1 Introduction↩︎

Quantum receivers can surpass the standard quantum limit (SQL) in binary optical-state discrimination by exploiting the structure of the signal set and appropriate measurement design. Beyond the extensively studied coherent-state case, displaced squeezed vacuum states provide an important nonclassical modulation format, since squeezing can improve discrimination performance when properly matched to the receiver [1]. For binary phase-shift-keyed displaced squeezed states (S-BPSK), the optimal squeezing fraction minimizes both the Helstrom bound and the SQL, making this alphabet a natural platform for studying squeezing-enhanced quantum communication [2]–[4].

An inverse-squeezing Kennedy (IS-Kennedy) receiver has recently been introduced for S-BPSK discrimination [1]. In the ideal case, the receiver first performs Kennedy nulling and then applies an inverse-squeezing operation, thereby mapping the squeezed on-off-keyed output into a coherent-state on-off-keyed alphabet before photon counting. This transformation converts the squeezing resource into an effective coherent-energy gain and allows the receiver to outperform the S-BPSK SQL [3] over a broad signal range.

In practice, however, the inverse-squeezing stage is highly sensitive to implementation imperfections. Deviations in the squeezing magnitude or phase may arise from optical loss, pump fluctuations, and phase-locking errors [5], [6], and even small mismatch can qualitatively change the receiver output statistics. This practical issue is particularly important because the performance advantage of the IS-Kennedy receiver relies on accurate cancellation of the input squeezing. In this letter, we investigate mismatch in the IS-Kennedy receiver for S-BPSK discrimination. We show that mismatch is equivalent to a residual squeezing after nulling, which makes the optimal maximum-a-posteriori (MAP) decision generally non-single-threshold. We further show that the receiver is much more sensitive to phase mismatch than to amplitude mismatch, and that under amplitude-only mismatch the saturation error with finite-resolution photon-number-resolving (PNR) detection exhibits a parity-step scaling. These results clarify the physical origin of mismatch-induced degradation and identify the phase-locking accuracy required for practical implementations of inverse-squeezing receivers.

2 Receiver modeling↩︎

We model the inverse-squeezing Kennedy receiver [1] as follows. Figure 1 schematically illustrates the receiver.

The S-BPSK signals to be discriminated are given by [3] \[\begin{align} &\text{hypothes}\;'0': \ket{\psi_0} = D(-\alpha)S(r)\ket{0} \sim \rho_0,\\ &\text{hypothes}\;'1': \ket{\psi_1} = D(+\alpha)S(r)\ket{0} \sim \rho_1. \end{align}\] Without loss of generality, \(\alpha\) is chosen to be real, and the mean photon numbers of the signals are \(N=|\alpha|^2 + \sinh^2(r)\). Throughout this letter, equal a priori probabilities are assumed, i.e., \(\pi_0 = \pi_1 = 0.5\).

Reference [3] defines the squeezing fraction \(\beta\) (the share of the total energy stored in squeezing) to quantify the advantage of S-BPSK: \[\beta \triangleq \frac{\sinh^2(r)}{N}.\] In addition, under ideal condition, the optimal squeezing fraction is \(\beta_\text{opt}=\frac{N}{2N+1}\), at which point Helstrom bound and SQL of S-BPSK are minimized [3]. In this letter, all S-BPSK employ the optimal squeezing fraction \(\beta_\text{opt}\): \[\alpha = \sqrt{N(1-\beta_\text{opt})},\quad r = \sinh^{-1}\sqrt{N\beta_\text{opt}},\quad \alpha \text{e}^r = \sqrt{N(N+1)}\]

2.1 Receiver under ideal condition↩︎

Under ideal condition, firstly, the received signal interferes with a local oscillator (LO) on a highly transmissive beam splitter (\(\tau \to 1\)) to perform the Kennedy nulling displacement \(D(\alpha)\). Consequently, the S-BPSK alphabet is mapped to a squeezed OOK (S-OOK) alphabet \(\sigma_i\): \[\sigma_i \sim D(\alpha)\ket{\psi_i} = \left\{\begin{align} &S(r)\ket{0}, & i = 0,\\ &D(2\alpha)S(r)\ket{0}, & i = 1. \end{align}\right. \label{eq:ISK95sigma95compact}\tag{1}\] Subsequently, \(\sigma_i\) passes through the IS module \(S(-r)\), mapping the S-OOK alphabet to a coherent-state OOK (C-OOK) alphabet \(\zeta_i\): \[\zeta_i = S(-r) \sigma_i S^\dagger(-r) \sim \left\{\begin{align} &S(-r)S(r)\ket{0} = \ket{0}, & i = 0,\\ &S(-r)D(2\alpha)S(r)\ket{0} = D(2\alpha \mathrm{e}^r)\ket{0} = \ket{2\gamma}. & i = 1.\\ \end{align} \right. \label{eq:ISK95zeta95compact}\tag{2}\] In the general case, \(\gamma = \alpha \cosh r + \alpha^* \sinh r\). In this letter \(\alpha \in \mathbb{R}_+\), so \(\gamma = \alpha e^r\). Therefore, we define: \[\gamma \triangleq \alpha e^r,\quad N_\text{eff} \triangleq |\gamma|^2 = |\alpha|^2 e^{2r}.\] Next, PNR detection is performed on \(\zeta_i\), yielding an output photon number \(n\). The positive operator-valued measure (POVM) for an ideal PNR detector with resolution M, denoted as PNR(M), is given by: \[\Pi^{(\text{M})}_n = \left\{\begin{align} &\Pi_n, \quad n =0,1,\cdots, \text{M}-1\\ &\mathbb{I} - \sum_{k=0}^{\text{M}-1} \Pi_k, \quad n = \text{M} \end{align}\right.\] where \(\Pi_n = \ket{n}\bra{n}, \;n = 0,1,2,\cdots\) and \(\mathbb{I}\) is the identity matrix in the Fock representation.

Therefore, the probability of detecting \(n\) photons given hypothes \('i'\) is given by \[P(n|i) = \text{Tr}\left\{\zeta\cdot\Pi_n^{(\text{M})}\right\} = \left\{\begin{align} &\text{Poiss}(n;\mu_i),\quad n =0,1,\cdots, \text{M}-1\\ &1 - \sum_{k=0}^{\text{M}-1}\text{Poiss}(k;\mu_i),\quad n = \text{M} \end{align}\right. \label{eq:zeta95n95prob}\tag{3}\] where \(\text{Poiss}(n;\mu) = \text{e}^{-mu}\frac{\mu^n}{n!}\); \(\mu_0 = 0\) and \(\mu_1 = 4N_\text{eff}\) are the mean photon number of \(\zeta_0\) and \(\zeta_1\), respectively. Then, the posterior probabilities are given by: \[P(i|n) = \frac{\pi_i P(n|i)}{\sum_{k} \pi_k P(n|k)} = \frac{P(n|i)}{\sum_{k} P(n|k)}\] The MAP decision rule is given by: \[\hat{i}(n) = \left\{\begin{align} &1,\quad P(i = 1|n) \ge P(i = 0|n),\\ &0,\quad \text{otherwise}. \end{align}\right.\] where \(\hat{i}\) is the decision of IS-Kennedy receiver. When the likelihood function \(L(n) = \ln\frac{P(n|i=0)}{P(n|i=1)}\) is monotonic, the MAP decision is equivalent to a single-threshold decision[1], with the optimal threshold denoted as \(n_\text{th}^*\). In addition, under ideal conditions, the optimal threshold is 1, which means that PNR detector does not provide a performance advantage over single photon detector (SPD) in this case [1]. So the error probability of IS-Kennedy is given by \[\begin{align} P_\text{err}^\text{ISK, ideal} &= \frac{1}{2}\left(P_\text{FA} + P_\text{Mi}\right) = \frac{1}{2}\left[P(n\ge n_\text{th}^*|i=0) + P(n< n_\text{th}^*|i=1)\right]\\ &= P_\text{err}^\text{K,ideal}(N_\text{eff}) = \frac{1}{2}\exp\left[-4N(N+1)\right], \end{align} \label{eq:isk95err95ideal}\tag{4}\] where \(P_\text{FA}\) and \(P_\text{Mi}\) are the false alarm probability (deciding 1 when 0 is sent) and the miss probability (deciding 0 when 1 is sent), respectively, and \(P_\text{err}^\text{K,ideal}\) is the ideal probability of the Kennedy receiver [7]. From this, it can be seen that the IS-Kennedy converts the squeezing resources of S-BPSK signals into higher coherent energy advantage (\(N\to N_\text{eff}=N(N+1)\)) through the IS module, thereby achieving a lower error probability. Compared to quantum receiver for coherent state discrimination, the energy gain brought by the IS-Kennedy takes the form \[G = 10\cdot \log_{10} \left(\frac{N_\text{eff}}{N}\right) = 10\cdot \log_{10} \left(N+1\right)\,(\text{dB})\]

2.2 Receiver under squeezing mismatch condition↩︎

In practical experimental setups, deviations between the receiver’s squeezing parameters and the input signal are inevitable due to factors such as optical loss, pump fluctuation, and phase locking errors. This phenomenon is referred to as squeezing mismatch. Without loss of generality, we adopt a simplified model where the transmitter performs perfect squeezing and the optical path is lossless, attributing all squeezing adaptation errors solely to the IS module. We denote the squeezing parameter applied by the IS module by \[z_s = \left(-r+\Delta r\right)\mathrm{e}^{j\Delta \theta},\quad j = \sqrt{-1},\] where \(\Delta r\) represents the magnitude mismatch and \(\Delta \theta\) represents the phase mismatch. Under ideal conditions, \(\Delta r = \Delta \theta = 0\).

Under mismatch conditions, the overall transformation of the receiver is given by: \[\zeta_i^{\text{sm}} \sim S\left(z_s\right) D(\alpha) \ket{\psi_i} = \left\{\begin{align} &S_\text{tot} \ket{0}, & i = 0,\\ &S_\text{tot} \ket{2\gamma}, & i = 1. \end{align}\right.\] Here, the composite squeezing operator is defined as \(S_\text{tot} \triangleq S\left(z_s\right) S(r)\), which reduces to the identity operator \(S_\text{tot} = \mathbb{I}\) in the ideal case. Thus, the essence of the mismatch is encapsulated within \(S_\text{tot}\).

Using the single-mode Gaussian operator decomposition, this operator can always be expressed as: \[S_\text{tot} = R(\vartheta)\,S(z_m), \label{eq:Stot95decomp}\tag{5}\] where \(R(\vartheta)=\exp(-j\vartheta a^\dagger a)\) is a phase rotation, and \(S(z_m)\) represents the equivalent residual squeezing with parameter \(z_m=r_m e^{j\theta_m}\). Since \(R(\vartheta)\) contributes only a phase factor in the Fock basis and does not alter the photon number distribution, the impact of mismatch on PNR statistics for a given energy \(N\) is determined solely by the residual squeezing parameters \((r_m,\theta_m)\).

Next, we determine \((r_m, \theta_m)\) using the Bogoliubov transformation. Letting \(r_s \triangleq -r+\Delta r\) and \(\theta_s \triangleq \Delta\theta\), the coefficients for the transformation \(S^\dagger_\text{tot} a S_\text{tot} = x a + y a^\dagger\) are derived as: \[x = \cosh(r_s)\cosh r + \mathrm{e}^{j\theta_s}\sinh(r_s)\sinh r,\qquad y = -\cosh(r_s)\sinh r - \mathrm{e}^{j\theta_s}\sinh(r_s)\cosh r,\] satisfying the unitarity condition \(|x|^2-|y|^2=1\). Consequently, we obtain: \[r_m=\mathrm{arcsinh}(|y|),\qquad \vartheta=-\arg(x),\qquad \theta_m=\arg(-y)-\arg(x). \label{eq:rm95theta95from95munu}\tag{6}\]

In practical scenarios characterized by small mismatches (\(\Delta r\ll 1,\;|\Delta\theta|\ll 1\)) [5], [6], a first-order approximation yields: \[y \approx -\Delta r + j\,\frac{\Delta\theta}{2}\sinh(2r), \label{eq:nu95approx}\tag{7}\] which leads to: \[\begin{align} &r_m \approx \sqrt{(\Delta r)^2+\left(\frac{\Delta\theta}{2}\sinh 2r\right)^2 }, \\ &\theta_m \approx \arg \left(\Delta r - j \frac{\Delta\theta}{2} \sinh\left(2r\right)\right) + \Delta \theta \sinh^2 r. \end{align} \label{eq:rm95approx}\tag{8}\] Equation 8 reveals that the total residual squeezing \(r_m\) is significantly more sensitive to the phase mismatch \(\Delta\theta\) than to the magnitude mismatch \(\Delta r\). This arises because the phase mismatch term is scaled by the coefficient \(\frac{1}{2}\sinh(2r)\), which grows rapidly with the squeezing parameter \(r\) (and thus the signal energy \(N\)). This leads to a “dominant masking effect”: in the presence of non-negligible phase mismatch, its contribution to \(r_m\) typically far exceeds that of the magnitude mismatch. Given the geometric relationship \(r_m \approx \sqrt{(\Delta r)^2+\left(K\Delta\theta\right)^2 }\), the value of \(r_m\) becomes almost entirely determined by the phase mismatch term. This explains the phenomenon observed in the simulation results, as illustrated in Fig. 2 (a) and (b), where the error probability curves for \((\Delta r, \Delta \theta) = (0.00, 0.03\pi)\) and \((0.02, 0.03\pi)\) nearly coincide. Consequently, in experimental systems, the requirement for phase locking precision is much more stringent than that for amplitude matching.

For phase-only mismatch (\(\Delta r=0\)), we obtain \[\begin{align} &r_m \approx \frac{|\Delta\theta|}{2}\sinh(2r) =|\Delta\theta|\,\frac{N(N+1)}{2N+1},\\ &\theta_m \approx -\frac{\pi}{2}\,\mathrm{sgn}(\Delta\theta) +\Delta\theta\,\sinh^2 r =-\frac{\pi}{2}\,\mathrm{sgn}(\Delta\theta)+\Delta\theta\,\frac{N^2}{2N+1},\;(\mathrm{mod}\;2\pi). \end{align}\] For amplitude-only mismatch (\(\Delta\theta=0\)), we obtain \[r_m \approx |\Delta r|,\qquad \theta_m \approx \begin{cases} 0, & \Delta r>0,\\ \pi, & \Delta r<0, \end{cases} \quad (\mathrm{mod}\;2\pi).\]

We now calculate the photon number probability distribution of \(\zeta_i^{\text{sm}}\). Ignoring the rotation \(R(\vartheta)\) which does not affect photon statistics, the equivalent output states derived from Eq. 5 are: \[\zeta_0^\text{sm} \sim S(z_m)\ket{0},\qquad \zeta_1^\text{sm} \sim S(z_m)\ket{2\gamma}.\] To utilize standard photon number distribution formulas, we rewrite \(\zeta_1^\text{sm}\) as a displaced squeezed vacuum state: \[\zeta_1^\text{sm} \sim S(z_m)\ket{2\gamma} = D(2\gamma_m)\,S(z_m)\ket{0}, \label{eq:sc95to95dss}\tag{9}\] where \[\gamma_m=\gamma\cosh r_m+\gamma^\ast \mathrm{e}^{i\theta_m}\sinh r_m. \label{eq:gamma95m95def}\tag{10}\] Consequently, the Poissonian distribution valid for the ideal IS-Kennedy (Eq. 3 ) no longer applies. For an ideal PNR(M) detector (considering only finite resolution), the probability of detecting \(n\) photons given hypothes \(i\) can be written as \[\label{eq:fock-population95mismatch95squeeze} P(n|i)=\text{Tr}\left(\Pi_n\,\zeta_i^{\text{sm}}\right)= \left\{\begin{align} &p_n^{\rm svs}(r_m), & i=0,\\ &p_n^{\rm dss}(2\gamma_m,r_m,\theta_m), & i=1, \end{align}\right.\tag{11}\] where the corresponding distribution is used for \(n \le \text{M}-1\), and the truncation/saturation bin is defined as \(P(n=\text{M}|i)=1-\sum_{n=0}^{\text{M}-1}P(n|i)\). Here, \(p_n^{\rm svs}(r_m)\) denotes the photon number distribution of the squeezed vacuum state (Eq. 12 ), and \(p_n^{\rm dss}(2\gamma_m,r_m,\theta_m)\) denotes that of the displaced squeezed vacuum state (Eq. 13 ).

The introduction of mismatch results in \(\zeta_0^\text{sm}\) becoming a squeezed vacuum state (containing only even photon numbers), while \(\zeta_1^\text{sm}\) becomes a displaced squeezed vacuum state whose Fock distribution generally exhibits even-odd oscillations, as shown in Fig. 2 (d). This implies that the likelihood function \(L(n)\) does not necessarily vary monotonically with the photon number \(n\). Consequently, the MAP decision generally becomes a set-based decision (potentially involving a multi-threshold structure) rather than a simple single-threshold test. Therefore, the minimum error probability (Eq. 4 ) is reformulated as: \[P_\text{err}^{\text{ISK},\text{sm}} = 1 - \frac{1}{2} \sum_{n=0}^{\text{M}} \max \left\{P(n|i=0),\;P(n|i=1)\right\}.\]

When employing an SPD, the observation yields only two outcomes: \(n=0\) (no-click) and \(n \ge 1\) (click). Under equal priors, the MAP decision is equivalent to a binary mapping of these outcomes. Since \(P(0|0) \ge P(0|1)\) (as the exponent in the latter is \(\le 0\)), the MAP rule maps to on–off detection: click decides hypothes 1, no-click decides hypothes 0. The false alarm probability is given by \[P_{\rm FA}=P(\text{click}|0)=1-P(n=0|i=0)=1-\frac{1}{\cosh r_m}.\] The miss detection probability is given by \[P_{\rm M}=P(n=0|i=1)=\frac{1}{\cosh r_m}\exp\left( -|2\gamma_m|^2+\mathrm{Re}\big[\mathrm{e}^{-i\theta_m}(2\gamma_m)^2\big]\tanh r_m \right).\] These equations clearly reveal that the residual squeezing \(r_m\) reduces the vacuum component of \(\zeta_0^\text{sm}\), thereby increasing the false alarm probability, while \(\theta_m\) modifies the vacuum overlap of \(\zeta_1^\text{sm}\) through the exponential term.

Furthermore, we consider the saturation characteristics in the case of amplitude-only mismatch (\(\Delta r \ne 0, \Delta \theta = 0\)). As the signal energy tends to infinity (\(N \to \infty\)), the mean photon number of the signal state \(\zeta_1^\text{sm}\) far exceeds the resolution M of the PNR detector, leading to \(P_\text{Mi} \to 0\). In this regime, the error probability of the IS-Kennedy is dominated by the false alarm probability. The MAP strategy is thus equivalent to a threshold decision with \(n_\text{th}^* = \text{M}\): decide hypothes 1 if the outcome falls into the saturation bin \(n=\text{M}\), and hypothes 0 otherwise. Since \(\zeta_0^\text{sm}\) is a squeezed vacuum state with a non-zero photon number distribution only for even terms, as given by Eq. 12 , the saturation error probability is given by \[P_\text{sat} \approx \frac{1}{2} P_\text{FA} = \frac{1}{2} \sum_{k = {\left \lceil M/2 \right \rceil }}^{\infty} p_{2k}^{\text{svs}}(r_m) = \frac{1}{2} \sum_{k = {\left \lceil M/2 \right \rceil }}^{\infty} \frac{(2k)!}{2^{2k}(k!)^2}\frac{\tanh^{2k}(r_m)}{\cosh(r_m)}.\] Under the condition of small mismatch (\(\Delta r \ll 1\)), we have \(|\Delta r| = r_m\) and \(\tanh r_m \approx r_m = |\Delta r|\), and the summation is dominated by the first term. With \(n_\text{min} = 2 \left \lceil \frac{\text{M}}{2} \right \rceil\), the error probability simplifies to: \[P_\text{sat} \approx \frac{1}{2} p_{n_\text{min}}^{\text{svs}}(r_m) = \frac{1}{2} \frac{n_\text{min}!}{2^{n_\text{min}}[(n_\text{min}/2)!]^2} \left(\Delta r\right)^{n_\text{min}}.\] This equation reveals the “parity step effect” of the PNR detector in resisting mismatch: increasing the detector resolution M does not continuously reduce the saturation error probability. Instead, the order of magnitude of the error probability transitions only when M crosses an even integer threshold. Substituting \(\text{M}=1\) and \(\text{M}=3\) confirms that the calculated results are consistent with the simulation results, as shown in Fig. 2 (a) and (b).

3 Simulation methods and results↩︎

The numerical results were obtained using QuTiP in Python [8]. We model the receiver output states in a truncated Fock space and compute the photon-number distributions from the corresponding density operators. Unless otherwise specified, the simulations use a cutoff \(N_\text{cut} = 100\). We verified numerical convergence by increasing the cutoff dimension until the resulting error probabilities changed negligibly. For each mean photon number \(N\), the S-BPSK states were generated with the optimal squeezing fraction \(\beta_\text{opt}=\frac{N}{2N+1}\).

Figure 2 (a) and Fig. 2 (b) show the error probability under SPD and PNR(3), respectively. In the ideal case, the error decreases rapidly with \(N\), reflecting the benefit of the inverse-squeezing transformation. Under amplitude-only mismatch, \(\left(\Delta r, \Delta \theta\right) = \left(0.02,0\right)\), the performance remains close to the ideal curve at low \(N\) but saturates at high \(N\), as expected from the residual squeezing acting on the nulled state \(\zeta_0^\text{sm}\). The saturation levels agree with Eq. (30): for SPD the error floor is about \(10^{-4}\), whereas for PNR(3) it is reduced to about \(3\times 10^{-8}\). This confirms the parity-step behavior predicted by the small-mismatch analysis, namely that increasing the PNR resolution suppresses the amplitude-mismatch-induced error only when the resolution crosses an even-photon threshold. The physical origin is illustrated in Fig. 2 (d): under mismatch, \(\zeta_0^\text{sm}\) becomes a squeezed vacuum with only even-photon components. As a result, the likelihood function is generally non-monotonic in photon number, and the MAP decision is no longer a simple single-threshold rule.

Phase mismatch causes a much stronger degradation. For \(\left(\Delta r, \Delta \theta\right) = \left(0,0.03\pi\right)\), the error begins to deviate from the ideal case at moderate \(N\) and increases significantly at higher energies. Moreover, the curves for \(\left(0,0.03\pi\right)\) and \(\left(0.02,0.03\pi\right)\) nearly overlap in both Fig. 2 (a) and Fig. 2 (b), showing that once phase mismatch is present, the additional effect of amplitude mismatch is comparatively weak. This is consistent with Eq. 8 , where the phase-mismatch contribution is amplified by \(\sinh \left(2r\right)/2\) and therefore dominates the residual squeezing strength \(r_m\) at sufficiently large squeezing (i.e. at high \(N\)). Although finite-resolution PNR clearly outperforms SPD in the mismatched regime, it does not remove the dominant sensitivity to phase mismatch.

Figure 2 (c) further plots the error-probability ratio in dB relative to the S-BPSK SQL, where negative values indicate performance below the SQL. In the ideal case, the IS-Kennedy receiver preserves a clear advantage as \(N\) increases. Under amplitude-only mismatch, this advantage is largely maintained over a broad range of \(N\), but is eventually limited by the saturation effect. By contrast, phase mismatch rapidly reduces the SQL advantage, and the error probability even continues to rise at large \(N\).

Figure 3 provides a more direct robustness metric by showing the maximum tolerable inverse-squeezing mismatch for which the IS-Kennedy receiver still outperforms the S-BPSK SQL, i.e., \(P_\text{err}^\text{ISK}<P_\text{SQL}^\text{S-BPSK}\). As shown in Fig. 3 (a), the tolerable amplitude mismatch \(\Delta r_\text{max}\) is non-monotonic in \(N\). It first increases,reaches its maximum in the low-to-intermediate energy region, and then decreases at higher \(N\). For PNR(1) and PNR(2), the admissible range rapidly shrinks at large \(N\), whereas PNR(3) retains an additional tolerance window. A similar trend is observed for the phase mismatch in Fig. 3 (b), but the tolerable phase error collapses more rapidly with increasing \(N\), in agreement with the enhanced phase sensitivity predicted by Eq. 8 . Moreover, the extra tolerance window preserved by PNR(3) is consistent with the parity-step behavior discussed above: increasing the PNR resolution improves mismatch robustness in a non-smooth, threshold-like manner rather than continuously.

Overall, the simulations show that the IS-Kennedy receiver is far more sensitive to phase mismatch than to amplitude mismatch, and that finite-resolution PNR improves robustness mainly against amplitude mismatch rather than imperfect phase locking.

4 Conclusion↩︎

In conclusion, we have investigated inverse-squeezing mismatch in the IS-Kennedy receiver for the discrimination of binary displaced squeezed vacuum states. We showed that imperfect inverse squeezing is equivalent to a residual squeezing after nulling, which modifies the output photon-number statistics and makes the optimal MAP decision generally non-single-threshold. The analysis further reveals a strong asymmetry between the two mismatch types: the receiver is much more sensitive to phase mismatch than to amplitude mismatch, because the phase error is effectively amplified by the squeezing process. Numerical results confirm that amplitude-only mismatch leads to a saturation error floor, and that finite-resolution PNR detection suppresses this floor in a parity-step manner rather than continuously with detector resolution. By contrast, phase mismatch rapidly degrades the receiver performance and can destroy its advantage over the S-BPSK SQL at high signal energies. The maximum-tolerable-mismatch results further show that the practically useful operating region of the IS-Kennedy receiver is ultimately limited by phase locking. These results clarify the physical origin of mismatch-induced degradation and provide guidance for the experimental design of inverse-squeezing receivers with finite-resolution photon counting.

5 Fock-basis Population of Squeezed Vacuum and Displaced Squeezed States↩︎

Let the Squeezed Vacuum state (SVS) and the Displaced Squeezed State (DSS) be denoted, respectively, as: \[S(z)\ket{0} \quad \text{and} \quad D(\alpha)S(z)\ket{0},\] where the squeezing parameter is complex, \(z = r\mathrm{e}^{j\theta}\) (with \(j=\sqrt{-1}\)).

The photon number distribution of the Squeezed Vacuum state is non-zero only for even photon numbers: \[p_{2k}^{\text{svs}}(r) = \frac{(2k)!}{2^{2k}(k!)^2} \frac{\tanh^{2k}(r)}{\cosh(r)}, \label{eq:pn95svs}\tag{12}\] \[p_{2k+1}^{\text{svs}}(r) = 0.\]

The photon number distribution of the displaced squeezed state can be expressed in terms of Hermite polynomials, \(H_n(\cdot)\): \[p_{n}^{\text{dss}} = \frac{1}{n!\cosh r} \left(\frac{\tanh r}{2}\right)^n \left| H_n\left( \frac{\alpha \mathrm{e}^{-j\theta/2}}{\sqrt{\sinh r \cosh r}} \right) \right|^2 \exp\left( -|\alpha|^2 + \mathrm{Re}\left[ \mathrm{e}^{-j\theta}\alpha^2 \right]\tanh r \right). \label{eq:pn95dss}\tag{13}\]

Data Availability↩︎

The data that support the findings of this study are available from the authors upon request.

Code Availability↩︎

The numerical simulations in this study were performed in Python using QuTiP 5.1.1. The custom scripts used to generate the results are available from the corresponding author on reasonable request.

Competing interests↩︎

The authors declare no competing interests.

References↩︎