Influence of multiphoton events on the quantum enhanced phase estimation

Mingran Zhang; Mingran Zhang; Mingran Zhang; Long Huang; Long Huang; Long Huang; Yang Liu; Yang Liu; Wei Zhao; Wei Zhao; Weiqiang Wang; Weiqiang Wang

doi:10.1364/OE.468727

1. Introduction

Quantum metrology can improve the sensing performance beyond classically achievable precision using quantum optical sources, as was demonstrated in squeezed state enhanced gravitational wave detection for instance [1,2]. Moreover, quantum metrology has attracted widespread attentions in quantum imaging [3], atom clock measuring [4,5] or biological sample probing [6], etc. In theory, the standard quantum limit is the upper limitation of metrology based on a classical laser source due to the intrinsic noise. Using an ideal quantum source, the metrology precision could approach Heisenberg limit ($1/n$, where $n$ is the total photon number) for a quantum metrology [7–14].

However, the Heisenberg limit is very difficult to reach due to the noise influence in quantum metrology experiments [14–20]. Many studies have concerned the influences of imperfect interferometers and detectors on quantum metrology precision [21–23]. For example, a lossy interferometer induces an extra information loss, which leads to the asymmetric photon distribution on the output arms of the interferometer [23–28]. The phase fluctuations between different modes result in the variation of off-diagonal elements of the interferometer density matrix [28–31]. Meanwhile, the limited detection efficiency and dark counts of single photon detectors result in photon event recording error [22,23]. And bunching events cannot be distinguished by commonly used single photon detectors [32,33]. To distinguish multiphoton events, transition-edge-sensor (TES) number-resolving detectors are developed for bunching events detection in visible and telecom bands [34–37], and pseudo photon number-resolving detectors (PNRDs) with on-off photon detector arrays are invented to detect bunching events with fundamental components [21,38–41]. However, the multiphoton events are rarely considered in quantum metrology to date [42,43]. For a nondeterministic single photon sources based on spontaneous parametric down-conversion (SPDC), multiphoton phenomena are commonly existing [44]. These imperfect components can deviate the probability distribution during projective measurements, leading to an extra information loss in phase estimation process.

In this paper, we theoretically analyze the influence of imperfect single photon source on quantum phase estimation. The classical Fisher information (FI) and quantum Fisher Information (QFI) are analyzed to evaluate the phase uncertainty bound of a quantum metrology system. Multiphoton events induce a stronger restriction of the quantum enhanced bound, and the upper limitation of multiphoton probability for quantum enhanced precision is obtained. Our results express the information loss caused by multiphoton events and provide a limitation of the multiphoton probability in quantum metrology.

2. Phase estimation process and Fisher information

A phase estimation system includes an initial probe state $\rho _0$ preparation, an unitary evolution $U(\boldsymbol {\theta })$, and a positive-operator valued measure (POVM) $\hat {E}$ acting on the evolved state $\rho (\boldsymbol {\theta } ) = U(\boldsymbol {\theta } ){\rho _0}{U^{\dagger} }(\boldsymbol {\theta } )$. The phases $\boldsymbol {\theta }=\{\theta _1,\theta _2,\ldots,\theta _M\}$ under test are estimated using a suitable estimator $\Lambda (x)$ based on measurement results $x = \{ {x_1},{x_2}, \ldots,{x_\nu }\}$, where $\nu$ is the number of measurements.

The accuracy of an estimation process is quantified by the Mean Square Error (MSE), which is defined by

(1)$${\text{MSE}}(\boldsymbol{\theta} ) = \sum_x {{{(\Lambda (x) - \boldsymbol{\theta} )}^2}P(x|\boldsymbol{\theta} )} =(\Delta\boldsymbol{\theta})^2+Bias(\boldsymbol{\theta})^2.$$

Where $P(x|\boldsymbol {\theta } )$ is the probability of detecting event $x$ under parameter $\boldsymbol {\theta }$. The MSE consists of bias $Bias(\boldsymbol {\theta })^2$ and variance $(\Delta \boldsymbol {\theta })^2$. For unbiased estimators, MSE is equal to the variance of the estimation value $(\Delta \boldsymbol {\theta })^2$, called phase variance. The square root of the unbiased estimator variance is termed as the phase uncertainty, i.e. $\Delta \boldsymbol {\theta }$. The phase uncertainty bound of a quantum phase estimation is restricted by the chain of inequalities:

(2)$$\sum_{i=1}^{M} {{{(\Delta {\theta_i})}^2}} \geqslant\frac{{{\text{Tr}}[{F^{ - 1}}]}}{\nu}\geqslant\frac{{{\text{Tr}}[{F_Q}^{ - 1}]}}{\nu},$$

Here $F$ is the classical Fisher information matrix (CFIM) [45]

(3)$${F_{i,j}} = \sum_{n=1}^{\nu} {\frac{1}{{p({x_n}|\boldsymbol{\theta} )}}} \frac{{\partial p({x_n}|\boldsymbol{\theta} )}}{{\partial {\theta _i}}}\frac{{\partial p({x_n}|\boldsymbol{\theta} )}}{{\partial {\theta _j}}}.$$

$F_Q$ is the quantum Fisher information matrix (QFIM)

(4)$${[{F_Q}]_{i,j}} = {\text{Tr[}}\rho (\boldsymbol{\theta} )\frac{{{L_i}{L_j} + {L_j}{L_i}}}{2}{\text{],}}$$

where $L_i$ is the symmetric logarithmic derivative (SLD) of the output state $\rho (\boldsymbol {\theta })$ with respect to the phase $\theta _i$,

(5)$$\frac{{\partial \rho (\boldsymbol{\theta} )}}{{\partial \theta_i }} = \frac{{L_i \rho (\boldsymbol{\theta} ) + \rho (\boldsymbol{\theta} )L_i}}{2}.$$

For single phase estimation, QFIM can be simplified to a single value named as QFI. The phase uncertainty $\Delta \theta$ limited by the QCRB represents:

(6)$$\Delta \theta\geqslant\frac{1}{\sqrt{\nu{F}}}\geqslant\frac{1}{\sqrt{\nu{F_Q}}}.$$

The lower bound can always be saturated by selecting the projectors set over the eigenstates of $L$ as the measurement operators [46,47]. QFI is determined by the output state $\rho (\theta )$. Through spectral decomposing the output state ${\rho (\theta )} = \sum _{i = 1}^M {{p_i}} |{\psi _i}\rangle \langle {\psi _i}|$, the QFI can be calculated by

(7)$${F_Q} = \sum_{i = 1}^M \frac{(\partial_{\theta} p_i)^2}{p_i}+\sum_{i = 1}^M 4p_i\langle \partial_{\theta}\psi_i|\partial_{\theta}\psi_i\rangle-\sum_{i,j = 1}^M \frac{8p_ip_j}{p_i+p_j}|\langle \psi_i|\partial_{\theta}\psi_j\rangle|^2.$$

A Mach-Zehnder interferometer (MZI) is always employed for single phase estimation, which is composed of two beam splitters (BS) and a phase shifter (PS). In theory, the transformation of an MZI on the initial state can be described as [45,48,49]

(8)$$\rho (\theta ) = U(\theta ){\rho _0}{U^{\dagger} }(\theta ) = {e^{{\text{i}}\theta {J_y}}}{\rho _0}{e^{ - {\text{i}}\theta J_y^{\dagger} }}.$$

Where $J_y$ is the linear Schwinger operator defined as $J_y=-\frac {\mathrm {i}}{2} (a_1^{\dagger} a_2-a_2^{\dagger} a_1)$, $a_i$ is the annihilation operator of the input mode $i$ [27,50]. Thus the generator operator $H$ of the parameterization process can be calculated as $H = i({\partial _\theta }{U^{\dagger} }(\theta ))U(\theta ) = - \frac {i}{2}(a_1^{\dagger} {a_2} - a_2^{\dagger} {a_1})$ [51].

The transformation of annihilation operators can be calculated by

(9)$$\left( {\begin{array}{c} {b_1^{\dagger} } \\ {b_2^{\dagger} } \end{array}} \right) = U(\theta )\left( {\begin{array}{c} {a_1^{\dagger} } \\ {a_2^{\dagger} } \end{array}} \right){U^{\dagger} }(\theta ) = \text{MZI}(\theta )\left( {\begin{array}{c} {a_1^{\dagger} } \\ {a_2^{\dagger} } \end{array}} \right).$$

Where $b_i^{\dagger}ger$ is the creation operator of the output mode $i$. $\text {MZI}(\theta )$ is the unitary matrix of MZI

(10)$$\text{MZI}(\theta) = \left( {\begin{array}{cc} {\cos \frac{\theta }{2}} & { - \sin \frac{\theta }{2}} \\ {\sin \frac{\theta }{2}} & {\cos \frac{\theta }{2}} \end{array}} \right).$$

A standard single photon detector cannot distinguish photon numbers at an output port, which corresponds to a measurement projector ${\hat S_{{b_i}}} = {({\mathbb {I}_{{b_i}}} - |0\rangle _{{b_i}{b_i}}}\langle 0|)$, where ${\mathbb {I}_{{b_i}}} = \sum _n | n{\rangle _{{b_i}{b_i}}}\langle n|$ is the completeness relation of Fock space [52].

For multiphase estimation processes, QFIM can be calculated with the output state $\psi _{\boldsymbol {\theta }}$ [13]:

(11)$${[{F_Q}]_{m,n}} = 4{\rm{Re}}[\langle \frac{{\partial {\psi _{\boldsymbol{\theta}} }}}{{\partial {\theta _m}}}|\frac{{\partial {\psi _{\boldsymbol{\theta}} }}}{{\partial {\theta _n}}}\rangle ] + 4\langle \frac{{\partial {\psi _{\boldsymbol{\theta} } }}}{{\partial {\theta _m}}}|{\psi _{\boldsymbol{\theta}} }\rangle \langle \frac{{\partial {\psi _{\boldsymbol{\theta}} }}}{{\partial {\theta _n}}}|{\psi _{\boldsymbol{\theta}}}\rangle$$

Maximum likelihood estimation (MLE) approach is the most commonly used estimator by maximizing the likelihood function

(12)$$L(X;\boldsymbol{\theta} ) = \prod_{n=1}^\nu P ({x_n}; \boldsymbol{\theta} ),$$

where $x_n$ is the $n$-th coincidence event, $\boldsymbol {\theta }$ is the $M$-phase sequence [45]. The logarithm of the likelihood function is defined as the log-likelihood function:

(13)$$\ln L(X;\boldsymbol{\theta} ) =\sum_n\ln P({x_n};\boldsymbol{\theta} )$$

MLE is an asymptotically unbiased estimator, i.e. $\mathop {\lim }_{\nu \to \infty } {\Lambda ^{{\text {MLE}}}}(\boldsymbol {\theta }) = \boldsymbol {\theta }$ for any $\boldsymbol {\theta }$ to be estimated. Thus Eq. (2) is satisfied for $\nu \gg 1$ under MLE.

3. Single phase estimation

For an ideal initial input state ${|\psi _0 \rangle } = |1{\rangle _{{a_1}}}|0{\rangle _{{a_2}}}$ as shown in Fig. 11(a), the expectation value $P$ of photon projector $\hat {S}_{b_1}$ is

(14)$${P =\langle \psi |\hat S|\psi \rangle = {\cos ^2}(\frac{\theta }{2}).}$$

Where $|\psi \rangle$ is the output state. Based on the inversion method, the phase can be estimated using the probability distribution $P$ as ${\theta _{{\text {est}}}} = \arccos (P)$ [53]. The phase uncertainty is deduced from the error propagation formula [7,54]

(15)$$\Delta \theta=\frac{\langle \Delta\hat{S}\rangle}{\left|\frac{\partial\langle \hat{S}\rangle}{\partial \theta}\right|}=\frac{ \sqrt{\langle\hat{S}^2\rangle-\langle\hat{S}\rangle^2}}{\left|\frac{\partial\langle \hat{S}\rangle}{\partial \theta}\right|}.$$

In this scenario, the phase uncertainty coincides with the QCRB

(16)$$\Delta \theta\geqslant \frac{1}{\sqrt{F_Q}}=1.$$

However, the multiphoton events in single photon sources result in a larger theoretical phase uncertainty. The purity of single photon source is evaluated using the second order autocorrelation function

(17)$${g^2}(0) = \frac{{\langle {a^{\dagger} }{a^{\dagger} }aa\rangle }}{{\langle {a^{\dagger} }a\rangle }} = 1 + \frac{{\langle \Delta {n^2}\rangle - \langle n\rangle }}{{{{\langle n\rangle }^2}}}.$$

$g^2(0)$, simplified as $g^2$, describes the probability of detecting photons impinging at both detectors at the same time. Generally, $g^2<0.5$ is a judgement of successfully generating single photon states. The two-photon state has much higher probability compared with other multiphoton states [42,44]. Therefore, two-photon state is mainly considered in our analysis. The non-ideal single photon probe state is described as $\rho = (1 - k)|1\rangle \langle 1| + k|2\rangle \langle 2|$, where $k$ is the generation probability of two photon state. On this condition, the two-photon state probability $k$ is expressed as

(18)$$k = \frac{{1 - {g^2} - \sqrt {1 - 2{g^2}} }}{{{g^2}}}.$$

For a non-ideal single photon source, the input state is changed to ${\rho _{\text {r}}} = (1 - k)|10\rangle \langle 10| + k|20\rangle \langle 20|$ which leads to a deviation of the expectation value $P$

(19)$$P = \langle \psi |\hat S|\psi \rangle = {\cos ^2}\frac{\theta }{2} + k{\cos ^2}\frac{\theta }{2}{\sin ^2}\frac{\theta }{2}.$$

There is an additional bias $\Delta P = - k{\cos ^2}(\theta /2)\sin ^2(\theta /2)$ due to multiphoton states. Thus, a bias term is introduced in MSE which is related to $k$ and the under test phase $\theta$. Fig. 22(c) shows the variation of Root Mean Squared Error (RMSE) versus the number of measurements. The calculation of RMSE can be expressed as follows:

(20)$$\text{RMSE}=\sqrt{(\Delta\theta)^2+Bias(\theta)^2}=\sqrt{1/\nu+Bias(\theta)^2}$$

As an example, a bias distribution versus $\theta$ for $g^2(0)=0.02$ is shown in Fig. 22(a). The maximal value of the bias is $5.1\times 10^{-3}\text {rad}$ at $\theta =\pi /2$, which limits the feasibility for a large number factorization using Shor’s algorithm [55]. To factorize a large number, the phase gate accuracy should be improved exponentially along with the employed qubits, e.g. an phase gate accuracy $\pi /2^9\approx 6.1\times 10^{-3}\text {rad}$ is demanded to factorize a number of 1024. To date, it is still a challenge to prepare an ideal single photon source to saturate the lower bound of phase estimation accuracy. An alternative method is considering the multiphoton effect during phase estimation. Based on the modified estimator, the phase uncertainty can be calculated by

(21)$$\Delta \theta = \frac{{\sqrt {{{\cos }^2}\frac{\theta }{2}\left( {1 + k{{\sin }^2}\frac{\theta }{2}} \right){{\sin }^2}\frac{\theta }{2}\left( {1 - k{{\cos }^2}\frac{\theta }{2}} \right)} }}{{|{\sin \theta - k\cos \theta \sin \theta}|/2}},$$

which coincides with the QFI only at $\theta =n\pi$, as shown in Fig. 33(a). In order to achieve the ideal estimate accuracy, the phase should be limited to a small range.

Fig. 1. Two-mode interferometer schemes. (a) Herald single photon input state. Entangled photon pairs are injected into the input port 1 and the trigger separately and being measured by a single photon detector at the output port 1. (b) Twin-Fock input state. Generated photon pair are injected into the different port of the interferometer respectively. The coincidence counts are measured by single photon detectors.

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Fig. 2. Bias error for imperfect photon source at $g^2=0.02$. (a) Bias error with single input state. (b) Bias error with twin-Fock input state. Bias error varies versus phase under test and reached its peak at $\theta =\pi /2$. (c) RMSE under different number of measurements $\nu$ at $\theta =\pi /2$. Blue line: perfect single-photon state. Red line: imperfect single-photon sources. RMSE converges to the phase deviation occurred by imperfect photon source, and can not be lower than this limit.

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Fig. 3. CRB for imperfect photon source. (a) Single photon input state model. Minimum phase uncertainty coincides with QCRB at $\theta =\pi$. (b) Twin-Fock input state model. Minimum phase uncertainty is larger than QFI with projective measurement of coincidence counts. Blue curve corresponds to the phase uncertainty when the estimated phase varies, red line to the QCRB in the ideal case, and black line to the QCRB in the imperfect case.

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Another single phase estimation model is represented in Fig. 11(b), where idler and signal photons are injected into the two ports of an MZI simultaneously. The twin-Fock state $|\psi \rangle = |N/2,N/2\rangle$ is used as an input state which can improve the phase estimation uncertainty to the Heisenberg limit with the total input photon number $N$ [18,56]. The lower bound of the phase uncertainty using twin-Fock states is

(22)$$\Delta \theta\geqslant\frac{1}{\sqrt{\nu F_Q }} = \frac{\sqrt{2}}{\sqrt{\nu N(N + 2)}}.$$

A photon number measurement is required to saturate the lower bound for $N>2$. Quantum state projection method has been developed to simulate number resolving detectors called PNRDs [14,17,39], where multiple photons are separated and detected by a single photon detector respectively. Unfortunately, it still can’t saturate QCRB with a set of projectors over the ideal output cases, which will be discussed later.

Using the simplest form of twin-Fock state $|1_{a_1},1_{a_2}\rangle$ as an input photon state, the coincidence count rate has the form

(23)$$\langle {1_{{a_1}}}{1_{{a_2}}}|b_1^{\dagger} b_2^{\dagger} {b_1}{b_2}|{1_{{a_1}}}{1_{{a_2}}}\rangle = \frac{{1 + \cos (2\theta )}}{2}.$$

The phase uncertainty correlating to the projector $\hat T = {({\mathbb {I}_{{b_1}}} - |0\rangle _{{b_1}{b_1}}}\langle 0|) \otimes {({\mathbb {I}_{{b_2}}} - |0\rangle _{{b_2}{b_2}}}\langle 0|)$ satisfies

(24)$$\Delta \theta = \frac{{\sqrt {{{\cos }^2}\theta - {{\cos }^4}\theta } }}{{2\cos \theta \sin \theta }} = \frac{1}{2}.$$

Imperfect photon source leads to an estimation bias of $\Delta {\theta _{{\text {max}}}} = 6.3 \times {10^{ - 2}}{\text {rad}}$ at $\theta =\pi /2$ due to the derivation of the coincidence count rate $\Delta P = k(1 - {\cos ^2}\theta - 3/4{\sin ^4}\theta )$, as shown in Fig. 22(b). When the influence of multi-photon states is considered, the phase uncertainty is

(25)$$\Delta \theta = \frac{{\sqrt {\left( {{{\cos }^2}\theta + k(1 - {{\cos }^2}\theta - \frac{3}{4}{{\sin }^4}\theta )} \right)\left( {{{\sin }^2}\theta - k(1 - {{\cos }^2}\theta - \frac{3}{4}{{\sin }^4}\theta )} \right)} }}{{|\sin 2\theta - k\sin 2\theta + 3k{{\sin }^3}\theta \cos \theta |}}.$$

The minimal phase variance $\Delta \theta = 0.2512$ is achieved at $\theta \approx 0.28\pi$, as shown in Fig. 33(b). However, due to the multi-photons information loss during measurement, it still cannot reach the theoretical lower phase variance bound ${(\Delta \theta )^2} = 0.2450$ for an imperfect photon source, as the QFI of the initial state $\rho _0 = (1 - k)|11\rangle \langle 11| + k|22\rangle \langle 22|$ is $F_Q=4+8k$. The necessary and sufficient conditions to achieve the lower bound of QFI are

(26)$$\operatorname{Im} [{\text{Tr}}({\rho _\theta }\hat EL)] = 0$$

and

(27)$$\frac{{\sqrt {\hat E} \sqrt {{\rho _\theta }} }}{{{\text{Tr}}[{\rho _\theta }\hat E]}} = \frac{{\sqrt {\hat E} L\sqrt {{\rho _\theta }} }}{{{\text{Tr}}[{\rho _\theta }\hat EL]}}.$$

Where $\hat {E}$ is the element of a POVM satisfying $\int {\text {d}}x{\hat E_x} = \mathbb {I}$, $\rho _\theta$ is the output state after parameter $\theta$ encoded and $p(x|\theta ) = {\text {Tr}}({\hat E_x}{\rho _\theta })$. For a perfect single photon source, the projector can be simplified to $\hat T = \hat E = |11\rangle \langle 11|$. Since the parameterized state ${\rho _\theta } = |\Psi \rangle \langle \Psi |$ is a pure state with $|\Psi \rangle = - {\sin \theta }/{\sqrt 2 }|20\rangle + {\sin \theta }/{\sqrt 2 }|02\rangle + \cos \theta |11\rangle$, the solution of $L$ has the form [57]

(28)$$L = 2(|{\partial _\theta }\Psi \rangle \langle \Psi | + |\Psi \rangle \langle {\partial _\theta }\Psi |).$$

By substituting Eq. (28) into Eq. (26) and Eq. (27), one can obtain,

(29)$${\text{Tr}}({\rho _\theta }\hat EL) = {\text{Re}}[{\text{Tr}}({\rho _\theta }\hat EL)] = {\partial _\theta }p(x|\theta ),$$

(30)$$\frac{{\sqrt {\hat E} \sqrt {{\rho _\theta }} }}{{{\text{Tr}}[{\rho _\theta }\hat E]}} = \frac{{|11\rangle \langle \Psi |}}{{\cos \theta }} = \frac{{\sqrt {\hat E} L\sqrt {{\rho _\theta }} }}{{{\text{Tr}}[{\rho _\theta }\hat EL]}}.$$

However, for an imperfect single photon, ${\rho _\theta } = (1 - k)|{\Psi _1}\rangle \langle {\Psi _1}| + k|{\Psi _2}\rangle \langle {\Psi _2}|$, where

(31)$$|{\Psi _1}\rangle ={-} \frac{{\sin \theta }}{{\sqrt 2 }}|20\rangle + \frac{{\sin \theta }}{{\sqrt 2 }}|02\rangle + \cos \theta |11\rangle ,$$

(32)$$|{\Psi _2}\rangle = \frac{{\sqrt 6 }}{4}[{\sin ^2}\theta (|40\rangle + |04\rangle ) - \sin 2\theta (|31\rangle + |13\rangle )] + (1 - \frac{3}{2}{\sin ^2}\theta )|22\rangle .$$

The expression of $L$ in the D-dimensional complete Hilbert space spanned by the eigenvectors of $\rho _\theta$ is

(33)$$L = \mathop \sum _{i = 1}^D \frac{{{\partial _\theta }{p_i}}}{{{p_i}}}|{\Psi _i}\rangle \langle {\Psi _i}| + 2\mathop \sum _{i,j = 1}^D \frac{{{p_j} - {p_i}}}{{{p_i} + {p_j}}}\langle {\Psi _i}|{\partial _\theta }{\Psi _j}\rangle |{\Psi _i}\rangle \langle {\Psi _j}|.$$

Where $|{\Psi _i}\rangle$ is the $i$-th eigenvector in the Hilbert space. By substituting Eq. (33) into Eq. (27), one can obtain

(34)$$\frac{{\sqrt {\hat E} \sqrt {{\rho _\theta }} }}{{{\text{Tr}}[{\rho _\theta }\hat E]}} = \frac{{\sqrt {1 - k} \cos \theta |\hat T\rangle \langle {\Psi _1}| + \sqrt k (1 - \frac{3}{2}{{\sin }^2}\theta )|\hat T\rangle \langle {\Psi _2}|}}{{(1 - k){{\cos }^2}\theta + k(1 - \frac{3}{4}{{\sin }^4}\theta )}},$$

(35)$$\begin{aligned}\frac{{\sqrt {\hat E} L\sqrt {{\rho _\theta }} }}{{{\text{Tr}}[{\rho _\theta }\hat EL]}} &= \frac{{\mathop \sum _{i = 1}^2 2{p_i}|\hat T\rangle \langle \hat T|{\partial _\theta }{\Psi _i}\rangle \langle {\Psi _i}|}}{{{\text{Tr}}[{\rho _\theta }\hat EL]}} \\ &= \frac{{2\sqrt {1 - k} ( - \sin \theta )|\hat T\rangle \langle {\Psi _1}| - 6\sqrt k \sin \theta \cos \theta |\hat T\rangle \langle {\Psi _2}|}}{{(1 - k)\sin 2\theta - 3k{{\sin }^3}\theta \cos \theta }}. \end{aligned}$$

The two formulas do not equal, which results in the lower bound of QFI not achievable under our measurement. Moreover, the condition equation of Eq. (27) is satisfied if and only if the measurement operators are projectors over the eigenstates of $L$ [47]. However, the eigenstates of $L$ are superpositions of different projection results. For example, one of the eigenstates is expressed as $|{L_\theta }\rangle = - 1/2U|31\rangle + 1/2U|13\rangle + 1/\sqrt 2 U|22\rangle$. This projector cannot be realized by adjusting PNRDs onto two-photon states. Therefore, the lower bound of QFI requires larger scale detector cascades due to the imperfect photon sources.

4. Multiphase estimation

Compared with two-mode MZIs, multimode interferometers have higher phase sensitivity in phase estimation [58,59]. Previous studies have shown an adaptive phase estimation strategy to optimize the precision of multiphase estimation [60–62]. The three-beam extension of a beam splitter, named tritter, has the mathematical description form [63]

(36)$$U = \sqrt {\frac{1}{3}} \begin{pmatrix} 1 & 1 & 1 \\ 1 & {{e^{{\text{i}}\frac{2}{3}\pi }}} & {{e^{ - {\text{i}}\frac{2}{3}\pi }}} \\ 1 & {{e^{ - {\text{i}}\frac{2}{3}\pi }}} & {{e^{{\text{i}}\frac{2}{3}\pi }}} \end{pmatrix}.$$

Tritter is a generalized beam splitter in the three-dimensional Hilbert space, i.e. each mode of a tritter has the same probability to detect an injected single photon. The schematic of a three-mode interferometer is shown in Fig. 44. A two-mode phase shifter is employed between two balanced tritters to control the phase difference,

(37)$$P=\begin{pmatrix} 1 & 0 & 0 \\ 0 & e^{\mathrm{i}\theta_1} & 0 \\ 0 & 0 & e^{\mathrm{i}\theta_2} \end{pmatrix}.$$

As a natural expansion of twin-Fock state, idler and signal photons can inject into different ports of the three-mode interferometer to enhance the phase estimation precision. Beam splitters are added for bunching events detection.

Fig. 4. Conceptual scheme of the three-mode interferometers with two-photon input states. The three-mode interferometer for two-phase estimation process is composed of two tritters and one inserted two-mode PS. Idler and signal photons are injected into the different mode of the three-mode interferometer simultaneously. Output states are measured by standard single photon detectors. Three additional 50:50 beam splitters are used for implementing PNRDs to detect bunching events.

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For a three-mode interferometer, since the unitary process without phase information does not effect the result of QFIM, the last tritter does not need to take into consideration [57]. The output state after the first tritter and phase encoded shifters can be described as

(38)$${\rm{|}}\Psi \rangle = \frac{1}{3}(\sqrt {2z} {e^{2i{\theta _1}}}|200\rangle + \sqrt 2 z{e^{2i{\theta _2}}}|020\rangle + \sqrt 2 {z^2}|002\rangle - z{e^{i({\theta _1} + {\theta _2})}}|110\rangle - {e^{i{\theta _1}}}|101\rangle - {e^{i{\theta _2}}}|011\rangle )$$

Where $z = {e^{i3/2\pi }}$. The partial derivative of two phases are

(39)$$\frac{{\partial |\Psi \rangle }}{{\partial {\theta _1}}} = \frac{1}{3}(2i\sqrt 2 z{e^{2i{\theta _1}}}|200\rangle - iz{e^{i({\theta _1} + {\theta _2})}}|110\rangle - i{e^{i{\theta _1}}}|101\rangle ) $$

(40)$$\frac{{\partial |\Psi \rangle }}{{\partial {\theta _2}}} = \frac{1}{3}(2i\sqrt 2 z{e^{2i{\theta _2}}}|020\rangle - iz{e^{i({\theta _1} + {\theta _2})}}|110\rangle - i{e^{i{\theta _2}}}|011\rangle ) $$

the QFIM can be analytically calculated by Eq. (11),

(41)$${F_Q} = \frac{4}{3}\left( {\begin{array}{cc} 2 & { - 1} \\ { - 1} & 2 \end{array}} \right).$$

The phase variance bound using entangled input states ${\text {tr}}[({F_Q})_{{\text {ent}}}^{ - 1}] = 1$ is smaller than the case of two separate photon input states ${\text {tr}}[({F_Q})_{{\text {sep}}}^{ - 1}] = 1.5$. Generally, it is not guaranteed to reach the limitation even using entangled input states due to the non-commutativity of the operators [13,64]. The calculated theoretical minimal phase variance is 1.301 which represents an enhancement accuracy compared with separate photons.

MLE requires a prior knowledge about the probability distribution $P(\mu )$ of the output measurement events $\mu$. For example, the likelihood function is $L(X;\boldsymbol {\theta } ) = \prod \nolimits _{p,q} {{P_i}{{(23 \to pq)}^{{n_{pq}}}}}$, where $n_{pq}$ is the number of measurements on output $(p, q)$, and ${P_i}(23 \to pq)$ is the ideal probability distribution. However, the probability distribution variation caused by imperfect single photon sources introduces an extra bias in the estimation process. The bias distribution results are shown in Figs. 55(a) and 55(b) under the assumption that the input state is a mixed state ${\rho _0} = (1 - k)|11\rangle \langle 11| + k|22\rangle \langle 22|$. At the point $({\theta _1},{\theta _2}) = \left ( {0.299, - 2.370} \right )$, the estimation of MLE is deviated from the real value due to the multiphoton influence. In order to eliminate the bias, multiphoton effects have to be considered. The probability distribution should be normalized as the multiphoton input states may trigger more than one coincident record at the same time. The modified likelihood function can be written as $L(X;\theta ) = \prod \nolimits _{p,q} {P{{(23 \to pq)}^{{n_{pq}}}}/[\sum \nolimits _{i,j} {P(23 \to ij)} ]}$ where $P(23 \to ij)$ is the real output probability to eliminate the bias. One might use post-selection to exclude multiphoton detecting events to neglect four-photon coincidence events. However, this method excludes separated single photon events that arrived at the same time resolution in error [65], and maintains some four-photon states unfiltered such as $|400\rangle$ or $|220\rangle$, etc. Furthermore, inefficiency TES detectors may misjudge the multiphoton states as single photon outputs. These states also lead to a probability deviation which is similar to our demonstration, and not discussed in this paper.

Fig. 5. Variation in estimation accuracy considering multiphoton states (a) The extra bias in MLE caused by the probability distribution deviation of imperfect single photon source. (b) Log-likelihood function distribution at $({\theta _1},{\theta _2}) = \left ( {0.299, - 2.370} \right )$. The region outlined in red highlights the scores which are higher than the value at the initial phase. Black point is the initial phase. Green point is the estimation result by MLE. (c) CRB distribution. The red areas highlight the improved performance compared with the QCRB with two distinguishable single-photon inputs. The white regions show the realistic improved performance under imperfect single photon scenario.

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Same as the case of the two-mode interferometer, extra multiphoton events also decrease the phase accuracy. The necessary and sufficient condition for saturating QCRB is [57]

(42)$${\text{Tr}}(\rho [{L_i},{L_j}]) = 0,\quad \forall i,j.$$

With the imperfect input state, one can obtain that

(43)$$\begin{aligned}&{\text{Tr}}(\rho [{L_i},{L_j}]) ={-} 2\sum_{l = 1}^2 {{p_l}} \sum_{m = 1}^D {(\langle } {\Psi _m}|{\partial _\theta }_{_j}{\Psi _l}\rangle \langle {\Psi _l}|{\partial _\theta }_{_i}{\Psi _m}\rangle - \langle {\Psi _l}|{\partial _\theta }_{_j}{\Psi _m}\rangle \langle {\Psi _m}|{\partial _\theta }_{_i}{\Psi _l}\rangle )\\ &={-} 2\sum_{l = 1}^2 {{p_l}} \sum_{m = 1}^D {(\langle } {\Psi _l}|{\partial _\theta }_{_i}{\Psi _m}\rangle \langle {\Psi _m}|({\partial _\theta }_{_j} - {\partial _\theta }_{_i}){\Psi _l}\rangle + \langle {\Psi _m}|{\partial _\theta }_{_i}{\Psi _l}\rangle \langle {\Psi _l}|({\partial _\theta }_{_i} - {\partial _\theta }_{_j}){\Psi _m}\rangle. \end{aligned}$$

Where $\rho =(1 - k)|{\Psi _1}\rangle \langle {\Psi _1}| + k|{\Psi _2}\rangle \langle {\Psi _2}|$ is the density matrix of the output state, $|{\Psi _i}\rangle$ is the $i$-th eigenvector of the complete D-dimensional Hilbert space expanded by $\rho$. Since ${\partial _{{\theta _1}}}{\Psi _l}$ and ${\partial _{{\theta _2}}}{\Psi _l}$ are unequal, Eq. (42) is not satisfied under the effect of multiphoton states. Furthermore, the multi-photons information loss during measurement will reduce the enhancement effect of quantum metrology. As shown in Fig. 55(c), the red regions highlight the improved performance for the ideal case. The white regions show the improved performance for the photon source with $g^2=0.04$, which is much smaller than the ideal case. The minimum phase variance of the imperfect scheme is 1.404, which is larger than the lower bound $\sum {(\Delta \boldsymbol {\theta } )^2}\geqslant 1.301$ of the ideal scheme. Projection measurement records the states of $|013\rangle$ and $|031\rangle$ as the same count $|011\rangle$. The correlation CFIM can be written as

(44)$$F_{m,n}^1 = \frac{1}{{f + g}}(\frac{{\partial f}}{{\partial {\theta _m}}} + \frac{{\partial g}}{{\partial {\theta _m}}})(\frac{{\partial f}}{{\partial {\theta _n}}} + \frac{{\partial g}}{{\partial {\theta _n}}}) + O,$$

where $f$ and $g$ are the probability of detecting output state $|013\rangle$ and $|031\rangle$ respectively, $O$ is the information provided by other output counts. Based on photon-number measurement, the relative CFIM has the form

(45)$$F_{m,n}^2 = \frac{1}{f}\frac{{\partial f}}{{\partial {\theta_m}}}\frac{{\partial f}}{{\partial {\theta _n}}} + \frac{1}{g}\frac{{\partial g}}{{\partial {\theta_m}}}\frac{{\partial g}}{{\partial {\theta_n}}} + O.$$

The information loss can then be evaluated by

(46)$$F_{m,n}^2 - F_{m,n}^1 = (\frac{1}{f}\frac{{\partial f}}{{\partial {\theta _m}}} - \frac{1}{g}\frac{{\partial g}}{{\partial {\theta _m}}})(\frac{1}{f}\frac{{\partial f}}{{\partial {\theta _n}}} - \frac{1}{g}\frac{{\partial g}}{{\partial {\theta _n}}})\frac{{fg}}{{f + g}}.$$

The diagonal elements are always nonnegative in the above formula, thus the total phase variance satisfies

(47)$$\min \mathop \sum _{j = 1}^n (\Delta \theta _j^1)^2 = \mathop \sum _{j = 1}^n \frac{1}{{F_{j,j}^1}} \geqslant \min \mathop \sum _{j = 1}^n (\Delta \theta _j^2) = \mathop \sum _{j = 1}^n \frac{1}{{F_{j,j}^2}}.$$

For single-phase estimation, Eq. (46) can be simplified to $F^2 - F^1 = (\frac {\partial f}{f\partial {\theta _m}} -\frac { \partial g}{g\partial {\theta _m}})\frac {fg}{(f + g)}$. For the case of $f = Cg$, the measurement information is lossless.

To achieve a phase uncertainty lower than separate photons, one needs to limit the above information loss in a small range. For two-mode interferometer model, the multiphoton probability $k$ should be lower than 0.5345. Three-mode interferometer requires a stricter region, $k$ cannot surpass 0.0406 which means only a close-to-pure state can generate quantum enhanced accuracy. The normalized relationship between phase variance and multiphoton probability $k$ are shown in Fig. 66. The multiphoton probability limitation is under the assumption that other parts of the measurement process is ideal. However, in realistic experiments, due to the photon loss and imperfect detector efficiency, the multiphoton states are more easily detected [42]. The limitation would be more restrict once other noises in the measurement system are considered.

Fig. 6. Normalized phase variance versus multiphoton probability $k$. Black dotted line represents the SQL in the ideal case. Green curve corresponds to the SQL with $k$ increasing

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5. Conclusion

In this paper, the influence of multiphoton events of single photon sources on quantum enhanced phase measurement is demonstrated. The multiphoton events are spontaneously and nondeterministic generated in parametric conversion single photon sources. It induces an extra bias which declines the phase estimation accuracy compared with ideal case. Meanwhile, the quantum enhanced phase estimation is restricted in a limited region for both single-phase and multi-phase scenarios. The phase estimator is also improved through considering the multiphoton events. Our results provide suggestions to eliminate multiphoton bias and analyzing information losses with imperfect single photon sources.

Funding

National Natural Science Foundation of China (62075238).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request

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Influence of multiphoton events on the quantum enhanced phase estimation

Abstract

1. Introduction

2. Phase estimation process and Fisher information

3. Single phase estimation

4. Multiphase estimation

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (47)

Optics Express