1. Introduction

Gaussian optical states are states whose Wigner functions are Gaussian [1–4]. Such states include strong and weak coherent states, which are used in conventional communication systems [5, 6] and quantum information science [7–10], respectively, and the photon-pair-like states produced by parametric sources [11, 12], which are also used in quantum information science [13, 14]. Such functions are specified completely by their mean vectors and covariance matrices, whose elements are quadrature variances and correlations.

Conventional systems can be modeled as sequences of transmission fibers (attenuators) and (linear) amplifiers. As optical signals propagate through such systems (links), noise is imparted to them by the attenuators and amplifiers. The standard measure of signal quality is the signal-to-noise ratio (SNR), which for direct (incoherent) detection is the square of the mean photon-number divided by the number variance and for homodyne (coherent) detection is the square of the mean quadrature divided by the quadrature variance. The standard measure of link performance is the noise figure, which is the input SNR divided by the output SNR, for either mode of detection. Knowledge of the noise figure allows one to estimate the bit-error-ratio associated with a particular signal format [5, 6] and the information capacity (maximal information-transfer rate) of the link [15, 16]. Noise-figure formulas have been calculated for a wide variety of transmission links [19–28]. For many links, the noise-noise contributions to the number variance can be neglected, in which case the direct and homodyne noise figures are equal [27], and are related to the input and output covariance matrices.

Many quantum information experiments involve single- or few-photon inputs. For such experiments, a better performance metric is the transmission fidelity [30], which is (roughly) the overlap of the input and output states. For Gaussian input states, the fidelity is also related to the input and output covariance matrices [31].

Although the noise figure and fidelity both characterize aspects of the output state, and are useful in certain contexts, neither metric specifies the state completely. It is difficult to determine the output state directly, because it is mixed and the number of basis states required to represent its density operator accurately increases rapidly with link length (distance). However, it is straightforward to determine the Wigner function of the output state (which is specified by the output covariance matrix), after which the associated density operator can be reconstructed by the evaluation of certain integrals.

This paper is organized as follows: In Sec. 2 the characteristic and Wigner functions used to describe optical states are introduced, and the basic properties of Gaussian states are mentioned briefly. The transmission fidelity is defined in terms of the density operators of the input and output states, and this definition is related to the associated Wigner functions. For Gaussian states, the fidelity formula involves only the input and output covariance matrices. In Sec. 3 the noise properties of transmission links are reviewed in detail for Gaussian input signals. Formulas are derived for the covariance matrices of the output signals, for links with lumped and distributed phase-insensitive (PI) and phase-sensitive (PS) amplification. A brief comparison is made of the noise figures of these links. In Sec. 4 the aforementioned covariance matrices are used to determine the transmission fidelities of one-mode links. The fidelities of two-mode links, which are related to the one-mode fidelities, are discussed briefly in Sec. 5. In Sec. 6 the covariance matrices are used to reconstruct the output density operators in the number-state representation. These density matrices are used to illustrate the decreases in coherence and changes in photon-number probabilities that are caused by transmission. Finally, in Sec. 7 the main results of this paper are summarized.

2. Representations and properties of Gaussian states

Suppose that W(p, q) is a classical probability density function (PDF) of the real variables p and q [29]. Then, if the PDF is normalized, the moments

In quantum mechanics, the characteristic function [2–4] is defined as the expectation value

Gaussian states are defined to be ones whose Wigner functions have the form

The Wigner functions of coherent [17] and squeezed [18] states are illustrated in Fig. 1. The Wigner function of the former state is symmetric, whereas the Wigner function of the latter is asymmetric. In Fig. 1(b), the p- and q-fluctuations are uncorrelated (C is diagonal), so the principal axes of the Wigner function are aligned with the quadrature axes. If the fluctuations were correlated (if C was full), the principal axes would be inclined at an angle with respect to the quadrature axes. The Wigner functions of some number states are illustrated in App. A.

 

Fig. 1 Wigner distribution function of (a) a coherent state with a p-quadrature mean of 2.5, which corresponds to a number mean of 3.0, and (b) a state whose p- and q-quadratures have been stretched and squeezed, respectively. Both distributions are Gaussian and positive.

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Gaussian distributions play an important role in the theory of optical transmission, because they are specified completely by their first two moments (mean vectors and covariance matrices), and because sequences of (linear) attenuators and amplifiers preserve their Gaussian natures [9]. Furthermore, it is relatively easy to calculate the effects of transmission on the aforementioned moments.

In quantum mechanics, the fidelity of two arbitrary states (1 and 2) is defined to be [30]

In most cases, the density operator of the output state is difficult to calculate directly. However, the trace formula [2] asserts that

For two distinct states, Eqs. (9) and (13) imply that

Formula (16) involves coefficient matrices and their determinants. In contrast, the Scutaru formula [31]

The preceding results apply to one-mode states. To extend them to d-mode states (which involve 2d real quadratures), one increases the dimensionality of the relevant vectors, matrices and integrals, and makes the replacement 2π → (2π)^d in Eqs. (4), (9), (10), (13) and (14). Factors of 2π remain absent from Eqs. (16) and (17).

If the density operator of a one-mode state is known, the characteristic and Wigner functions are determined by Eqs. (4) and (5). Conversely, if the Wigner function is known, the density operator can be reconstructed. It is often convenient to use the number-state representation, in which the density operator

3. Noise properties of transmission links

Optical transmission links are sequences of fibers (attenuators) and amplifiers, and their noise properties were studied in detail [19–28]. In this section the properties of one-signal-mode phase-sensitive (PS) links, one-signal-mode phase-insensitive (PI) links and two-signal-mode PS links will be reviewed briefly, for convenience. (Introductions to quantum optics and parametric amplification are provided in [4, 34].)

Let â be the signal-mode amplitude operator, which satisfies the boson commutation relation [â, â^†] = 1, where square brackets denote a commutator and † denotes a hermitian conjugate. Then the (beam-splitter-like) input–output (IO) relation for attenuation is â′ = τâ + ρŵ, where ′ denotes an output quantity, ŵ is the loss-mode operator, and the transmission coefficient τ and reflection coefficient ρ satisfy the auxiliary equation |τ|² +|ρ|² = 1. (Here ρ is a reflection coefficient, not a density operator.) In general, the transmission and reflection coefficients are complex. However, by defining the input and output reference phases judiciously, one can rewrite the IO relation in such a way that only the moduli of these coefficients appear. By defining the real and imaginary signal-mode quadrature operators p̂ = (â^† + â)/2^1/2 and q̂ = i(â^† − â)/2^1/2, which satisfy the commutation relation [p̂, q̂] = i, one obtains the quadrature IO relations

The IO relation for one-mode parametric amplification is â′ = μâ + νâ^†, where μ and ν are gain coefficients that satisfy the auxiliary equation |μ|² − |ν|² = 1. In general, the gain coefficients are complex, but by defining the reference phases judiciously, one obtains the quadrature IO relations

The IO relations for two-mode parametric amplification are ${\widehat{a}}_1^{\prime }=\mu {\widehat{a}}_1+\nu {\widehat{a}}_2^{†}$ and ${\widehat{a}}_2^{\prime }=\mu {\widehat{a}}_2+\nu {\widehat{a}}_1^{†}$, where the subscripts 1 and 2 denote the signal and idler modes, respectively. If the input idler is absent, the output signal and idler operators are proportional to the input signal operator (or its conjugate), and the constants of proportionality are independent of the input signal phase, so the amplification process is PI. Conversely, if the input idler is present, the output operators depend on both input operators, so the amplification process is PS. By defining the reference phases judiciously, one obtains the quadrature IO relations

3.1. One-mode phase-sensitive links

Consider the first stage of a one-mode PS link, in which loss is followed by PS gain. By combining Eqs. (23) and (24), one obtains the composite IO relations

A balanced stage (v′_p = v_p) requires that λ₊τ = 1, in which case λ₋ = τ and

We are particularly interested in links with distributed amplification [35, 36], which can be modeled as concatenations of many short stages. For such a stage, ${\lambda }_{\pm }^2\approx 1\pm \gamma \delta z$, ρ² ≈ αδz and τ² ≈ 1 − αδz, where α and γ are loss and gain coefficients, respectively, and δz is the stage length. By making these substitutions in Eqs. (28) and taking the limit as δz → 0, one obtains the differential equations

3.2. Two-mode phase-insensitive links

Consider the first stage of a two-mode PI link. In the first part of the stage, a signal mode is attenuated. In the second part, the signal interacts with an idler (and a strong pump) and is amplified. The input idler is a vacuum state, so the amplification process is PI. At the end of the stage, the idler is discarded. Although PI amplification is a two-mode process, a two-mode PI link transmits only one signal mode. By combining Eqs. (23) and (25), one obtains the composite IO relations

For a short stage, τ² ≈ 1 − αδz, ρ² ≈ αδz, μ² ≈ 1 + γδz and ν² ≈ γδz. By making these substitutions in Eq. (37) and taking the limit as δz → 0, one obtains the differential equation

Notice that one can write Eq. (42) in the form of Eq. (38) by setting the loss factor l = 1+αz, which is equivalent to setting μ² = 1 + αz, ν² = αz, τ² = 1/(1 + αz) and ρ² = αz/(1 + αz). This result is consistent with a theorem in quantum information [37], which states that every PI Gaussian channel (link) is equivalent to a lumped attenuator followed by a lumped PI amplifier, with suitably chosen loss and gain parameters. It should be emphasized that in this example, the loss parameter depends only linearly on the loss distance αz (in contrast to the physical loss factor, which depends exponentially on distance).

3.3. Two-mode phase-sensitive links

Now consider the first stage of a two-mode PS link. In the first part of the stage the signal and idler modes both are attenuated. In the second part, the signal and idler interact (with a strong pump) and are amplified. By combining Eqs. (23), (25) and (26), one obtains for the p-quadratures the composite IO relations

Define the variances $v_j=〈{\widehat{p}}_j^2〉-{〈{\widehat{p}}_j〉}^2$ and correlation c₁₂ = 〈p̂₁p̂₂〉 − 〈p̂₁〉 〈p̂₂〉. Then, by combining Eqs. (43) and (44), and using the fact that the loss-mode quadratures are uncorrelated, one obtains the variance and correlation (moment) equations

If the input signal and idler means are equal, a balanced stage requires that (μ + ν)τ = 1. If the input fluctuations have the same variance (v_j = v₀), but are uncorrelated (c₁₂ = 0), then the (common) output variance and correlation are

For a short stage, τ² ≈ 1 − αδz, ρ² ≈ αδz and (μ + ν)² ≈ 1 + γδz. Hence, (μ − ν)² ≈ 1 − γδz and 2µν ≈ γδz. [These relations also imply that ν ≈ γδz/2 and μ ≈ 1 + (γδz)²/8.] By making these substitutions in Eqs. (45)–(47), one obtains the differential equations

It is well known that a two-mode amplification process can be reformulated as two independent one-mode processes. Specifically, define the real sum- and difference-mode quadratures p̂_± = (p̂₁ ± p̂₂)/2^1/2, respectively, and the associated variances $v_{\pm }=〈{\widehat{p}}_{\pm }^2〉-{〈{\widehat{p}}_{\pm }〉}^2$. Then the superposition modes evolve independently, and their variances v_± = s₁₂ ± c₁₂. By adding and subtracting Eqs. (58) and (59), one obtains Eqs. (32) and (33), respectively: The real sum-mode quadrature is stretched, whereas the real difference-mode quadrature is squeezed.

One can deduce results for the q-quadratures from the preceding results by making the substitution ν → −ν, which is equivalent to changing the sign of c₁₂ in Eqs. (45)–(47) and their solutions. In particular, the imaginary sum-mode quadrature is squeezed, whereas the imaginary difference-mode quadrature is stretched.

3.4. Noise figures

The standard metric for the performance of a link is its noise figure. For homodyne detection, the SNR of a signal is the square of its mean quadrature divided by its quadrature variance, and the noise figure of the link is the input SNR divided by the output SNR. Higher noise figures correspond to lower output SNRs, so the noise figure is a figure of demerit.

The SNR of a coherent state (CS) does not depend on its phase (the relative magnitude of p and q). It is well known that a CS subject to attenuation remains a CS: Its mean quadratures decrease, whereas its quadrature variances remain constant (because τ² + ρ² = 1). Hence, the noise figure of a CS subject to attenuation only (direct transmission) is the loss factor L = exp(αz), where α is the power-loss rate and z is the distance (link length).

By combining the preceding definition with the results of the preceding sections, one finds that the noise figures of one-mode PS links with in-phase CS inputs (p₁ ≠ 0 and q₁ = 0) are L and 1 + logL, for lumped and distributed amplification, respectively [Eqs. (29) and (34)]. For one-mode PI links with arbitrary CS inputs (p₁ ≠ 0 and q₁ ≠ 0), the noise figures are 1 + 2(L − 1) and 1 + 2logL, for lumped and distributed amplification, respectively [Eqs. (38) and (42)].

The noise figures of the aforementioned links are plotted as functions of loss in Fig. 2. (None of them depend on the power of the input signal.) Figure 2(a) shows that the noise figure of direct transmission increases linearly with the loss factor L (exponentially with the loss distance αz). Figure 2(b) shows that the noise figures of links with lumped amplification also increase linearly with loss, because attenuation preceeds amplification. The noise figure of the PS link equals the noise figure of direct transmission, because the PS amplifier adds no excess noise, whereas the noise figure of the PI link exceeds that of direct transmission by 3 dB, because the PI amplifier adds as much noise as the attenuator. Figure 2(b) also illustrates the benefits of distributed amplification: The noise figure of the distributed PS link is 3-dB lower than that of the distributed PI link, for the reason explained above, and the noise figures of both distributed links are lower than those of the lumped links, because the former noise figures increase only logarithmically with loss (linearly with distance). For high-loss links, this performance improvement is significant.

 

Fig. 2 Noise figure plotted as a function of loss for links (a) without gain and (b) with gain. The dotted blue (green) curve represents a link with a single phase-sensitive (insensitive) amplifier, whereas the solid blue (green) curve represents a link with distributed phase-sensitive (phase-insensitive) amplification. Links with distributed amplification perform significantly better than links with lumped amplification.

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The noise figures of two-mode PS links are 6-dB lower than those of two-mode PI links. However, if one were to redefine the noise figures based on the total input power (signal and idler), they would be only 3-dB lower [23,24]. This result is consistent with the fact that a two-mode amplification process involving physical modes can be reformulated as two independent one-mode processes involving superposition modes (Sec. 3.3). Thus, the noise figures of two-mode PS links do not need to be illustrated separately.

4. One-mode fidelities and purities

In this section fidelity formulas will be derived for a single mode subject to attenuation and (PI and PS) amplification. However, to put these results in perspective, the fidelity formula for a single mode subject to attenuation only (direct transmission) is derived first. Consider an input coherent state (CS). It is well known that attenuation diminishes the mean quadratures, but maintains the quadrature variances at their vacuum level of 1/2 [Eqs. (32) and (33) with γ = 0.] Hence, in Eq. (17) the covariance matrices C₁ = C₂ = diag(1/2, 1/2) and C₁₂ = diag(1, 1), so the determinant D₁₂ = 1. The exponent is $-\left(p_{12}^2+q_{12}^2\right)/2$, where p₁ is an input quadrature mean, p₂ = τp₁ is an output quadrature mean and p₁₂ = p₁ − p₂ is the quadrature difference. (Similar definitions apply to q₁, q₂ and q₁₂.) Thus, for the attenuation of a coherent state, the fidelity

Now consider an input single-photon state. In the beam-splitter model of attenuation, the input state is |1, 0〉 and the output state is τ|1, 0〉−ρ|0, 1〉, where the first and second elements of the ket-vectors represent the signal and loss mode, respectively, and the attenuation coefficients were assumed to be real, for simplicity. Hence, for the attenuation of a single-photon state, the fidelity

The fidelity of attenuation is plotted as a function of the loss factor in Fig. 3. The 1-photon CS curve decreases slowly towards the asymptote 0.37. This relatively-high fidelity reflects the fact that there is little difference between CS with the same quadrature fluctuations, and number means of 1 and 0. In contrast, the many-photon CS curves decrease rapidly to low asymptotic fidelitites and the 1-photon number-state curve also decreases rapidly towards the asymptote 0. These results show the limitations of direct transmission.

 

Fig. 3 Fidelity of attenuation plotted as a function of the loss factor. The dotted, dashed, dot-dashed and solid orange curves represent input coherent states with photon-number means of 1, 3, 10 and 30, respectively, whereas the dotted red curve represents an input number state with 1 photon. For most input states, the fidelity decreases rapidly as the loss increases.

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The object of amplification is to compensate for attenuation, so the following discussions apply to balanced links, for which the output means of the quadratures that transport information (in principle) are equal the input means. First, consider a one-mode link with distributed PS amplification. For such a link the in-phase quadrature is preserved, so we choose p₁ ≠ 0 and find that p₂ = p₁. However, the out-of-phase quadrature is squeezed, so we choose q₁ = 0 and find that q₂ = 0. In both cases (but for different reasons) the output mean equals the input mean, so the exponent in Eq. (17) is zero. For a CS both input variances are 1/2, so C₁ = diag(1/2, 1/2). The output quadrature variances are specified by Eqs. (34) and (35), from which it follows that C₂ = diag[(1 + logL)/2, (1 + 1/L²)/4]. By combining these results, one obtains the fidelity

Second, consider a one-mode link with distributed PI amplification. For such a link both quadratures are preserved, so we choose p₁ ≠ 0 and q₁ ≠ 0, and find that p₂ = p₁ and q₂ = q₁. In both cases the output mean equals the input mean, so the exponent in Eq. (17) is zero. For a CS both input variances are 1/2, so C₁ = diag(1/2, 1/2). The output quadrature variances are specified by Eq. (42), from which it follows that C₂ = diag(1/2 + logL, 1/2 + logL). By combining these results, one obtains the fidelity

The fidelity of transmission is plotted as a function of link loss in Fig. 4, for a variety of links with CS inputs. Also plotted are the attenuation fidelities of 1-photon-mean CS and 1-photon number-state inputs. Figure 4(a) shows that the transmission fidelity of a link with distributed PI amplification (which is the best PI link) is not significantly higher than the fidelity of a single photon subject to direct transmission. This result shows that fidelity is a more sensitive measure of link performance than noise figure (Sec. 3.4). In contrast, the transmission fidelity of a link with distributed PS amplification is higher than both of the direct-transmission fidelities for a wide range of loss factors. Consequently, we will not discuss PI links (with CS inputs) further, except to state that both types of link maintain the photon flux, which might be more important than the fidelity for some applications. Figure 4(b) shows the transmission fidelities of links with 1, 2 and 4 discrete PS amplifiers, which are placed at the ends of stages whose lengths equal the link length, and one-half and one-quarter of the link length, respectively. These results show that one can obtain good link performance with only a modest number of PS amplifiers

 

Fig. 4 Fidelity of transmission plotted as a function of link loss. The solid blue and green curves represent links with distributed phase-sensitive and phase-insensitive amplification respectively, whereas the dotted, dashed and dot-dashed blue curves represent links with 1, 2 and 4 phase-sensitive amplifiers, respectively. The inputs to these links are coherent states. The dotted orange and red curves are the attenuation fidelities of 1-photon-mean coherent-state and 1-photon number-state inputs, respectively. Phase-sensitive links with coherent-state inputs perform well.

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Since PS amplifiers stretch and squeeze the signal quadratures, it is natural to consider squeezed-state (SS) inputs. For such states C₁ = diag(λ/2, 1/2λ), where λ is the (variance) squeezing factor. As explained before Eq. (64), for balanced links the fidelity does not depend on the mean (in-phase) quadrature. For a link with distributed PI amplification, C₂ = diag(λ/2 + αz, 1/2λ + αz), from which it follows that

For a link with distributed PS amplification, C₂ = diag[(λ + αz)/2, e^−2αz/2λ + (1 − e^−2αz)/4]. For short distances (αz ≪ 1), D₁₂ ≈ 1 + (λ + 1/λ − 2)αz/2, which depends symmetrically on λ and its inverse. As αz increases, the second entry of C₂ tends rapidly to 1/4, at which point

The fidelity of transmission is plotted as a function of loss in Fig. 5, for a link with distributed phase-sensitive amplification and a variety of SS inputs. The transmission fidelity of the same link with a CS input is also plotted for comparison. The results show that squeezing the in-phase quadrature always degrades the transmission fidelity. Stretching the in-phase quadrature decreases the fidelity for low losses, but increases the fidelity slightly for high losses. However, the loss value at which stretching improves the performance of the link is an increasing function of the stretching factor and high-loss links have low fidelities that decrease their usefulness.

 

Fig. 5 Fidelity of transmission plotted as a function of loss for a link with distributed phase-sensitive amplification. The dotted, dashed, dot-dashed and solid purple curves represent inputs with stretching parameters of 0.1, 0.3, 3, and 10, respectively, whereas the solid blue curve represents an unstretched (coherent-state) input. Stretched inputs perform slightly better than the unstretched input for high losses The solid black curve represents the maximal fidelity (displaced upward by 0.01 for clarity).

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It is also instructive to consider the purities of the output states. As mentioned in Sec. 2, the purity is tr(ρ̂²). For a single-photon state subject to attenuation, the output density matrix is τ²|1〉 〈1| + ρ²|0〉 〈0|, so the output purity is τ⁴ + ρ⁴ = 1 − 2τ²ρ². For a Gaussian state, the output purity is $1/2D_2^{1/2}$, where D₂ is the determinant of the output covariance matrix. For all the examples considered in this section, the output covariance matrices were stated previously. It is easy to calculate their determinants (because they are diagonal), so there is no need to state the purity formulas explicitly.

The output purity is plotted as a function of loss in Fig. 6, for a variety of links without and with gain. For a single-photon state subject to attenuation only, the purity decreases from 1 to 1/2, then increases to 1: The state evolves from a pure state with 1 photon to a pure state with 0 photons. For a 1-photon-mean CS, the purity is constant. This result (which is independent of the number mean) reflects the fact that an attenuated CS is still coherent and pure. For a link with distributed PI amplification, the purity decreases rapidly: Not only is the output state mixed, rather than pure, it also bears little resemblance to the input state [Fig. 4(a)]. Thus, PI links do not transmit quantum signals faithfully. For links with PS amplification, the purity decreases slowly as the loss increases. For a 1-stage link, the purity is constant. This result (which is also independent of the number mean) reflects the fact that a stretched coherent state is pure. However, the output state bears little resemblance to the input state [Fig. 4(b)]. As the number of stages increases, the purity decreases. However, this decrease in purity corresponds to an increase in fidelity [Fig. 4(b)]. Thus, one can transmit a pure state that is nothing like the input state, or a mixed state that is a reasonable facsimile of the input state.

 

Fig. 6 Purity of the output state plotted as a function of loss for links (a) without gain and (b) with gain. The dotted orange and red curves are the output purities of 1-photon-mean coherent-state and 1-photon number-state inputs, respectively. An attenuated coherent state is pure. The solid blue and green curves represent links with distributed phase-sensitive and phase-insensitive amplification respectively, whereas the dotted, dashed and dot-dashed blue curves represent links with 1, 2 and 4 phase-sensitive amplifiers, respectively. The inputs to these links are coherent states. Phase-sensitive links degrade the purity less than phase-insensitive links, and a stretched coherent state is pure.

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The output purity is plotted as a function of loss in Fig. 7, for a link with distributed PS amplification and a variety of SS inputs. The output purity of a CS input is also plotted for comparison. For this example also, the output covariance matrix was stated previously. The associated determinant is the product of two factors, the variances of the in-phase and out-of-phase quadratures. The former variance increases monotonically as the loss increases, whereas the latter variance increases monotonically for λ > 2 and decreases monotonically for λ < 2. In both cases, the latter variance tends rapidly to its asymptotic value of 1/4. Consequently, if the in-phase quadrature is stretched, the purity tends to its asymptotic values [2/(λ + αz)]^1/2 monotonically, whereas if the in-phase quadrature is squeezed, it tends to its asymptotic values nonmonotonically. For high losses, the purity depends only weakly on the stretching parameter.

 

Fig. 7 Purity of the output state plotted as a function of loss for a link with distributed phase-sensitive amplification. The dotted, dashed, dot-dashed and solid purple curves represent inputs with stretching parameters of 0.1, 0.3, 3, and 10, respectively, whereas the solid blue curve represents an unstretched (coherent-state) input. For high losses, the output purity depends only weakly on the stretching parameter.

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5. Two-mode fidelities and purities

First, consider the fidelity of two-mode attenuation. If the input state is the product state |ψ_a〉 ⊗ |ψ_b〉, then the input density matrix ρ̂ = ρ̂_a ⊗ ρ̂_b, where ⊗ denotes a direct product. Such a density matrix obeys the trace rule tr(ρ̂) = tr(ρ̂_a)tr(ρ̂_b) and two such density matrices obey the product rule ρ̂₁ρ̂₂ = ρ̂_1aρ̂_2a ⊗ ρ̂_1bρ̂_2b. Each attenuation process is independent of the other. By combining these facts with Eq. (12), it is easy to show that the two-mode fidelity of attenuation is just the product of the one-mode fidelities. [The one-mode fidelities of attenuation for input coherent and single-photon states were specified by Eqs. (62) and (63), respectively.] Likewise, the two-mode output purity is the product of the one-mode output purities.

The fidelity of two-mode attenuation is plotted as a function of loss in Fig. 8(a), for a variety of input states. Although the two-mode fidelities are smaller than their one-mode counterparts (Fig. 3), their relative size and dependence on loss are similar.

 

Fig. 8 (a) Fidelity of two-mode attenuation plotted as a function of loss. The dotted, dashed, dot-dashed and solid orange curves represent input coherent states with photon-number means of 1, 3, 10 and 30, respectively, whereas the dotted red curve represents an input number state with 1 photon. (b) Fidelity of transmission plotted as a function of loss. The solid blue curve represents a link with distributed phase-sensitive amplification, whereas the dotted, dashed and dot-dashed blue curves represent links with 1, 2 and 4 phase-sensitive amplifiers, respectively. Two-mode phase-sensitive links with coherent-state inputs perform well.

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For two-mode PS links with CS inputs, the input and output covariance matrices

The fidelity of two-mode transmission is plotted as a function of loss in Fig. 8(b), for a variety of links with CS inputs. Although the two-mode fidelities are smaller than their one-mode counterparts [Fig. 4(b)], their relative size, and their dependence on loss and the number of amplifiers are similar. In every case considered, the two-mode fidelity (69) equals the square of the one-mode fidelity (64). This result reflects the fact that in the superposition-mode basis, two-mode transmission also is a separable process (Sec. 3.3). For this reason, there is no need to repeat the study of the fidelities associated with SS inputs, or the purities associated with arbitrary inputs.

Throughout this paper, the fidelity [Eqs. (12) and (17)] was used to quantify the similarities of the input and output states (of the same signal). In this context, high fidelities correspond to similar states. However, one could also use the fidelity to quantify the differences between two states (of different signals). In this context, low fidelities correspond to distinguishable states. For example, at the input, a vacuum state and a few-photon CS are easily distinguishable. During transmission through a balanced link, their mean quadratures stay the same, but their fluctuations increase (perhaps to the point of overlap). Hence, the output states are less distinguishable than the input ones. To quantify distinguishability, one must use the full fidelity formula of [31], because both output states are mixed.

6. Output-state reconstruction

In Secs. 3–5, two simple metrics (noise figure and fidelity) were used to characterize the performances of optical transmission systems. The first metric quantifies how much noise (in the form of quadrature fluctuations) a system adds to the signal, whereas the second quantifies the overlap between the input and output signal states. Although these metrics provide some information about the output signal (as does the purity), they do not describe its state completely. Remarkably, a complete description is available. All of the (linear) systems considered in Sec. 3 preserve the Gaussian nature of signal states. The Wigner functions of such states are specified completely by their covariance matrices, which were calculated in Sec. 3 and can be measured [44]. One can use these Wigner functions to calculate all of the quadrature (number) moments, which collectively specify the signal state, or reconstruct the associated density matrices [44]. In the number-state representation (20), the matrix elements ρ_mn are specified by integrals (21). In this section we take the latter approach, which we validated by comparing the integral results to the well-known formulas for coherent and squeezed states.

Array plots of the density matrices associated with different states are shown in Fig. 9. The input state [Fig. 9(a)] is coherent (pure), with a p-quadrature mean of 2.5 and a number mean of 3.0. Consequently, its density matrix has many large non-diagonal elements, which correspond to numbers (m or n) near the mean value. The output state produced by an attenuator with a loss factor of 3 dB [Fig. 9(b)] is also coherent. Its density matrix still has many large non-diagonal elements, which correspond to lower numbers. The output state produced by a balanced PI link with 3 dB of loss and lumped gain [Fig. 9(c)] has a density matrix with fewer large non-diagonal elements, which shows that the output state is less coherent (more mixed) than the input state. Although the output state produced by a balanced PS link [Fig. 9(d)] differs from the input state, its density matrix has many more large non-diagonal elements than the previous matrix, which shows that a PS link degrades the coherence of an input state less than the corresponding PI link, because it adds less noise to the in-phase quadrature and subtracts noise from the other (out-of-phase) quadrature.

 

Fig. 9 Density matrices in the number-state representation. Darker squares denote higher values of |ρ_mn|, whereas lighter squares denote lower values. (a) input coherent state with a number mean of 3.0, (b) output coherent state produced by a 3-dB attenuator, (c) output state produced by a balanced phase-insensitive link with 3 dB of loss and lumped gain, and (d) output state produced by a balanced phase-sensitive link. The former link degrades the coherence of the state more than the latter.

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For the preceding cases, the theoretical purities (Sec. 4) are 1, 1, 1/3 and 1, respectively, and the numerical purities were 1.00, 1.00, 0.33 and 0.99. (An evaluation of the last purity based on a larger basis resulted in the value 1.00.) The first two values confirm that coherent and attenuated coherent states are pure. The third value shows that a PI link with only 3 dB of loss degrades the purity significantly. The fourth value shows that the state produced by a (one-stage) PS link is pure. This result reflects the fact that the input to the (first) PS amplifier is a coherent state, so the output from the amplifier is a squeezed coherent state [18], which is also pure.

Number distributions associated with the same states are shown in Fig. 10. The distribution associated with the input coherent state [Fig. 10(a)] is localized near the mean number 3.0, whereas the distribution associated with the attenuated coherent state [Fig. 10(b)] is localized near the mean number 1.5. In contrast, the distributions associated with the other output states are spread over broader ranges. This spreading is more significant for a PI link than a PS link, because the former link adds more noise to each quadrature.

 

Fig. 10 Number distribution of (a) an input coherent state with a number mean of 3.0, (b) the output coherent state produced by a 3-dB attenuator, (c) the output state produced by a balanced phase-insensitive link with 3 dB of loss and lumped gain, and (d) the output state produced by a balanced phase-sensitive link. The former link distorts the number distribution more than the latter.

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A link with distributed PS gain (not shown) adds even less noise to the signal than the link with lumped PS gain, but for moderate loss factors, the performance improvement is also moderate. It is worth noting that the numerical purity associated with the distributed PS link is 0.97, which is slightly lower than the purity associated with the lumped PS link: Although less noise is added, the output state is not quite pure.

Loss and gain factors of 3 dB are enough to produce moderate changes in the signal state. However, typical factors are larger and can produce significant changes. To illustrate these changes, array plots of the density matrices produced by balanced links are shown in Fig. 11. For a link with 10 dB of loss and lumped PI gain [Fig. 11(a)], the matrix is almost completely diagonal, which means that the output state is almost completely incoherent. (The theoretical and numerical purities are 0.05.) In contrast, for a link with 10 dB of distributed loss and PS gain [Fig. 11(b)], the matrix still has many large non-diagonal elements, which means that the output state retains a high degree of coherence. (The theoretical and numerical purities are 0.77.)

 

Fig. 11 Density matrices in the number-state representation. Darker squares denote higher values of |ρ_mn|, whereas lighter squares denote lower values. The output states are produced by balanced links with (a) 10 dB of loss and lumped gain, and (b) 10 dB of distributed loss and phase-sensitive gain. The state produced by the former link is nearly incoherent, whereas the state produced by the latter is nearly coherent.

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The number distributions produced by the same links are shown in Fig. 12. Both distributions differ noticeably from the input distribution [Fig. 10(a)]. For the lumped PI link [Fig. 12(a)], the probability of a substate with 20 photons is appreciable, despite the fact that the input state had only 3 photons! In contrast, for the distributed PS link [Fig. 12(b)], the probabilities of substates with 1–5 photons remain high, and the probabilities of substates with more than 12 photons are low, so the overlap of the input and output states is large (Fig. 4). For the stated loss factor, the distributed PS link performs significantly better than the corresponding lumped PS link (not shown).

 

Fig. 12 Number distributions of the output states produced by balanced links with (a) 10 dB of loss and lumped phase-insensitive gain, and (b) 10 dB of distributed loss and phase-sensitive gain. The former link distorts the number distribution much more than the latter.

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7. Summary

In this paper, the effects of transmission on Gaussian optical states were studied in detail. Such states include strong and weak coherent states, which are used in conventional communication systems and quantum information experiments, respectively, and the photon-pair-like states produced by parametric sources, which also are used in quantum information experiments. Gaussian states are specified completely by their quadrature means, and the variances and correlations of their quadrature fluctuations.

In Sec. 3, equations were derived that model the spatial evolution of these quantities in transmission links with loss and gain. One-signal-mode links with phase-sensitive (PS) amplification were considered in Sec. 3.1, one-signal-mode links with (two-mode) phase-insensitive (PI) amplification were considered in Sec. 3.2 and two-mode links with PS amplification were considered in Sec. 3.3. In each section, formulas for the quadrature variances and correlations (elements of the covariance matrices) were derived for links with arbitrary numbers of discrete amplifiers (lumped links) and links with distributed amplification (distributed links). These formulas allow one to determine the noise figures of the links and their fidelities of transmission. The noise figures were compared in Sec. 3.4. In general, links with many weak amplifiers have lower noise figures than links with a few strong amplifiers. The noise figures of links with lumped amplification increase exponentially with distance (loss), whereas the noise figures of links with distributed amplification increase only linearly with distance. PI amplifiers add excess noise, whereas PS amplifiers do not. These facts are well known.

For a Gaussian state, there is a simple relation between its input and output covariance matrices, and the transmission fidelity [Eq. (17)]. In Sec. 4, this relation was used to determine the fidelities of one-signal-mode states transmitted through PI and PS links. For direct transmission (attenuation without amplification), the fidelity decreases rapidly with distance, because the quadrature means decrease. The fidelity associated with a distributed PI link is higher than that associated with direct transmission, but the margin is small. Although the link maintains the quadrature means, PI amplification adds excess noise to both quadratures, and even modest amounts of noise lower the fidelity appreciably. (For this link, fidelity is a more-sensitive performance metric than noise figure.) The fidelities associated with PS links are significantly higher. The distributed PS link has the highest fidelity, but even links with moderate numbers of discrete PS amplifiers have reasonable fidelities. One-mode PS links perform well because the noise in the (unused) out-of-phase quadrature is squeezed (clamped at half the vacuum level) and excess noise is not added to the stretched quadrature by the amplifiers (only by the attenuators). The fidelities of two-mode states transmitted through PS links were studied in Sec. 5. Because a two-mode state can be represented in terms of superposition modes, which are transmited independently, the two-mode fidelities are just the squares of the one-mode fidelitlies studied in the previous section. Thus, two-mode PS links also perform well.

It is possible to reconstruct the density operator of a Gaussian signal, in the number-state representation, from its covariance matrix [Eqs. (9), (11), (21) and (22)]. This density matrix allows one to determine the coherence (purity) of the output state and the probability distribution of photons within it (Sec. 6). For a balanced link, as distance increases the coherence decreases and the distribution spreads over a broader range of photon numbers. These changes develop much more rapidly for links with lumped PI amplification than for links with lumped or distributed PS amplification. In summary, based on the aforementioned considerations, links with distributed PS amplification perform significantly better than other links.

Appendix A: Wigner function

In this appendix an alternative formula for the Wigner function is derived and some of its properties are discussed. The trace in Eq. (5) is basis independent. In the q-basis, the characteristic function

(70)$$C(k,l)=\int 〈q\left|\widehat{\rho }\text{exp}\left[-i(k\widehat{p}+l\widehat{q})\right]\right|q〉dq.$$

By using the Baker–Campbell–Hausdorff formula [2–4]

(71)$$\text{exp}(\widehat{x}+\widehat{y})=\text{exp}(\widehat{x})\text{exp}(\widehat{y})\text{exp}\left(-\left[\widehat{x},\widehat{y}\right]/2\right),$$

for operators x̂ and ŷ that commute with their commutator, one finds that

(72)$$\begin{array}{lll}\text{exp}\left[-i(k\widehat{p}+l\widehat{q})\right]\hfill & =\hfill & \text{exp}(-ik\widehat{p})\text{exp}(-il\widehat{q})\text{exp}(-ikl/2)\hfill \\ \hfill & =\hfill & \text{exp}(-il\widehat{q})\text{exp}(-ik\widehat{p})\text{exp}(ikl/2).\hfill \end{array}$$

Further progress requires the identities

(73)$$\text{exp}(-ik\widehat{p})|p〉=\text{exp}(-ikp)|p〉,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{exp}(-ik\widehat{p})|q〉=|q+k〉,$$

(74)$$\text{exp}(-il\widehat{q})|p〉=|p-l〉,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{exp}(-il\widehat{q})|q〉=\text{exp}(-ilq)|q〉.$$

The first and fourth of these identities follow from the definitions of quadrature ket-vectors, whereas the second and third follow from the generalized commutation relations

(75)$$\left[\widehat{q},{\widehat{p}}^m\right]=im{\widehat{p}}^{m-1},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[\widehat{p},{\widehat{q}}^n\right]=-in{\widehat{q}}^{n-1},$$

and are consistent with the replacements p̂ = −i∂_q and q̂ = i∂_p, respectively. By combining definition (70) with the first of Eqs. (72) and second of Eqs. (73), one finds that

(76)$$C(k,l)=\int 〈q-k/2\left|\widehat{\rho }\right|q+k/2〉\text{exp}(-ilq)dq.$$

By combining Eqs. (4) and (76), one obtains the Wigner formula [1]

(77)$$W(p,q)=\int 〈q-k/2\left|\widehat{\rho }\right|q+k/2〉\text{exp}(ikp)dk/2\pi ,$$

which implies that the Wigner function is real. One also can derive a formula for the Wigner function by using the p-basis. The result is

(78)$$W(p,q)=\int 〈p+l/2\left|\widehat{\rho }\right|p-l/2〉\text{exp}(ilq)dl/2\pi .$$

The Wigner function is not a true PDF because it can have negative values (otherwise the overlap integrals of orthogonal states could not be zero). However, the marginal distributions

(79)$$P(p)=\int W(p,q)dq=〈p\left|\widehat{\rho }\right|p〉,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Q(q)=\int W(p,q)dp=〈q\left|\widehat{\rho }\right|q〉$$

are both non-negative and can be regarded as true PDFs. Formula (77) resembles a local Fourier transform, which indicates that the Wigner function contains information about the local wavenumber (instantaneous frequency) p at the position (time) q. Similar phase-space distribution functions have many applications in classical optics [38, 39].

Of particular importance is the Wigner function associated with the operator |n〉 〈m|. It follows from Eq. (77) that

(80)$$W_{nm}(p,q)=\int {\psi }_n(q-k/2){\psi }_m^{*}(q+k/2)\text{exp}(ikp)dk/2\pi ,$$

where the number-state wavefunction [2–4]

(81)$${\psi }_n(q)=〈q|n〉=H_n(q)\text{exp}(-q^2/2)/{\left(2^{n}n!\right)}^{1/2}{\pi }^{1/4}.$$

By combining Eqs. (80) and (81), completing the square and defining the variable l = k/2 − ip [3], one finds that

(82)$$\begin{array}{lll}{W}_{nm}(p,q)\hfill & =\hfill & \left[\text{exp}(-p^2-q^2)/{\left(2^{m+n}m!n!\right)}^{1/2}{\pi }^{3/2}\right]\hfill \\ \hfill & \hfill & \times \int H_m(q+ip+l)H_n(q-ip-l)\text{exp}(-l^2)dl.\hfill \end{array}$$

Further progress requires the identity H_n(−x) = (−1)ⁿH_n(x) [33] and formula 7.377 of [40], which states that

(83)$$\int H_m(x+y)H_n(x+z)\text{exp}(-x^2)dx=2^m{\pi }^{1/2}n!y^{m-n}{L}_n^{m-n}(-2yz)$$

for the case in which m ≥ n. By using these results to evaluate the integral in Eq. (82), one obtains the explicit formula

(84)$$W_{nm}(p,q)={\left(-1\right)}^n{\left(n!/m!\right)}^{1/2}{\left[2^{1/2}(q+ip)\right]}^{m-n}\text{exp}\left(-p^2-q^2\right)L_n^{m-n}\left(2p^2+2q^2\right)/\pi .$$

Equation (84) is the first of Eqs. (22). For the complementary case in which m < n, one obtains the related formula

(85)$$W_{nm}(p,q)={\left(-1\right)}^m{(m!/n!)}^{1/2}{\left[2^{1/2}(q-ip)\right]}^{n-m}\text{exp}\left(-p^2-q^2\right)L_m^{n-m}\left(2p^2+2q^2\right)/\pi ,$$

which is the second of Eqs. (22).

The Wigner functions of the first four number states |n〉 (density operators |n〉 〈n|) are illustrated in Fig. 13. In contrast to the Wigner functions of Gaussian states, which are centered at arbitrary points in the pq-plane and decrease monotonically (Fig. 1), these Wigner functions are centered at the origin and all but one decrease in an oscillatory manner.

 

Fig. 13 Wigner distribution functions of number states with 0, 1, 2 and 3 photons. The vacuum distribution (a) is Gaussian and positive, whereas the other distributions are non-Gaussian, and have both positive and negative values. As the photon number increases, so also does the number of oscillations.

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The right side of Eq. (13) involves the double integral of two Wigner functions, each of which is defined by Eq. (77). Observe that

(86)$$\begin{array}{l}\iint \iint 〈q-k_1/2\left|{\widehat{\rho }}_1\right|q+k_1/2〉〈q-k_2/2\left|{\widehat{\rho }}_2\right|q+k_2/2〉\text{exp}\left[i(k_1+k_2)p\right]d{k}_{1}d{k}_{2}dpdq/2\pi \\ =\iiint 〈q-k_1/2\left|{\widehat{\rho }}_1\right|q+k_1/2〉〈q-k_2/2\left|{\widehat{\rho }}_2\right|q+k_2/2〉\delta (k_1+k_2)d{k}_{1}d{k}_{2}dq\\ =\iint 〈q-k/2\left|{\widehat{\rho }}_1\right|q+k/2〉〈q+k/2\left|{\widehat{\rho }}_2\right|q-k/2〉dkdq\\ =\int 〈q\left|{\widehat{\rho }}_1{\widehat{\rho }}_2\right|q〉dq,\end{array}$$

from which follows the trace formula (13). This formula applies to hermitian and non-hermitian operators alike.

Appendix B: Mathematical identity

It is required to prove that

(87)$$(A+B)-(A-B)S^{-1}(A-B)=4{\left(A^{-1}+B^{-1}\right)}^{-1},$$

where A and B are arbitrary square matrices and S = A + B is their sum. Suppose that the required matrix inverses exist. Then the relevant matrix product is

(88)$$\begin{array}{ll}\hfill & \left(A^{-1}+B^{-1}\right)\left[A+B-\left(A-B\right)S^{-1}(A-B)\right]\hfill \\ =\hfill & \left(A^{-1}+B^{-1}\right)\left[\left(A-A{S}^{-1}A\right)+\left(B-B{S}^{-1}B\right)+A{S}^{-1}B+B{S}^{-1}A\right].\hfill \end{array}$$

By expanding the first term in Eq. (88), one finds that

(89)$$\left(A^{-1}+B^{-1}\right)\left(A-A{S}^{-1}A\right)=I-S^{-1}A+B^{-1}A-B^{-1}\left(S-B\right)S^{-1}A=I.$$

The expansion of the second term is similar. By expanding the third term, one finds that

(90)$$\left(A^{-1}+B^{-1}\right)A{S}^{-1}B=S^{-1}B+B^{-1}(S-B)S^{-1}B=I.$$

Alternatively, one can deduce Eq. (90) from Eq. (89) by using the fact that AS⁻¹B = (S − B)S⁻¹B = B − BS⁻¹B. The expansion of the fourth term is similar. By combining Eqs. (89) and (90), one obtains the required result. Notice that this derivation does not depend on the dimension of the matrices involved (which is twice the number of participating modes).

Appendix C: Matrix elements

In this appendix, formulas are derived for the characteristic function C_nm(ξ) and Wigner function W_nm(α) associated with the density operator |n〉 〈m|. The former quantity is the matrix element 〈m|D̂(ξ)|n〉, where D̂(ξ) = exp(ξâ^† − ξ^*â) is the displacement operator, and the latter is the Fourier transform of the former. (Here D̂ is a displacement operator, not a determinant.) These formulas were derived by Cahill [41, 42], whose notations we follow loosely.

By using the Baker–Campbell–Hausdorff formula (71), one can rewrite the displacement operator in the alternative forms

(91)$$\begin{array}{lll}\widehat{D}(\xi )\hfill & =\hfill & \text{exp}(\xi {\widehat{a}}^{†})\text{exp}\left(-{\xi }^{*}\widehat{a}\right)\text{exp}\left(-{\left|\xi \right|}^2/2\right)\hfill \\ \hfill & =\hfill & \text{exp}\left(-{\xi }^{*}\widehat{a}\right)\text{exp}\left(\xi {\widehat{a}}^{†}\right)\text{exp}\left({\left|\xi \right|}^2/2\right),\hfill \end{array}$$

which are described as normally and antinormally ordered, respectively. It is easy to verify that

(92)$$〈m|\text{exp}(\xi {\widehat{a}}^{†})=\sum _{k=0}^m{\left(\begin{array}{c}m\\ k\end{array}\right)}^{1/2}\frac{{\xi }^k〈m-k|}{{\left(k!\right)}^{1/2}},$$

(93)$$\text{exp}\left(-{\xi }^{*}\widehat{a}\right)|n〉=\sum _{l=0}^n{\left(\begin{array}{c}n\\ l\end{array}\right)}^{1/2}\frac{{\left(-{\xi }^{*}\right)}^l|n-l〉}{{\left(l!\right)}^{1/2}}.$$

Suppose that m = n + d, where d ≥ 0. Then, by combining the first of Eqs. (91) with Eqs. (92) and (93), and using the fact that 〈m − k|n − l〉 is only nonzero if k = l + d, one finds that

(94)$$C_{nm}(\xi )={\xi }^d\text{exp}\left(-{\left|\xi \right|}^2/2\right)\sum _{l=0}^n\frac{{\left[n!(n+d)!\right]}^{1/2}{\left(-{\left|\xi \right|}^2\right)}^l}{l!(l+d)!(n-l)!}.$$

The associated Laguerre polynomials [33] are defined by the power series

(95)$$L_n^m(x)=\sum _{l=0}^n\left(\begin{array}{c}n+m\\ l+m\end{array}\right)\frac{{\left(-x\right)}^l}{l!}.$$

By using this definition, one can rewrite the characteristic function (94) in the compact form

(96)$$C_{nm}(\xi )={\left(n!/m!\right)}^{1/2}{\xi }^{m-n}\text{exp}\left(-{\left|\xi \right|}^2/2\right)L_n^{m-n}\left({\left|\xi \right|}^2\right).$$

Conversely, suppose that n = m + d, where d > 0. Then, by repeating the preceding derivation, one obtains the related formula

(97)$$C_{nm}(\xi )={\left(m!/n!\right)}^{1/2}{\left(-{\xi }^{*}\right)}^{n-m}\text{exp}\left(-{\left|\xi \right|}^2/2\right)L_m^{n-m}\left({\left|\xi \right|}^2\right).$$

If 〈m|D̂(ξ)|n〉 is known for m > n, then its conjugate 〈n|D̂^†(ξ)|m〉 = 〈n|D̂ (−ξ)|m〉, where n < m. Hence, one can deduce Eq. (97) from Eq. (96) by conjugating the characteristic function, replacing ξ by −ξ and interchanging the symbols m and n. For the characteristic function (96), conjugation and replacement are equivalent to the substitution ξ → −ξ^*.

The characteristic and Wigner functions are related by the Fourier transforms [41, 42]

(98)$$W_{nm}(\alpha )=\int C_{nm}(\xi )\text{exp}\left(\alpha {\xi }^{*}-{\alpha }^{*}\xi \right)d^2\xi /\pi ,$$

(99)$$C_{nm}(\alpha )=\int W_{nm}(\xi )\text{exp}\left(\xi {\alpha }^{*}-{\xi }^{*}\alpha \right)d^2\alpha /\pi ,$$

where d²ξ is an abbreviation for dξ_rdξ_i, and the subscripts r and i denote real and imaginary parts, respectively. Notice that in these definitions, the real and imaginary parts of the transform variables appear in the exponents with opposite signs, and the (common) denominator is π. In polar coordinates, α = re^iθ, ξ = ρe^iϕ and d²ξ = ρdρdϕ. (Here ρ is a radius, not a density operator.) By combining Eq. (96), which is valid for m ≥ n, with Eq. (98), one finds that

(100)$$\begin{array}{lll}{W}_{nm}(\alpha )\hfill & =\hfill & {\left(n!/m!\right)}^{1/2}\iint {\rho }^{d+1}\text{exp}\left(-{\rho }^2/2\right)L_n^d({\rho }^2)\hfill \\ \hfill & \hfill & \times \text{exp}\left[id\varphi +i2r\rho \text{sin}(\theta -\varphi )\right]d\rho d\varphi /\pi ,\hfill \end{array}$$

where d = m − n ≥ 0. The angular part of this expression is

(101)$$\int \text{exp}(id\theta )\text{exp}\left[id(\varphi -\theta )-i2r\rho \text{sin}(\varphi -\theta )\right]d(\varphi -\theta ).$$

By using the integral representation [33]

(102)$$J_p(x)={\int }_{-\pi }^{\pi }\text{exp}\left[i(pt-x\text{sin}t)\right]dt/2\pi ,$$

where p is an integer, one finds that

(103)$$W(r,\theta )=2{\left(n!/m!\right)}^{1/2}\text{exp}(id\theta )\int {\rho }^{d+1}\text{exp}(-{\rho }^2/2)L_n^d({\rho }^2)J_d(\rho y)d\rho ,$$

where y = 2r. Formula 7.421.4 of [40] implies that

(104)$$\int x^{d+1}\text{exp}(-x^2/2)L_n^d(x^2)J_d(xy)dx={(-1)}^n{y}^d\text{exp}(-y^2/2)L_n^d(y^2),$$

from which it follows that

(105)$$W_{nm}(r,\theta )=2{\left(-1\right)}^n{(n!/m!)}^{1/2}{(2r)}^d\text{exp}(id\theta -2r^2)L_n^d(4r^2).$$

This polar formula is equivalent to the cartesian formula [41–43]

(106)$$W_{nm}(\alpha )=2{\left(-1\right)}^n{\left(n!/m!\right)}^{1/2}{\left(2\alpha \right)}^{m-n}\text{exp}\left(-2{\left|\alpha \right|}^2\right)L_n^{m-n}\left(4{\left|\alpha \right|}^2\right).$$

By repeating the preceding derivation for m < n, one obtains the related formula

(107)$$W_{nm}(\alpha )=2{\left(-1\right)}^m{\left(m!/n!\right)}^{1/2}{\left(2{\alpha }^{*}\right)}^{n-m}\text{exp}\left(-2{\left|\alpha \right|}^2\right)L_m^{n-m}\left(4{\left|\alpha \right|}^2\right).$$

Notice that in contrast to Eq. (97), there is no factor of (−1)^n−m in Eq. (107). This result is a consequence of the fact that

(108)$$\begin{array}{lll}{W_{nm}(\alpha )|}_{m<n}\hfill & =\hfill & \int {C_{mn}\left(-{\xi }^{*}\right)|}_{n>m}\text{exp}\left(\alpha {\xi }^{*}-{\alpha }^{*}\xi \right)d^2\xi /\pi \hfill \\ \hfill & =\hfill & \int {C_{mn}\left(-{\xi }^{*}\right)|}_{n>m}\text{exp}\left[{\alpha }^{*}{\left(-{\xi }^{*}\right)}^{*}-\alpha \left(-{\xi }^{*}\right)\right]d^2\left(-{\xi }^{*}\right)/\pi \hfill \\ \hfill & =\hfill & {W_{mn}\left({\alpha }^{*}\right)|}_{n>m}.\hfill \end{array}$$

By comparing the complex-argument transforms used in this appendix to the real-argument transforms used in the text, one finds that

(109)$$C_r(k,l)=C_c\left(k/2^{1/2},-l/2^{1/2}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{W}_r(x,y)=W_c\left(y/2^{1/2},x/2^{1/2}\right)/2\pi ,$$

where the subscripts r and c denote real and complex, respectively, and the arguments of C_c and W_c are the real and imaginary parts of ξ and α, respectively. It follows from the real version of the trace formula [Eq. (13)] that

(110)$$\text{tr}({\rho }_1{\rho }_2)=\iint W_1(\alpha )W_2(\alpha )d^2\alpha /\pi ,$$

where the subscript c was omitted for simplicity.

Appendix D: Quadrature measurements

One can measure the photon number of a weak (quantum) signal by using a detector. One can measure the mode quadratures of the signal by combining it with a strong (classical) local oscillator at a beam splitter [2–4,23]. The difference between the output numbers is proportional to the input signal quadrature, whose phase is determined by the local-oscillator phase (which can be varied). Thus, homodyne detection allows one to measure the means and variances of the principal quadratures (in separate experiments). However, the quadrature correlation

(111)$$〈\widehat{p}\widehat{q}〉=i〈{\left({\widehat{a}}^{†}\right)}^2-{\widehat{a}}^2+1〉/2,$$

which is not real. This result reflects the fact that p̂ and q̂ do not commute, so cannot be measured simultaneously [45].

If one were to pass the signal through a beam splitter prior to detection, the output mode-operators would be

(112)$${\widehat{a}}_1^{\prime }=\tau \widehat{a}+\rho \widehat{w},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{a}}_2^{\prime }=-{\rho }^{*}\widehat{a}+{\tau }^{*}\widehat{w},$$

where ŵ is the input operator of the second beam-splitter mode (which is a vacuum state). For simplicity, we assume that τ and ρ are real, but our conclusions do not depend on this assumption. Equations (112) imply that the principal output quadratures are

(113)$${\widehat{p}}_1^{\prime }=\left[\tau \left({\widehat{a}}^{†}+\widehat{a}\right)+\rho \left({\widehat{w}}^{†}+\widehat{w}\right)\right]/2^{1/2},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{q}}_2^{\prime }=i\left[-\rho \left({\widehat{a}}^{†}-\widehat{a}\right)+\tau \left({\widehat{w}}^{†}+\widehat{w}\right)\right]/2^{1/2}.$$

It is easy to verify that p̂′₁ commutes with q̂′₂, because the non-commuting contributions from â and ŵ cancel. Hence, p̂′₁ and q̂′₂ can be measured simultaneously, and their correlation

(114)$$〈{\widehat{p}}_1^{\prime }{\widehat{q}}_2^{\prime }〉=-i\tau \rho 〈{\left({\widehat{a}}^{†}\right)}^2-{\widehat{a}}^2〉/2=-\tau \rho 〈{\left(\widehat{p}\widehat{q}\right)}_s〉,$$

where the subscript s denotes symmetric ordering. Notice that the correlation measurement is not affected by the vacuum fluctuations of the second beam-splitter mode. However, the variances

(115)$$〈{\left({\widehat{p}}_1^{\prime }\right)}^2〉={\tau }^2〈{\widehat{p}}^2〉+{\rho }^2/2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}〈{\left({\widehat{q}}_2^{\prime }\right)}^2〉={\rho }^2〈{\widehat{q}}^2〉+{\tau }^2/2$$

do include contributions from the aforementioned fluctuations. Thus, by using a beam splitter prior to measurement, one can measure both signal quadratures, and their variances and correlation, simultaneously. The price one pays for this ability is the addition of noise to the variance measurements. Although this excess noise is comparable to the intrinsic noise of a coherent-state signal, it is much weaker than the excess noise added to that signal by a transmission link.

Acknowledgments

We acknowledge useful discussions with A. Agarwal, J. Dailey, D. James, N. Peters and P. Toliver. We also thank the reviewers for bringing to our attention [10] and [44]. This work was supported by the Defense Advanced Research Projects Agency under contract W31P4Q-13-C-0069. The views, opinions and findings contained in this article are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

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