Inefficient eight-port homodyne detection and covariant phase space observables

# Inefficient eight-port homodyne detection and covariant phase space observables

## Abstract.

We consider the quantum optical eight-port homodyne detection scheme in the case that each of the associated photon detectors is assigned with a different quantum efficiency. We give a mathematically rigorous and strictly quantum mechanical proof of the fact that the measured observable (positive operator measure) in the high-amplitude limit is a smearing of the covariant phase space observable related to the ideal measurement. The result is proved for an arbitary parameter field. Furthermore, we investigate some properties of the measured observable. In particular, we show that the state distinguishing power of the observable is not affected by detector inefficiencies.
PACS numbers: 03.65.-w, 03.67.-a, 42.50.-p

Keywords: eight-port homodyne detector, covariant phase space observable, detector inefficiency

## 1. Introduction

The eight-port homodyne detection scheme has been investigated extensively ever since it was introduced in the realm of quantum optics. The significance of this scheme comes from the fact that it provides a means to study many fundamental questions in quantum mechanics. Among these are the problems of quantum state reconstruction and approximate joint measurements of quadrature observables. The usefulness of this setup is due to the fact that it provides a quantum optical realization of the measurement of any covariant phase space observable [13]. With regard to the aforementioned problems, these observables are of great importance. On one hand, since the work of [1], a large class of covariant phase space observables are known to possess the property that the measurement outcome statistics determine the state uniquely. On the other hand, the quadrature observables are approximately jointly measurable exactly when there exists a covariant phase space observable which is their approximate joint observable [4]. Thus, it is natural to investigate the detailed structure of the observable measured with this specific scheme.

Since any realistic measurement involves detectors with non-unit quantum efficiencies, it is important to study also the effects of detector inefficiencies in detail. As reported in the recent review on single-photon detectors [7], the efficiencies of available detectors range from very high to as low as a few percents. It is therefore clear that in most cases the effect of inefficiencies is far from being negligible. In the eight-port homodyne detection scheme, it was shown in [16] that with the specific choice of a vacuum parameter field and an overall quantum efficiency for the detectors, the measured probability distribution is a smoothed version of the -function of the signal field. The smoothing is caused by a Gaussian convolution which is due to the precence of the non-unit quantum efficiencies. Up to our knowledge, this analysis has not yet been done in the case of an arbitrary parameter field, or with different quantum efficiencies for each of the detectors.

The purpose of this paper is to give a mathematically rigorous derivation of the high-amplitude limit observable measured with an inefficient eight-port homodyne detector. The derivation is done strictly within the framework of quantum mechanics without any classicality assumptions. The result is that whenever detector inefficiencies are present, the measured observable is a smearing of the ideal one. Furthermore, we study some basic properties of the measured observable. In particular, we find that the state distinguishing power does not depend on the associated quantum efficiencies. More specifically, we show that the measurement statistics of the ideal observable can always be reconstructed from the smeared statistics. The paper is organized as follows. We start by giving the basic framework for our study in section 2. In section 3 we derive the high-amplitude limit observable. First, we consider the high-amplitude limit in an inefficient balanced homodyne detector, and then use the results to obtain the measured observable in an inefficient eight-port homodyne detector in the high-amplitude limit. The basic properties of the high-amplitude limit observable are studied in section 4, and the conclusions are given in section 5.

## 2. Preliminaries

Let be a complex separable Hilbert space associated with a single mode electromagnetic field, and let be an orthonormal basis of . Let , and denote the creation, annihilation and number operators associated with this basis. Let and denote the sets of bounded and trace class operators on . The states of the system are represented by positive trace class operators with unit trace, density operators, and the pure states correspond to the one-dimensional projections , , . Among the pure states are the coherent states defined by

 |z⟩=e−|z|22∞∑n=0zn√n!|n⟩.

For , the corresponding coherent state has the position representation

 ψz(x)=(1π)1/4e−iqp2eipxe−12(x−q)2,

and the subspace is dense in .

The observables are represented by normalized positive operator measures. Among these are the standard quadrature observables , where stands for the Borel -algebra of subsets of . That is, and are the spectral measures of the quadrature operators and , where the bar stands for the closure of an operator. For each we define the rotated quadrature observable by

 Qθ(X)=eiθNQ(X)e−iθN,X∈B(R),

so that in particular and . For each positive trace class operator with unit trace , we define the phase space observable by

 (1) GS(Z)=12π∫ZWqpSW∗qpdqdp,Z∈B(R2),

where is the Weyl operator. The operator is called the generating operator of the observable. The mapping is an irreducible projective unitary representation of , and each is covariant with respect to in the sense that

 WqpGS(Z)W∗qp=GS(Z+(q,p))

for all and . Furthermore, each covariant phase space observable is of the form (1) for some generating operator [9, 19] (for recent alternative proofs, see [5, 11]).

For a quantum system in a state , the measurement statistics of an observable is given by the probability measure . It follows that for each phase space observable the associated probability measure has the density . For a pure state , we use the notation for the probability measure related to the observable , and the notation for the observable generated by . Any two observables are informationally equivalent if their ability to distinguish between states is equal. If the measurement statistics of an observable determine the state uniquely, the observable is said to be informationally complete.

## 3. Measurement scheme

### 3.1. Inefficient balanced homodyne detector

The balanced homodyne detector involves two modes, the signal field with the Hilbert space and an auxiliary field of the local oscillator with the Hilbert space . We denote by the state of the signal field and the auxiliary field is in the coherent state . These fields are coupled via a lossless beam-splitter which is described by a unitary operator satisfying

 (2) U|α⟩⊗|β⟩=|1√2(α−β)⟩⊗|1√2(α+β)⟩

for all . Here the first term in the tensor product refers to the signal field, and the second term to the auxiliary field. The scheme involves two photon detectors and with quantum efficiencies and , respectively. With these efficiencies, each of the detectors now measures the smeared photon number, given by the detection observable (see, for instance, [15, pp. 79-83] or [3, pp. 177-180])

 (3) n↦Eϵjn=∞∑m=n(mn)ϵnj(1−ϵj)m−n|m⟩⟨m|.

We are interested in the scaled photon number differences so that the set of possible measurement outcomes is taken to be

 Ω={1√2|z|(nϵ2−mϵ1)∣∣∣m,n∈N}.

This specific choice for the scaling is motivated by the fact that it assures that for a coherent signal state the first moment of the probability measure remains finite in the limit . The detection statistics is thus represented by the observable ,

 Eϵ1,ϵ2(X)=∑XEϵ1m⊗Eϵ2n

where the summation is now over those for which . The signal observable measured with this setup is now completely determined by the relation

 tr[ρEzϵ1,ϵ2(X)]=tr[Uρ⊗|z⟩⟨z|U∗Eϵ1,ϵ2(X)]

for all states and all , that is, the observable can be written as

 Ezϵ1,ϵ2(X)=V∗zU∗Eϵ1,ϵ2(X)UVz,X∈B(R)

where is the linear isometry .

To consider rigorously the high-amplitude limit in this measurement scheme, we need to be specific about what we mean by the limit of the associated observables. First of all, we recall that a sequence of probability measures converges weakly to a probability measure if for all bounded continuous functions . According to [2, Theorem 2.1], the weak convergence is equivalent to the condition for all such that , where denotes the boundary of . This is the motivation for the following definition, used also in [12].

###### Definition 1.

A sequence of observables converges to an observable weakly in the sense of probabilities if

 limk→∞Ek(X)=E(X)

in the weak operator topology for all such that .

Several equivalent conditions for this convergence are given in [12, Proposition 10]. In particular, this convergence happens if and only if there exists a dense subspace such that for all unit vectors , the corresponding sequence of probability measures converges weakly to . Note that since the weak limit of a sequence of probability measures is unique [2, Theorem 1.3], it follows that a sequence of observables can converge to at most one observable weakly in the sense of probabilities. Furthermore, according to the continuity theorem [2, Theorem 7.6], the weak convergence of probability measures is equivalent to the pointwise convergence of the corresponding characteristic functions. We will use these facts with the choice to prove our result.

We fix the phase of the local oscillator and take an arbitrary sequence of positive numbers such that . Let , so that we obtain a sequence of observables . Suppose that or , and define the probability density by

 (4) fϵ1,ϵ2(x)=√2ϵ1ϵ2π(ϵ1−2ϵ1ϵ2+ϵ2)e−2ϵ1ϵ2ϵ1−2ϵ1ϵ2+ϵ2x2.

Let be the probability measure determined by , that is, for all . We wish to extend the definition of to include also the case of ideal detectors, and thus we define as the Dirac measure concentrated at the origin. We will prove in the next proposition, that the smeared rotated quadrature observable defined as the weak integral

 (5) (μϵ1,ϵ2∗Qθ)(X)=∫μϵ1,ϵ2(X−x)dQθ(x),X∈B(R),

is the high-amplitude limit in this measurement scheme. Note that . We start with a lemma.

###### Lemma 1.

For all we have

 limx→∞[ax2(1−e−iax)+bx2(1−eibx)]=12(1a+1b).
###### Proof.

Using the change of variables and l’Hospital’s rule twice we have

 limx→∞[ax2(1−e−iax)+bx2(1−eibx)] = limy→0+a(1−cos(y/a))+b(1−cos(y/b))y2+ilimy→0+asin(y/a)−bsin(y/b)y2 = limy→0+12(1acos(y/a)+1bcos(y/b))+ilimy→0+12(1bsin(y/b)−1asin(y/a)) = 12(1a+1b)

###### Proposition 1.

For all the sequence converges to weakly in the sense of probabilities.

###### Proof.

The case has been proved in [12], so we may assume that or . We need to show that

 (6) limk→∞∫eitxd⟨α|Ezkϵ1,ϵ2(x)|β⟩=∫eitxd⟨α|(μϵ1,ϵ2∗Qθ)(x)|β⟩

for all and . For the equation is clearly true, so we assume now that .

First note that for all we have

 ⟨α|V∗zkU∗Eϵ1m⊗Eϵ2nUVzk|β⟩ = ⟨1√2(α−zk)|Eϵ1m|1√2(β−zk)⟩⟨1√2(α+zk)|Eϵ2n|1√2(β+zk)⟩ = 1m!n!(ϵ12(¯¯¯¯α−¯¯¯zk)(β−zk))m(ϵ22(¯¯¯¯α+¯¯¯zk)(β+zk))n ×e−12|α|2−12|β|2−|zk|2e12(1−ϵ1)(¯¯¯α−¯zk)(β−zk)+12(1−ϵ2)(¯¯¯α+¯zk)(β+zk)

so that

 ∫eitxd⟨α|Ezkϵ1,ϵ2(x)|β⟩=∞∑m,n=0eit√2|zk|(nϵ2−mϵ1)⟨α|V∗zkU∗Eϵ1m⊗Eϵ2nUVzk|β⟩ = ∞∑m,n=01m!n!(e−it√2ϵ1|zk|)m(eit√2ϵ2|zk|)n(ϵ12(¯¯¯¯α−¯¯¯zk)(β−zk))m(ϵ22(¯¯¯¯α+¯¯¯zk)(β+zk))n ×e−12|α|2−12|β|2−|zk|2e12(1−ϵ1)(¯¯¯α−¯zk)(β−zk)+12(1−ϵ2)(¯¯¯α+¯zk)(β+zk) = e−12|α|2−12|β|2+¯¯¯αβe−ϵ12(1−exp(−it√2ϵ1|zk|))(¯¯¯α−¯zk)(β−zk)e−ϵ22(1−exp(it√2ϵ2|zk|))(¯¯¯α+¯zk)(β+zk) = e−12|α|2−12|β|2+¯¯¯αβe−ϵ12(1−cos(t√2ϵ1rk)+isin(t√2ϵ1rk))(¯¯¯αβ−rk(¯¯¯αeiθ+βe−iθ)) ×e−ϵ22((1−cos(t√2ϵ2rk)−isin(t√2ϵ2rk))(¯¯¯αβ+rk(¯¯¯αeiθ+βe−iθ))e−ϵ12r2k(1−exp(−it√2ϵ1rk))−ϵ22r2k(1−exp(it√2ϵ2rk))

Now we may use lemma 1 and standard limit results for trigonometric functions to calculate

 limk→∞∫eitxd⟨α|Ezkϵ1,ϵ2(x)|β⟩=e−12|α|2−12|β|2+¯¯¯αβeit√2(¯¯¯αeiθ+βe−iθ)e−t28(1ϵ1+1ϵ2).

We still need to show that this is the right-hand side of equation (6).

Since for all we have

 ⟨α|Qθ(X)|β⟩=⟨α|eiθNQ(X)e−iθN|β⟩=⟨e−iθα|Q(X)|e−iθβ⟩,

we may express the density of the measure as

 x↦∫fϵ1,ϵ2(x−y)d⟨α′|Q(y)|β′⟩,

where and . Putting and we find that in the position representation

 ∫eitxd⟨α|(μϵ1,ϵ2∗Qθ)(x)|β⟩=∫eitx(∫fϵ1,ϵ2(x−y)d⟨α′|Q(y)|β′⟩)dx = 1π√2ϵ1ϵ2ϵ1−2ϵ1ϵ2+ϵ2∫eitx(∫e−2ϵ1ϵ2ϵ1−2ϵ1ϵ2+ϵ2(x−y)2ei2(qp−uv)eiy(v−p)e−12(y−q)2−12(y−u)2dy)dx = e−12|α|2−12|β|2+¯¯¯αβeit√2(¯¯¯αeiθ+βe−iθ)e−t28(1ϵ1+1ϵ2).

It follows that

 limk→∞∫eitxd⟨φ|Ezkϵ1,ϵ2(x)φ⟩=∫eitxd⟨φ|(μϵ1,ϵ2∗Qθ)(x)φ⟩

for all unit vectors , so the claim follows from [12, Proposition 10] and the continuity theorem [2, Theorem 7.6].

### 3.2. Inefficient eight-port homodyne detector

The eight-port homodyne detector involves four input modes, four beam splitters, a phase shifter, and four photon detectors (see figure 2). If , , is the Hilbert space of the th input mode, then the Hilbert space of the entire four mode field is . We denote by the state of the signal field and by the state of the parameter field. If the coherent local oscillator is in the state , the initial state of the four-mode field is

 ρ⊗S⊗|0⟩⟨0|⊗|√2z⟩⟨√2z|.

In this case we use the notation for the unitary transform representing the beam splitter. Here the subscripts refer to the primary and secondary input modes, that is, the first and second components of the tensor product in equation (2). The dashed lines in figure 2 represent the primary input modes. The phase shifter with phase shift is modelled with the unitary operator .

We assign to each detector a quantum efficiency , so that each detector measures the observable defined in equation (3). The detection is represented by the biobservable

 (X,Y)↦Eϵ1,ϵ3(X)⊗Eϵ2,ϵ4(Y)=∑X,YEϵ1k⊗Eϵ2l⊗Eϵ3m⊗Eϵ4n,

where the summation is now taken over those for which and . The state of the entire four-mode field before detection is

so that the detection statistics are given by the probability bimeasures

 (X,Y)↦tr[σρ,S,z,ϕEϵ1,ϵ3(X)⊗Eϵ2,ϵ4(Y)].

Now there exists a unique signal observable such that

 tr[ρES,z,ϕ(X×Y)]=tr[σρ,S,z,ϕEϵ1,ϵ3(1√2X)⊗Eϵ2,ϵ4(1√2Y)],

where the scaling has been chosen for later convenience. In order to calculate the high-amplitude limit we wish to express in terms of the unsharp homodyne detection observables and . In fact, after simple calculations we find that

 tr[ρES,z,ϕ(X×Y)]=tr[U12(ρ⊗S)U∗12Ezϵ1,ϵ3(1√2X)⊗Ezeiϕϵ2,ϵ4(1√2Y)]

for all . Denote again , where is fixed and is an arbitrary sequence of positive numbers such that . It follows from proposition 1 and the boundedness of the associated operators that for all such that the boundaries and are of zero Lebesgue measure, we have the convergence

 limk→∞tr[ρES,zk,ϕ(X×Y)]=tr[U12(ρ⊗S)U∗12(μϵ1,ϵ3∗Qθ)(1√2X)⊗(μϵ2,ϵ4∗Qθ+ϕ)(1√2Y)].

Note that the condition of zero Lebesgue measure follows from the fact that each is unitarily equivalent to which is absolutely continuous with respect to the Lebesgue measure. In particular, we may choose and to obtain the limit

 limk→∞tr[ρES,rk,π2(X×Y)]=tr[U12(ρ⊗S)U∗12(μϵ1,ϵ3∗Q)(1√2X)⊗(μϵ2,ϵ4∗P)(1√2Y)].

Now we still need to find the explicit form of the high-amplitude limit observable.

Let be the unique probability measure satisfying

 (7) μϵ(X×Y)=μϵ1,ϵ3(1√2X)μϵ2,ϵ4(1√2Y)

for all . Here we have chosen a collective symbol to represent the involved quantum efficiencies . This probability measure has a density which we denote by if and only if both and have densities given by (4). In the rest of the paper we will indicate explicitly when we assume the existence of the density . Let denote the conjugation map and let be as before. The high-amplitude limit observable is now given by the following proposition, in which the smeared phase space observable is defined as a weak integral similar to (5).

###### Proposition 2.

The sequence converges to weakly in the sense of probabilities.

###### Proof.

We begin by showing that

 (8) tr[U12(ρ⊗S)U∗12(μϵ1,ϵ3∗Q)(1√2X)⊗(μϵ2,ϵ4∗P)(1√2Y)]=tr[ρ(μϵ∗GCSC−1)(X×Y)]

for all .

Let and be unit vectors. First note that where is the Fourier-Plancherel operator. Furthermore, the relation

 (I⊗F)U12(φ⊗ψ)(x,y)=1√π⟨W√2x,√2yCψ|φ⟩

holds for all and almost all (see, e.g., the proof of [13, Lemma 2]). Now a direct calculation shows us that

 tr[U12(P[φ]⊗P[ψ])U∗12(μϵ1,ϵ3∗Q)(1√2X)⊗(μϵ2,ϵ4∗P)(1√2Y)] = ⟨(I⊗F)U12(φ⊗ψ)|(μϵ1,ϵ3∗Q)(1√2X)⊗(μϵ2,ϵ4∗Q)(1√2Y)(I⊗F)U12(φ⊗ψ)⟩ = ∫μϵ1,ϵ3(1√2X−x)μϵ2,ϵ4(1√2Y−y)∣∣((I⊗F)U12φ⊗ψ)(x,y)∣∣2dxdy = 1π∫μϵ1,ϵ3(1√2X−x)μϵ2,ϵ4(1√2Y−y)∣∣⟨W√2x,√2yCψ|φ⟩∣∣2dxdy = 12π∫μϵ1,ϵ3(1√2(X−x′))μϵ2,ϵ4(1√2(Y−y′))∣∣⟨Wx′,y′Cψ|φ⟩∣∣2dx′dy′ = 12π∫μϵ(X×Y−(x′,y′))∣∣⟨Wx′,y′Cψ|φ⟩∣∣2dx′dy′ = ⟨φ|(μϵ∗GCψ)(X×Y)φ⟩

for all , so that equation (8) holds for and . Since both sides of equation (8) depend linearly and continuously on and , the validity of the equation in the general case follows by using the spectral representations for and .

Now let be such that and are of zero Lebesgue measure, so that according to the previous discussion we have the convergence

 limk→∞tr[ρES,rk,π2(X×Y)]=tr[ρ(μϵ∗GCSC−1)(X×Y)]

for any state . Since the family of sets of the form where the boundaries of and are of zero Lebesgue measure is closed under finite intersections and includes a neighbourhood base of any point , it follows from [2, Corollary 1, p. 14] that for any state , the sequence of probability measures converges weakly to the probability measure . This completes our proof. ∎

## 4. Some properties of the high-amplitude limit observable

In [13] it was shown that in the case of ideal photon detectors the high-amplitude limit observable is the covariant phase space observable . Proposition 2 now implies that the presence of inefficiencies causes a Gaussian smearing of the observable so that the actually measured observable is . In this section we consider some properties of this smeared observable.

The first important observation is given in the next proposition which shows that the covariance is not lost in the process of smearing. That is, the observable is of the form for some generating operator . In fact, the operator can always be expressed as a convolution of the operator and the probability measure , defined as the weak integral [19]

 (9) μϵ∗CSC−1=∫WqpCSC−1W∗qpdμϵ(q,p),

which is clearly a positive operator with unit trace.

###### Proposition 3.

The high-amplitude limit observable is a covariant phase space observable with the generating operator , that is, .

###### Proof.

The definition of implies that for all , and the covariance of