Two-particle indistinguishability and identification of boson-and-fermion species: a Fisher information approach
Su-Yong Lee Jeongho Bang Jaewan Kim











S.-Y. Lee^⋆ (papercrane79@gmail.com) J. Bang^⋆ (jbang@kias.re.kr) J. Kim School of Computational Sciences, Korea Institute for Advanced Study, Hoegi-ro 85,Dongdaemun-gu, Seoul 02455, Korea 

              ^⋆ Corresponding authors





Two-particle indistinguishability and identification of boson-and-fermion species: A Fisher information approach




    Su-Yong Lee Jeongho Bang Jaewan Kim
    Received: date / Accepted: date
====================================================================================================================



















We present a study on two-particle indistinguishability and particle-species identification by introducing a Fisher-information (FI) approach—in which two particles pass through a two-wave mixing operation and the number of particles is counted in one of the output modes. In our study, we first show that FI can reproduce the Hong-Ou-Mandel (HOM) effect with two bosons or two fermions. In particular, it is found that even though bosons and fermions exhibit different physical behavior (i.e., “bunching" or “anti-bunching") due to their indistinguishability, the aspects of HOM-like dip are quantitatively same. We then provide a simple method for estimating the degree of two-particle indistinguishability in a Mach-Zehnder interferometer-type setup. The presented method also enables us to identify whether the particles are bosons or fermions. Our study will provide useful primitives for various study of  boson and fermion characteristics.











§ INTRODUCTION



Fisher information (FI), originally introduced by R.A. Fisher  <cit.>, is often regarded as a measure of indeterminacy and has been used to derive the lower bound for the mean-squared error of an unbiased estimator <cit.>. The FI is exploited to determine a small deviation from a true value of a parameter with a fixed measurement <cit.>. Given a specific measurement basis, we can readily evaluate the FI by processing the measurement outcomes. In quantum theory, on the other hand, the FI can be maximized by considering all possible measurement bases, and it presents quantum Fisher information (QFI) which is an intrinsic quantity for an input quantum state. 
The QFI was studied in quantum metrology, aiming at achieving better precision in a parameter estimation and enhancing its sensitivity <cit.>. 
The QFI is widely used for various metrological tasks <cit.>, 
but this paper focuses on the (classical) FI to get a better insight in quantum phenomena and sensitivity of parameters. Except quantum metrology, previously there are only few quantum tasks studied by the classical FI <cit.>.

The FI is related with the relative entropy, i.e., a measure of the statistical difference between two probability distributions P(X) and P(X_0) <cit.>.  For Δ X≡ X-X_0 ≪ 1, the relative entropy, also called “Kullback-Leibler divergence”, can be approximated as 

    D(P(X)||P(X_0))=∑_y P(y|X)lnP(y|X)/P(y|X_0)≈(Δ X )^2/2F(X_0),

where P(y|X) and P(y|X_0) denote the conditional probabilities of getting the measurement outcomes y for the parameters X and X_0. Here, the FI of X_0 is defined as

    F(X_0)=∑_y1/P(y|X_0)(∂ P(y|X_0)/∂ X_0)^2.

From the two neighborhood probability distributions, the relative entropy is thus approximated as FI of X_0, multiplied by Δ X^2/2 <cit.>.


  

Using the above-described properties of FI and its relation to the relative entropy, we present a FI approach to yield useful primitives for understanding “boson” and “fermion” characteristics. Here, we utilize the indistinguishability of the identical particles. Due to the quantum interference of the identical particles, the particle indistinguishability was exploited to enhance quantum information protocols <cit.>. Our approach is based on the use of a two-wave mixing (TWM) operation, e.g., beam-splitter or Mach-Zehnder interferometer (MZI). Specifically, we feed the particles into a TWM operation, say Û_ab, and after passing through, the FI is evaluated by measuring only one of the output modes (see Fig. <ref>). Our study is two-fold: (i) Firstly, we show that FI exhibits the Hong-Ou-Mandel (HOM) effect with both bosons and fermions which exhibit the same FI. (ii) Secondly, we propose a simple method to estimate the degree of the two-particle indistinguishability, identifying their species, namely whether they are bosons or fermions. This is enabled by using the MZI-type operation. All these features are exhibited consistently for both bosons and fermions with different underlying physics, i.e., “bunching” and “anti-bunching.” 
Our study presents useful insight on the issues raised by other related works <cit.>.





§ EQUIVALENCE BETWEEN GLOBAL AND LOCAL FISHER INFORMATION



We indicate that in our scenario (as depicted in Fig. <ref>) the evaluation of FI can be carried out by two different ways: i.e., global and local ways. Here,  the“global”  means that the FI is constructed by measuring both output modes. On the other hand, we can also evaluate the FI by counting the particle number in only one of the output modes. We define this “local” FI. In this section, we prove that the global and local FIs are quantitatively equivalent. Firstly, let us consider the output after a TWM operation on two particles. Without loss of the generality, the state of the output is given as

    a(θ,ϕ)|1,1⟩_ab+b(θ,ϕ)|2,0⟩_ab+c(θ,ϕ)|0,2⟩_ab,

where |a(θ,ϕ)|^2+|b(θ,ϕ)|^2+|c(θ,ϕ)|^2=1. 
The state of |n,m⟩_ab represents n and m particles in the two modes a and b, respectively. 
θ and ϕ are the parameters of the TWM.
Subsequently, by considering the conditional probabilities

    P_ab(n_a,m_b|θ,ϕ)    =   ⟨n̂_a⊗m̂_b⟩, 
    
    P_a(n_a|θ,ϕ)    =   ⟨n̂_a⊗_b⟩, 
    
    P_b(m_b|θ,ϕ)    =   ⟨_a⊗m̂_b⟩,

we can verify the following properties:

    {[ P_ab(1,1|θ,ϕ)=P_a(1|θ,ϕ)=P_b(1|θ,ϕ)=|a(θ,ϕ)|^2,; P_ab(2,0|θ,ϕ)=P_a(2|θ,ϕ)=P_b(0|θ,ϕ)=|b(θ,ϕ)|^2,; P_ab(0,2|θ,ϕ)=P_a(0|θ,ϕ)=P_b(2|θ,ϕ)=|c(θ,ϕ)|^2. ].

P_ab(n_a,m_b|θ,ϕ) represents the probability of detecting n  and m particles in the modes a and b, respectively. 
P_a(b)(n_a(b)|θ,ϕ) means the probability of detecting n particles in the mode a(b).
Then, we arrive at

    F_ab(θ)=F_a(θ)=F_b(θ) and F_ab(ϕ)=F_a(ϕ)=F_b(ϕ).

This feature can save the cost with less experimental efforts, because it allows us to explore the particle characteristics by the local FI taking into account the favorable parameter setting. Thus, hereafter we focus on the local FI.









§ REPRODUCING HONG-OU-MANDEL EFFECT WITH FISHER INFORMATION



Now we consider a scenario that a parameter is encoded in one input mode. A small difference between the two particles is quantifiable by the Hong-Ou-Mandel effect, which originally exhibits bunching effect of two bosonic particles <cit.>. Initially, two identical single photons enter coherently into a 50:50 beam splitter, one on each side. 
The beam-splitting transformation is represented by 

    â^†→â^†+b̂^†/√(2) and b̂^†→b̂^†-â^†/√(2),

where â and â^† (b̂ and b̂^†) are bosonic annihilation and creation operators in mode a (mode b), respectively. Using this beam-splitting transformation, we can have a two-mode output state 

    B̂_ab|1, 1⟩_ab=|0,2⟩_ab-|2,0⟩_ab/√(2).

Given a time delay between the two input single photons, we have a half probability of counting each photon in both output modes. With no time delay, the each photon counting event drops into zero by bunching effect. Using the probability of each photon counting event, we can observe HOM-like dip as a function of the time delay between the two input single photons. The indistinguishability of the two single photons determines the depth of the dip, which is bounded between zero and the maximum number of counts.

  

We illustrate that the original HOM effect can be observed with bosons and fermions by means of the FI. Firstly, let us consider the bosonic case. We replace the time delay by the polarization degree-of-freedom which exhibits the indistinguishability of the photons. We then introduce a control parameter, say Δ, in order to control the HOM-like dip. Note that such an adoption of the polarization degree-of-freedom with Δ is physical and does not restrict the generality of our analysis. In this setting, two polarized photon states |φ_1⟩_a=√(e^-gΔ^2)|H⟩_a+√(1-e^-gΔ^2)|V⟩_a and |φ_2⟩_b=|H⟩_b are fed into a 50:50 beam splitter as

    B̂_ab(√(e^-gΔ^2)|H⟩_a+√(1-e^-gΔ^2)|V⟩_a)|H⟩_b,

where a factor g is related to the length of the photon wave packet in the original experiment. By counting the number of photons in the output mode b, the photon-number probability distributions are characterized as

    P_b(0|Δ)    =    P_b(2|Δ)=1/4(1+e^-gΔ^2), 
    
    P_b(1|Δ)    =   1/2(1-e^-gΔ^2).

Consequently, from Eq. (<ref>), the FI of Δ can be evaluated as

    F(Δ)=4g^2Δ^2/e^2gΔ^2-1,

where 0 ≤ F(Δ) ≤ 2g. 
The minimum and the maximum values are obtained with different and same polarized states, respectively. Specifically, the value of F(Δ) exactly represents `indistinguishability' of the two input photons—here, the indistinguishability is defined as the fidelity between the two input particle states ⟨φ_1|φ_2⟩^2. This feature is equivalent to that of the original HOM effect. In Fig. <ref>(a), we depict the graph of F(Δ) with respect to the indistinguishability of the two input photons for g=3. Additionally, we also investigate the relation between F(Δ) and an input-output fidelity ⟨φ_1,φ_2|B̂_ab|φ_1,φ_2⟩^2. It is found that F(Δ) is inversely proportional to the input-output fidelity (blue solid curve in Fig. <ref>(b)). Obviously, this is attributed to the `bunching' effect of the bosonic system. Based on the above results, we can produce HOM-like dip, by taking minus sign on F(Δ) (see Fig. <ref>(c)). Here, the factor g determines the depth of the dip.


The same behaviors of the HOM effect can occur even in fermionic systems. In this case, two electron-spin states |φ_1⟩_a=√(e^-gΔ^2)|↓⟩_a+√(1-e^-gΔ^2)|↑⟩_a and |φ_2⟩_b=|↓⟩_b are fed into a 50:50 electronic beam-splitter as 

    B̂^e_ab(√(e^-gΔ^2)|↓⟩_a+√(1-e^-gΔ^2)|↑⟩_a)|↓⟩_b,
 
where the electronic beam splitting transformation is defined by <cit.>

    f̂^†_a →f̂^†_a+f̂^†_b/√(2)  and  f̂^†_b →f̂^†_b-f̂^†_a/√(2).

f̂_a and f̂_a^† (f̂_b and f̂_b^†) are fermionic annihilation and creation operators in mode a (mode b), respectively. Here, the control parameter Δ and the factor g are for the single-electron state, and they play the same role as the parameters of the bosonic case. By counting the number of electrons in the output mode b, we can characterize the probability distributions as

    P_b(0|Δ)   =   P_b(2|Δ)=1/4(1-e^-gΔ^2), 
    
    P_b(1|Δ)   =   1/2(1+e^-gΔ^2),

and then the corresponding FI of Δ exhibits the same formula as the Eq. (<ref>) of the bosonic case. Thus, F(Δ) evaluated by the fermions increases monotonically with respect to the indistinguishability of two electron states,  as shown in Fig. <ref>(a). However, it is proportional to the input-output fidelity (red dashed curve in Fig. <ref>(b)), contrary to the case of bosons. Actually, this is attributed to the `anti-bunching' effect, which is a property of fermionic system, namely from the Pauli exclusion principle. Thus, the HOM-like dip also appears near the completely same electron-spin states (see Fig. <ref>(d)), even though the underlying physics is complete different from that of bosons. 






§ MEASURE OF INDISTINGUISHABILITY AND PARTICLE SPECIES



We consider a parameter of a two-wave mixing operation: specifically, a phase-shifting parameter in a MZI-type operation, which is defined as

    Û_MZI=B̂_abe^iϕâ^†âB̂_ab,

where e^iϕâ^†â is a single-mode operation for shifting a phase ϕ∈ (0, π][Here we do not consider the case of ϕ=0.]. This MZI operation is defined consistently for both bosons and fermions <cit.>.

 

Firstly, let us consider the bosonic case, where we use two photons—one is assumed to be in the state of |φ_1⟩_a=√(1-β^2)|V⟩_a + β|H⟩_a and the other is in |φ_2⟩_b=|H⟩_b. Then, we can obtain the probability distributions P(n_H_a',m_V_a'; k_H_b', l_V_b') by counting the number of photons and identifying their polarization, where n_H_a' and m_V_a' (or k_H_b' and l_V_b') are the number of H-polarized and V-polarized photons in the output a'-mode (or b'-mode), respectively.  
However, owing to Eq. (<ref>), it is enough to characterize the following conditional probabilities:

    P_b'(1_H|ϕ)   =   1-β^2/161-e^iϕ^4+β^2/41+e^i 2ϕ^2, 
    
    P_b'(1_V|ϕ)   =   1-β^2/161+e^iϕ^4, 
    
    P_b'(2_H|ϕ)   =   β^2/81-e^i 2ϕ^2, 
    
    P_b'(1_H,1_V|ϕ)   =   1-β^2/161-e^i 2ϕ^2,

where P_b'(0|ϕ) = P_b'(2_H|ϕ)+P_b'(1_H,1_V|ϕ). With these probability distributions, we can directly obtain the FI of the phase ϕ as the function of the indistinguishability, i.e., ⟨φ_2|φ_1⟩^2=β^2, such that

    F(ϕ)    =   ( 1-β^2 ) + ( 1+3β^2 ) cos^2ϕ
           +[ 2(1-β^2)sinϕ - (1+3β^2 )sin2ϕ]^2/4( 1-β^2 )( 1-cosϕ)^2 + 8β^2( 1+cos2ϕ).

For a given β^2, let us define the maximum value of F(ϕ) over ϕ as F^max≡max_ϕ∈ (0, π]F(ϕ). Then, it is found that

    F^max = 2 ( 1 + β^2 ) for ϕ = π,

which indicates that F^max monotonically increases from 2 up to 4 with respect to the indistinguishability of the two bosonic particles (see Fig. <ref>(a)). This feature is due to the bunching property of the bosons.

Secondly, in the fermionic case, we feed two electrons into the electronic MZI operation <cit.>. Here, the two electrons are assumed to be in the states of |φ_1⟩_a=√(1-β^2)|↑⟩_a + β|↓⟩_a and |φ_2⟩_b = |↓⟩_b, respectively.  The probability distributions P(n_↑_a',m_↓_a'; k_↑_b', l_↓_b') are also defined by counting the number of electrons and identifying the spin. Here, in order to evaluate FI, we can also characterize the useful conditional probabilities:

    P_b'(1_↑|ϕ)   =   1-β^2/161+e^i ϕ^4, 
    
    P_b'(1_↓|ϕ)   =   β^2+1-β^2/161-e^iϕ^4, 
    
    P_b'(1_↑,1_↓|ϕ)   =   1-β^2/161-e^i2ϕ^2=P_b'(0|ϕ).

Using these probability distributions, we can obtain the FI of the phase ϕ as

    F(ϕ)=( 1- β^2 )(1+cos^2ϕ)+( 1- β^2 )(1-cosϕ)^2sin^2ϕ/(1-cosϕ)^2+ 4β^2/1- β^2.

With the above-obtained F(ϕ), we can define F^max = max_ϕ∈ (0, π]F(ϕ), and it is found that

    F^max = 2 ( 1 - β^2 ) for ϕ = π.

On the contrary to the bosonic case, it is observed that the value of F^max monotonically decreases from 2 to 0 with respect to the indistinguishability of the two fermionic particles (see Fig. <ref>(b)). This is caused by the anti-bunching property of the fermions. 

 

On the basis of the above observations, we define a quantity[We note that such a definition by using the maximum FI occurs, frequently, in some physical contexts (see Refs. <cit.>).]

    Γ≡1/2max_ϕF(ϕ) - 1.

which is bounded as -1 ≤Γ≤ 1. This quantity Γ is indeed useful. First, we can estimate the two-particle indistinguishability; specifically, for both bosons and fermions, we draw a new definition with the absolute value of Γ:

    Degree of indistinguishability≡abs(Γ).

Here, the particles are completely indistinguishable (or distinguishable) when abs(Γ)=1 (or abs(Γ)=0). Furthermore, in our approach, the question of whether the (unknown) input particles are bosons or fermions can also be answered by evaluating sign(Γ) ∈{+, -}; namely, we can identify the particle species, such that

    {[           Boson  when 0 < Γ≤ 1,;         Fermion when -1 ≤Γ < 0. ].

This feature originates from the different properties (bunching and anti-bunching) of the particle species. Currently, developing different methods of assessing the indistinguishability is a growing interest, as potentially it could be useful as a resource of quantum tasks <cit.>. 


Before closing, we briefly note that one may raise a question of how to distinguish a pure input state √(1-β^2)|V⟩_a+β|H⟩_a from a mixed state (1-β^2)|V⟩_a⟨V|+β^2|H⟩_a⟨H|. Actually, such a task is possible by adjusting the state of the other input mode b; specifically, by changing the parameter θ in the state of cosθ|H⟩_b + sinθ|V⟩_b, we can identify the (degree of) interference term in the state, and hence, can evaluate which of these, i.e., purity or mixedness, is relevant. We defer the details to Appendix A.









§ SUMMARY AND DISCUSSION



We have presented a study on boson-and-fermion characteristics by using FI. Our FI approach was developed under a two-wave mixing (TWM) operation (as depicted in Fig. <ref>). Using the symmetric behavior of TWM operation, we investigated the particle characteristics by measuring only one of the output modes and taking into account the parameter(s) that we are interested in. We firstly showed that the FI can reproduce the Hong-Ou-Mandel-like effect, consistently for both bosons and fermions. Under the degree of indistinguishability of the input particles, however, bosons and fermions are related to different physical phenomena, namely of the bunching and anti-bunching, respectively. We then proposed a simple method to estimate the degree of two-particle indistinguishability and to identify their particle species in the MZI-type setup. It is expected that our study will provide different viewpoints in terms of the FI and be extended to many-particle scenarios. For many-particle systems, we may take a symmetric transformation that can present a relation between the local and global FIs. It will be a good candidate to consider multi-port beam splitters <cit.>, including the idea of an integrated tritter <cit.>.






S.-Y. L. and J.B. would like to thank Changhyoup Lee, Jaeyoon Cho, Tomasz Paterek, Junghee Ryu, and Marcin Wieśniak for useful discussions and comments. We acknowledge the financial support of the Basic Science Research Program through the National Research Foundation of Korea (NRF) (No. 2018R1D1A1B07048633).



Appendix A: Distinguishing an input pure state from an input mixed state by Fisher information 


 


For simplicity, here we count the number of particles without discriminating their polarizations or spins. For bosons, we start with the single-photon of mode b in the superposed state of cosθ|V⟩_b+sinθ|H⟩_b. When the single-photon is in a pure state α|V⟩_a+β|H⟩_a, we obtain the probability distributions by counting the number of photons only in the output mode b',  such that:

    P_b'(0|ϕ)   =   P_b'(2|ϕ)=1-cos2θ/4(c_0+c_3/2), 
    
    P_b'(1|ϕ)   =   1/2( 1 + c_1 cos2ϕ+ c_2cos^2ϕ-c_3).

With the above probability distributions, we can have the FI of the phase parameter ϕ as

    F_pure(ϕ)=2( c_0 + c_3/2)^2 sin^22ϕ[ 1/(c_0+c_3/2)(1-cos2ϕ)+1/2-2(c_0+c_3/2)sin^2ϕ],

where c_0=1/2(1+c_1), c_1=α^2cos^2θ+β^2sin^2θ, c_2=α^2sin^2θ+β^2cos^2θ, and 
c_3=(αβ^∗+βα^∗)sinθcosθ.  Here, note that the factor c_3 is related to the interference, and hence the purity, of the input state of the mode a. On the other hand, for a mixed state α^2|V⟩_a⟨V|+β^2|H⟩_a⟨H|, the FI is given by

    F_mix(ϕ)=2c_0^2 sin^22ϕ[1/c_0 (1-cos2ϕ) +1/2-2c_0 sin^2ϕ].

 Consequently, we can figure out the difference between Eq. (<ref>) and Eq. (<ref>) in terms of the existence of the interference with c_3 (as long as ϕ≠π/2 and ϕ≠π) As an example, in Fig. <ref> we depict the 3D-graphs of (left) F_pure(ϕ) and (right) F_mix(ϕ) for ϕ=0.1.

 

For fermions, we start with the single-electron of mode b in the superposed state of cosθ|↑⟩_b+sinθ|↓⟩_b. When the single-electron of mode a is in a pure state of α|↑⟩_a+β|↓⟩_a, we obtain the probability distributions by counting the number of electrons only in the output mode b',  such that:

    P_b'(0|ϕ)   =   P_b'(2|ϕ)=1-cos2θ/8(c_2+c_3), 
    
    P_b'(1|ϕ)   =   c_0+c_2 cos^2ϕ/2 - c_3sin^2ϕ/2.
 
Then, from the above probabilities, the FI of ϕ is given as

    F_pure(ϕ)=(c_2+c_3)^2sin^22ϕ[1/(c_2+c_3)(1-cos2ϕ)+1/4-2(c_2+c_3)sin^2ϕ].

On the other hand, for a mixed state α^2|↑⟩_a⟨↑|+β^2|↓⟩_a⟨↓| of the single-electron, the FI is given as

    F_mix(ϕ)=c_2^2sin^22ϕ[1/c_2(1-cos2ϕ)+1/4-2c_2sin^2ϕ].

As such, we can also figure out the difference of FIs in Eq. (<ref>) and Eq. (<ref>) with c_3 (as long as ϕ≠π/2 and ϕ≠π) In Fig. <ref>, we also present the graphs of (left) F_pure(ϕ) and (right) F_mix(ϕ) for ϕ=0.1, which exhibits the same behavior as the case of boson.[Note that c_2 (and c_1) ∈ [0,1], whereas c_0 ∈[1/2, 1].]








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