a configuration of θ i such that the QFIM is nonsingular for τ i = 0 (i = 1, 2, 3), a fact that mimics the observation of [38] that no EZC point can be found under this condition.
From the perspective of realistic application, the current illustration described by the GHOM interferometry with a set of unknown time delays is one of representative problems of multi-phase estimation, which could inspire many applications like developing the monitoring of terrain deformation or the photogrammetry by virtue of the Interferometric Synthetic Aperture Radar (InSAR) techniques [39,40].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).