Maximal quantum Fisher information for phase estimation without initial parity

Mach-Zehnder interferometer is a common device in quantum phase estimation and the photon losses in it are an important issue for achieving a high phase accuracy. Here we thoroughly discuss the precision limit of the phase in the Mach-Zehnder interferometer with a coherent state and a superposition of coherent states as input states. By providing a general analytical expression of quantum Fisher information, the phase-matching condition and optimal initial parity are given. Especially, in the photon loss scenario, the sensitivity behaviors are analyzed and specific strategies are provided to restore the phase accuracies for symmetric and asymmetric losses.


I. INTRODUCTION
Phase estimation is an important task in quantum metrology. Classically the measurement precision in optical phase estimation is bounded by the standard quantum limit (SQL), 1/ √ N , where N is the total number of photons. Such sensitivity, however, can be further enhanced by exploiting spin-squeezing techniques [1]. In 1981, Caves [2] pointed out that, by feeding a high intensity coherent state in one input port of a interferometer and a low intensity squeezed vacuum state in the other, the phase sensitivity can approach the Heisenberg scaling, 1/N . The SQL can also be beat by employing nonclassical entangled states [3,4]. The realization of surpassing the SQL as well as approaching the Heisenberg limit have brought about significant progress in recent years [5][6][7].
In Ref. [2], Caves also pointed out that, in order to enhance precision of measurement, the phase difference of the input states should satisfy a certain relation. Such relation is called the phase-matching condition (PMC). Recently a more general PMC for even (odd) states has been investigated [8]. However, it remains unknown whether non-even (odd) states also have a general PMC invariance. Therefore, the major purpose of this paper is to study the PMC of non-even (odd) states, especially under particle losses.
In this paper, we discuss a specific Mach-Zehnder interferometer. One of the input port is injected by a coherent state and the other is injected by a coherent superposition state whose parity depends on an adjustable parameter. We first give an analytic expression of the QFI without particle losses and determine a PMC to optimize the parameter precision. Then we consider a setting where particle losses occur in both arms with the same transmission coefficient and give the expression of the QFI as well as the PMC. Based on the expressions of the QFIs, we prove that the PMC keeps unchanged for any transmission coefficients, whether input state is with parity or * Electronic address: xgwang@zimp.zju.edu.cn not.

II. MACH-ZEHNDER INTERFEROMETER
Mach-Zehnder (MZ) interferometer is a well-known optical apparatus in quantum metrology. It can be used to study many counterintuitive phenomena in quantum mechanics [9,10]. A general MZ interferometer mainly consists of beam splitters and phase shifts. A SU(2) MZ interferometer can be described by Schwinger operators, which are [11][12][13] were a, a † , b, b † are annihilation and creation operators for ports A and B. Above Schwinger operators satisfy the commutation relations where ǫ ijk is the Levi-Civita symbol. A 50:50 beam splitter can be described by a unitary transformation [11][12][13] and the phase shift is where θ is the phase difference between two arms after phase shift. With Eqs. (3) and (4), the total transformation performed by the MZ interferometer can be expressed by A vivid picture for the performance of the SU(2) MZ interferometer can be given [11] from the angular momentum description of the Schwinger operators. The operators B x , B † x and P z can also be treated as the rotation operation to the input states about the x axis and the z axis, respectively. Combining the three operations in sequence, the transformation of total setup is equivalent to a rotation about the y axis.

III. QUANTUM FISHER INFORMATION
Fisher information, initially introduced by Fisher [14] in 1925, is a very important quantity in statistics. Quantum Fisher information (QFI) is the quantum extension of Fisher information and of great significance in quantum technology and quantum metrology [7,[15][16][17]. In quantum metrology, it is a central concept to describe the lowest bound for the precision of a parameter under estimation. The variance of an unbiased estimatorθ (i.e., θ = θ) for the parameter θ is theoretically bounded by the quantum Cramér-Rao inequality: Var(θ) ≥ 1/νF [18][19][20][21], where Var(·) is the variance, ν is the repeated times of experiments and F refers to quantum Fisher information. The quantum Fisher information is defined by F = Tr(ρL 2 θ ) [18,19], where L θ is so-called symmetric logarithmic derivative (SLD) and is determined by the equation Recently, it has been found that the quantum Fisher information for a non-full rank density matrix is determined by its support [22][23][24]. Denote the spectral decom- where M is the dimension of the support, and p i , |ψ i are the ith eigenvalue and eigenstate for ρ, respectively. In this representation, the quantum Fisher information can be expressed by [22][23][24]

IV. PMC WITHOUT PHOTON LOSSES
First we consider an ideal Mach-Zehnder interferometer without particle losses. For a pure input state ρ in = |φ φ|, the output state is ρ out = U MZ |φ φ|U † MZ . Based on Eq. (5), one can easily find that the QFI is Utilizing the expression of J y , the QFI can be expressed by are average photon numbers for arm A and B, respectively. The expected value · := φ| · |φ .
In this paper, we choose the input state in port A to be a coherent state |iαe iφ iαe iφ | and that in port B to be a coherent superposition state |α + α|, where |α + = N α (|α + e iω | − α ) with |α also a coherent state and Here the relative phase between two input ports is φ + π/2, and φ + π/2 ∈ [0, π), so φ ∈ [− π 2 , π 2 ]. Notice that when ω = 0, ρ B owns parity, i.e., it is a even state satisfying + α|b|α + = 0. After obtaining the terms, the QFI becomes From this equation, it is easy to see that with fixed α, the PMC to maximum the QFI will always be satisfied as φ = 0, regardless of the valus of ω, that is, no matter input state has parity or not, the PMC does not change.
When φ = 0, the maximal QFI F m becomes Note that under the PMC, F m only depends on the average photon numbers of both ports:n A andn B . And to investigate the relationship between the QFI and the phase sensitivity limit, we can also express F m in terms of total photon number N by denoting N =n A +n B . With Eq. (9), N can be expressed as Then F m becomes Large F is required for the enhancement of phase sensitivity, and this needs high intensity of input states, which means |α| ≫ 1. Then with high intensity and suppose the total photon number is fixed, it is not difficult to find that F m ∝ N + N 2 . Under this condition, F m can lead to approach the Heisenberg scaling [8]. More interestingly, when |α| is large enough, the term which contains cos ω will become neglectable, indicating that the parity of input states not only does not affect the PMC, but also makes no difference to the value of F m under high intensity.

V. PMC WITH PHOTON LOSSES
In section IV we have discussed the PMC without particle losses, which is an ideal scenario. Now we will take into account the effects of photon losses on PMC. Essentially, one can use a fictitious beam splitter to describe the photon losses process. In this paper we will  Figure 1: (Color online) Numeric and analytic results of QFI under particle losses. In figure (a) and (b) ω is fixed, and is chosen to be 0 and 6π/7. The '+' dots represent the numerical results while the lines represent the analytic ones. Each plot has ten lines, which separately has, from bottom to top, the transmission coefficient T varying from 0.1 to 1. Figure (c) shows the φm for different ω with T being fixed at 0.83. The straight line demonstrates that φm is invariant against ω, namely, the PMC is unaffected by the input state's parity. α is set to be 0.3 in all three graphs.
to describe both real and fictitious beam splitters, where T is the so-called transmission coefficient. As for fictitious beam splitters, when T = 1, there are no particle losses, which goes back to our previous discussion; when T = 0, all the photons leak out of the interferometer. For convenience we also define the reflection coefficient R, which satisfies T + R = 1. Note that the operator B T actually has no difference from the general form of Eq. (3), that is We just substitute τ with arccos √ T , i.e., T = cos 2 τ 2 . Essentially, the whole photon loss setting can be converted into a neat scenario, which includes three steps: (1) the input state imports into a MZ interferometer without any loss on photons and passes the first beam splitter; (2) the output state goes through a particle loss channel; (3) the final state after particle loss undergoes a phase shift and passes the second beam splitter.
For simplicity, we assume that the transmission coefficient T are the same on both arms and the input state is separable: ρ in = ρ A ⊗ ρ B . Then after the first 50:50 beam splitter B 1 2 we have where B 1 2 = exp[i(2 arccos 1 2 )J x ]. Then the state goes through a particle loss process, after which we have where with c, c † , d, d † the annihilation and creation operators of mode C and D. Trace operation is conducted here to eliminate the information on mode C and mode D. And consequently we need to cope with mixed states afterwards. Since the second beam splitter merely causes a rotation of ρ in the Hilbert space, it does not affect the QFI, and we shall omit it in the following discussion.
Now we continue to use the previous states as input states, that is, ρ A = |iαe iφ iαe iφ | and ρ B = |α + α|, where |α + = N α (|α + e iω | − α ) and N 2 α = 1/(2 + 2e −2|α| 2 cos ω), ω ∈ [0, π). After some algebra, we can express ρ in the following form (see Appendix A) where η = N 2 α (1 + 2e −2|α| 2 cos ω + p 2 t ), ξe iτ = N 2 α (p r e −iω + p t ) 1 − p 2 t . Here p t = exp(−2|α| 2 T ), p r = exp(−2|α| 2 R). In addition, notice that since the second beam splitter is omitted from the calculation, now we should use the operator J z instead of J y , and the expression of the QFI becomes where λ i and |λ i are eigenvalues and eigenstates of ρ. It is easy to find that [25] where detρ = N 4 α 1 − p 2 t 1 − p 2 r is the determinant of ρ, |A and |A ⊥ are orthogonal basis vectors for the matrix of ρ, and As α increases, the total photon number N also increases. The '+' dots represent N 2 while the ' * ' dots represent N as functions of ω. Each graph has five lines representing the Fm, which separately has, from bottom to top, the transmission coefficient T varying from 0.6 to 1. A line that is above the '+' dots indicates a surpass of the Heisenberg limit, while a line that is above the ' * ' dots indicates a surpass of the standard quantum limit(SQL).
Then through some calculation (see Appendix B) we have the analytic expression of the QFI as where X = 2p t N 2 To have a more intuitional vision of the result, we have plotted it as shown in Fig. 1. From this figure we can see that the analytic results coincide well with the numerical ones as the transmission coefficient T changes, the PMC does remain the same where φ = 0. More importantly, such PMC invariance will be achieved both for input states with and without parity, that is, regardless of the value of ω. When φ = 0, the maximal QFI F m becomes In Fig. 2 we have plotted F m under different particle loss situation against ω as well as the corresponding N and N 2 . From the figure (a) we can see that with low intensity of input states, the QFI can lead to the surpass of the Heisenberg limit even photon losses are involved. As the total photon number N increases, in figure (b) the Heisenberg limit almost can only be beated by the QFI without any photon losses. However, it is still possible to surpass the SQL when T is smaller than 1. From figure (a) to figure (b) we can see that as |α|(or N) increases, F m gradually becomes independent on ω, that is, when α is large enough, the parity of input states almost does not affect the value of the QFI. As N continue increasing(see figure (c)), the difference between with and without particle losses becomes more and more significant, even a small portion (10%) of photon leak will cause the QFI to decrease dramatically. The difference between figure (b) and (c) also indicates that the higher |α|(or N) is, the more difficult it is to surpass the standard quantum limit (SQL).
Now we consider what will happen to each term of F m as |α| becomes larger and larger. This will give us explanations for the phenomena mentioned above. When T =1, both p r and p t go to 0, and N 2 α goes to 1/2. Consequently, all terms of F m that have the factor 4T |α| 4 go to 0, so F m ≈ 2T |α| 2 , which is independent of ω, as we have concluded from Fig. 2. When T=1, however, R=0 and p r =1, and the value of detρ changes significantly. Recall that detρ = N 4 α 1 − p 2 t 1 − p 2 r , so as T changes from a number that is slightly smaller than 1 to exact "1", detρ instantly changes from 1/4 to 0. This leads to a nonzero term with the factor 4T |α| 4 , that is Z(=1). Therefore, when T=1, F m ≈ 2|α| 2 + 4|α| 4 ≈ 4|α| 4 , which is also independent of ω, as expected. It is also not difficult to see from Eq. (14) that with large |α|, N 2 ≈ 4|α| 4 . This means F m and N 2 are almost the same with high intensity of input state, as shown in Fig. 2. Moreover, the abrupt change of the F m caused by T also explains why that even a small portion of photon losses can lead to a significant decrease of the QFI.

VI. CONCLUSION
In summary, we have considered a general scenario of a Mach-Zehnder interferometer. The input state in one mode is a coherent state and the other is a coherent superposition state whose parity can be controlled by an adjustable parameter. We have discussed both scenarios with and without particle losses and provided analytic expressions of the QFI under both circumstances. We have also shown the numeric results of the QFI with particle losses. The results indicate that the PMC is independent of phase difference between input states of the two modes and the parity of input states will not affect the QFI provided high intensity of input states is guaranteed. In this appendix we will utilize the following formula: , |α and |β are coherent states of mode A and mode B. This formula can be proven by expressing |α in the form and the same goes with |β . Then taking advantage of the Hermiticity of B T as well as the equation where τ = 2 arccos √ T , one will find it not difficult to derive Eq. (A.1).
Equipping with this useful formula, we can now begin the calculation. Denote |in = |iαe iφ A ⊗ N α (|α + e iω | − α ) B , (A.4) then after the first beam splitter we have After that the state undergoes photon losses process, and we have To have a more consice and compact form we denote that which is orthogonal to |A . Then we have ρ =N 2 α (1 + 2e −2|α| 2 cos ω + p 2 t )|A A| + N 2 α (p r e −iω + p t ) 1 − p 2 t |A A ⊥ | + N 2 α (p r e iω + p t ) 1 − p 2 t |A ⊥ A| + N 2 α (1 − p 2 t )|A ⊥ A ⊥ |. (A.10) Now we can obtain the matrix form of ρ as shown in Eq. (19).