Inverse-squeezing receivers for squeezed-state pulse-position modulation under ideal and phase-diffusion conditions

Chen Dong; Enhao Bai; Fengkai Sun; Huankai Zhang; Jian Peng; Tianyi Wu; Zhenrong Zhang

arxiv: 2606.31541 · v1 · pith:YWPBD5HUnew · submitted 2026-06-30 · 🪐 quant-ph

Inverse-squeezing receivers for squeezed-state pulse-position modulation under ideal and phase-diffusion conditions

Enhao Bai , Fengkai Sun , Tianyi Wu , Huankai Zhang , Jian Peng , Chen Dong , Zhenrong Zhang This is my paper

Pith reviewed 2026-07-01 04:59 UTC · model grok-4.3

classification 🪐 quant-ph

keywords squeezed statespulse-position modulationinverse squeezingquantum communicationphase diffusionoptical receiversphoton efficiencyconditional pulse nulling

0 comments

The pith

Inverse squeezing maps squeezed-state PPM into an equivalent high-energy coherent-state PPM, giving a closed-form error probability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces squeezed-state pulse-position modulation in which empty slots are squeezed vacuum states and the occupied slot is a displaced squeezed state. It proposes an inverse-squeezing conditional pulse-nulling receiver that, under ideal conditions, converts the modulated signal into an ordinary coherent-state PPM waveform whose pulse energy is greatly increased. This equivalence supplies an exact analytic expression for the receiver's error probability. Numerical comparisons show the new receiver beats the conventional conditional pulse-nulling receiver at the same mean photon number and continues to do so when phase diffusion or finite photon-number resolution is added.

Core claim

In the ideal case, inverse squeezing maps S-PPM into an equivalent coherent-state PPM signal with a large pulse energy, leading to a closed-form expression for the receiver error probability. The same receiver is then analyzed under common phase diffusion by a finite-path MAP rule that uses phase-averaged likelihoods. Numerical results show that IS-CPN outperforms conventional CPN under the same energy constraint and remains advantageous under phase noise and finite photon-number resolution.

What carries the argument

The inverse-squeezing conditional pulse-nulling (IS-CPN) receiver, which applies an inverse-squeezing operation to convert the incoming S-PPM waveform into an equivalent coherent-state PPM waveform before performing conditional nulling.

If this is right

A closed-form error probability is available for the ideal IS-CPN receiver.
IS-CPN achieves lower error rates than conventional CPN at identical average energy.
The performance advantage persists when common phase diffusion is present.
The advantage also persists when the detector has only finite photon-number resolution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same inverse-squeezing step could be applied to other squeezed-state modulation formats to obtain analytic performance bounds.
Laboratory tests with existing squeezed-light sources would directly check whether the ideal-case gain survives realistic squeezing efficiency and loss.
The mapping suggests that photon-efficient links could be built by trading squeezing for higher effective pulse energy without increasing transmitted power.

Load-bearing premise

Ideal noiseless squeezing and perfect inverse-squeezing operations are required for the exact mapping to an equivalent coherent-state PPM with boosted energy.

What would settle it

Measure the symbol-error probability of an IS-CPN receiver on laboratory squeezed vacuum and displaced squeezed states at known mean photon number; any statistically significant deviation from the closed-form expression derived under the ideal mapping would falsify the central equivalence.

Figures

Figures reproduced from arXiv: 2606.31541 by Chen Dong, Enhao Bai, Fengkai Sun, Huankai Zhang, Jian Peng, Tianyi Wu, Zhenrong Zhang.

**Figure 1.** Figure 1: FIG. 1. The DD and Hom error probabilities of S-PPM signals of different modulations. Negative values indicate that homodyne [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. The structure of inverse-squeezing conditional pulse nulling receiver, where [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The decision tree of ideal inverse-squeezing conditional pulse-nulling receiver ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. In the ideal phase-matched case, the performance of the inverse-squeezing conditional pulse nulling receiver (IS-CPN) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Under phase-diffusion condition ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Under phase-diffusion condition ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Maximum tolerable phase noise [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Under phase-diffusion condition, the ratio [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

read the original abstract

We introduce a squeezed-state pulse-position modulation (S-PPM) format, where the empty slots are squeezed vacuum states and the pulse slot is a displaced squeezed state. Based on this property, we propose an inverse-squeezing conditional pulse-nulling (IS-CPN) receiver. In the ideal case, inverse squeezing maps S-PPM into an equivalent coherent-state PPM signal with a large pulse energy, leading to a closed-form expression for the receiver error probability. We further analyze IS-CPN under common phase diffusion using a finite-path MAP formulation with phase-averaged likelihoods. Numerical results show that IS-CPN outperforms conventional CPN under the same energy constraint and remains advantageous under phase noise and finite photon-number resolution. These results demonstrate that combining squeezed-state modulation with inverse-squeezing conditional nulling can improve photon-efficient optical communication.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

S-PPM with inverse-squeezing CPN receiver yields closed-form error probability in the ideal case and numerical gains over standard CPN under phase diffusion.

read the letter

The paper defines squeezed-state PPM with squeezed vacuum in empty slots and a displaced squeezed state in the pulse slot, then introduces an inverse-squeezing conditional pulse-nulling receiver. In the ideal case this maps the problem exactly onto coherent-state PPM with large pulse energy, which supplies the closed-form error probability. They also treat phase diffusion via a finite-path MAP rule that averages the likelihoods over phase.

The new element is the specific S-PPM format paired with the IS-CPN construction and its phase-diffusion extension; the mapping and the numerical comparisons under fixed energy, phase noise, and finite photon resolution are the concrete contributions. The ideal-case reduction is clean and the reported outperformance over conventional CPN is consistent with the stated assumptions.

The closed-form result holds only when the receiver squeezing parameter matches the transmitter exactly and the operations are lossless; any mismatch or loss breaks the equivalence, as the paper itself notes by restricting the claim to the ideal case. The finite-path approximation for phase diffusion is reasonable but its accuracy depends on the number of paths chosen, and the paper does not appear to explore how sensitive the gains are to realistic squeezing levels or additional loss. The energy constraint comparison is standard, but the advantage is tied to how the mean photon number is counted across formats.

This is for people working on quantum optical communication and photon-efficient modulation. A reader who needs concrete receiver architectures or analytic benchmarks for squeezed-light PPM will get direct value.

Send it to peer review. The combination of a new format, an explicit receiver, and both analytic and numerical results is enough to justify referee time even if the ideal-case restriction requires clarification.

Referee Report

1 major / 1 minor

Summary. The paper introduces squeezed-state pulse-position modulation (S-PPM), with the pulse slot as a displaced squeezed state and empty slots as squeezed vacuum. It proposes an inverse-squeezing conditional pulse-nulling (IS-CPN) receiver. In the ideal case, inverse squeezing maps S-PPM to equivalent coherent-state PPM with large pulse energy, yielding a closed-form receiver error probability. The analysis extends to phase diffusion via a finite-path MAP formulation using phase-averaged likelihoods. Numerical results indicate that IS-CPN outperforms conventional CPN under identical energy constraints and retains advantages under phase noise and finite photon-number resolution.

Significance. If the ideal-case mapping is rigorously established, the work advances photon-efficient optical communication by combining squeezed-state modulation with receiver-side inverse squeezing. The closed-form error probability provides analytical utility, and the MAP treatment of phase diffusion addresses a realistic impairment. Numerical comparisons under phase noise and finite resolution strengthen the practical implications. Credit is given for the explicit ideal-case derivation leading to the closed-form expression and for the reproducible numerical evaluation across conditions.

major comments (1)

[ideal-case mapping and closed-form derivation] The central claim of a closed-form error probability rests on the exact mapping under ideal inverse squeezing (abstract). The derivation must explicitly compute the post-inverse-squeezing states for the displaced squeezed pulse slot and the squeezed-vacuum empty slots to confirm reduction to standard coherent-state PPM (with the stated large pulse energy) without residual squeezing or mismatch terms; any unaccounted deviation would invalidate the closed-form expression.

minor comments (1)

Notation for the squeezing parameter r and the inverse-squeezing operation should be introduced with explicit definitions before the mapping is invoked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for highlighting the need for greater explicitness in the ideal-case derivation. We address the single major comment below and will incorporate the requested details in the revision.

read point-by-point responses

Referee: [ideal-case mapping and closed-form derivation] The central claim of a closed-form error probability rests on the exact mapping under ideal inverse squeezing (abstract). The derivation must explicitly compute the post-inverse-squeezing states for the displaced squeezed pulse slot and the squeezed-vacuum empty slots to confirm reduction to standard coherent-state PPM (with the stated large pulse energy) without residual squeezing or mismatch terms; any unaccounted deviation would invalidate the closed-form expression.

Authors: We agree that an explicit computation of the post-inverse-squeezing states strengthens the rigor of the central claim. In the revised manuscript we will add a dedicated paragraph (new subsection 2.1) that applies the inverse squeezing operator S(-r) to both the displaced squeezed state |α, r⟩ (pulse slot) and the squeezed vacuum |0, r⟩ (empty slots). This yields a coherent state |α e^r⟩ for the pulse slot (with effective energy scaled by the squeezing factor) and the vacuum state |0⟩ for the empty slots, with no residual squeezing or mismatch terms remaining under ideal conditions. The resulting signal is therefore exactly equivalent to coherent-state PPM with large pulse energy, directly justifying the closed-form error-probability expression already stated in the paper. We will also include the corresponding operator algebra steps for reproducibility. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard quantum optics mapping

full rationale

The paper's central claim derives a closed-form error probability by applying the inverse-squeezing operator to map ideal S-PPM (displaced squeezed pulse + squeezed vacuum) onto an equivalent coherent-state PPM whose energy then invokes the known PPM formula. This step is constructed from the definitions of the squeezing operator and displacement operator under the explicit ideal (lossless, matched-r) assumption stated in the abstract; it does not reduce to a fitted parameter, a self-citation chain, or an ansatz imported from the authors' prior work. No load-bearing step in the given text equates a prediction to its own input by construction, and the numerical comparisons under phase diffusion use a separate finite-path MAP formulation. The derivation is therefore self-contained against external quantum-optics benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum-optics assumptions for squeezed states and phase diffusion; no free parameters or invented entities are identifiable from the abstract alone.

axioms (2)

standard math Squeezed vacuum and displaced squeezed states obey the standard bosonic commutation relations and can be transformed by ideal squeezing operators.
Invoked implicitly when stating that inverse squeezing maps S-PPM to coherent-state PPM.
domain assumption Phase diffusion can be modeled by a finite-path average over discrete phase values in the MAP detector.
Used in the phase-diffusion analysis section referenced in the abstract.

pith-pipeline@v0.9.1-grok · 5687 in / 1484 out tokens · 35504 ms · 2026-07-01T04:59:08.816540+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 1 canonical work pages · 1 internal anchor

[1]

=|⟨0|0, r⟩| 2 = 1 coshr , v0 =p(n̸= 0|H ′
[2]

= 1−u 0, u1 =p(n= 0|H ′
[3]

=|⟨0|α, r⟩| 2 = 1 coshr exp −α2 (1 + tanhr) , v1 =p(n̸= 0|H ′
[4]

(13) Sinceu 1 < u0 for anyα >0, one hasv 1 > v0

= 1−u 1. (13) Sinceu 1 < u0 for anyα >0, one hasv 1 > v0. Therefore, under equal priors, the MAP rule reduces to the following intuitive decision strategy: if at least one slot clicks, choose uniformly among the clicked slots; if no slot clicks, choose uniformly among allMhypotheses. Without loss of generality, suppose the true hypothesis isH 1, so that t...
[5]

= 1√2πVx exp − x2 2Vx , p(x|H ′
[6]

no-click

= 1√2πVx exp −(x−α) 2 2Vx ,(17) whereV x = 1 4 e−2r. For equal priors, the optimal threshold isx=α/2. The false-alarm and miss probabilities are then identical and are given by ϵ=Q α/2√Vx =Q(αe r) =Q p Neff = 1 2 erfc r Neff 2 ! (18) whereQ(·) is the standard Gaussian Q-function. Then we adopt the following classical post-processing rule: if exactly one s...
[7]

= Z R pσϕ(ϕ)p(x|H ′ 0, ϕ) dϕ, ¯p(x|H′
[8]

(45) Letx th be the single-slot decision threshold

= Z R pσϕ(ϕ)p(x|H ′ 1, ϕ) dϕ. (45) Letx th be the single-slot decision threshold. The receiver declares the slot to be a pulse slot whenx > x th, and declares it to be an empty slot otherwise. The corresponding false-alarm and miss probabilities are pF (xth) = Z ∞ xth ¯p(x|H′
[9]

dx= Z R pσϕ(ϕ)Q xthp Vϕ ! dϕ, pM(xth) = Z xth −∞ ¯p(x|H′
[10]

(46) Unlike the ideal phase-matched case, these two probabilities are generally not equal

dx= Z R pσϕ(ϕ)Q αcosϕ−x thp Vϕ ! dϕ. (46) Unlike the ideal phase-matched case, these two probabilities are generally not equal. Therefore, the optimal homodyne threshold must be found numerically. TheM-ary decision is obtained using the same post-processing rule as in Sec. II. If exactly one slot is declared as the pulse slot, that slot is selected. If mo...
[11]

Formally, the stateρ i+1 is obtained fromρ i through a symmetry operatorTthat pushes each Kronecker factor by one position (modulo M) to the left

GUS reduction of the phase-diffused ensemble The geometrically uniform symmetry (GUS) structure of the PPM, whether C-PPM or S-PPM, constellation is apparent [11]. Formally, the stateρ i+1 is obtained fromρ i through a symmetry operatorTthat pushes each Kronecker factor by one position (modulo M) to the left. ρm ∼ |Ψ m⟩=T m−1 |Ψ1⟩, m= 1,· · ·, M.(A1) Unde...
[12]

SRM benchmark for high-order phase-diffused PPM To obtain a tractable quantum benchmark for high-order phase-diffused S-PPM, we employ the square-root mea- surement associated with the mixed-state ensemble{ρ pd m }M m=1. This measurement is not assumed to be the exact Helstrom measurement for general mixed-state ensembles; rather, it provides a physically...
[13]

curse of dimensionality

Efficient Numerical Construction of the Gram Matrix Although the SRM error probability has been analytically reduced to operations on the Gram matrixG, constructing Gexplicitly still suffers from the “curse of dimensionality” because the inner productsζ † i ζj reside in an exponentially large tensor-product space (N cut)M. Here we introduce an efficient n...
[14]

I. A. Burenkov, M. V. Jabir, and S. V. Polyakov, Practical quantum-enhanced receivers for classical communication, AVS Quantum Sci.3, 025301 (2021)

2021
[15]

C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys.1, 231 (1969)

1969
[16]

S. Dolinar Jr, A near-optimum receiver structure for the detection of m-ary optical ppm signals, The Telecommunications and Data Acquisition Progress Report 42-72, October–December 1982 , 30 (1982)

1982
[17]

M. G. A. Paris, Nearly ideal binary communication in squeezed channels, Phys. Rev. A64, 014304 (2001)

2001
[18]

Chesi, S

G. Chesi, S. Olivares, and M. G. A. Paris, Squeezing-enhanced phase-shift-keyed binary communication in noisy channels, Phys. Rev. A97, 032315 (2018)

2018
[19]

Wang and X

Y. Wang and X. Chen, Joint modulation of 3-ppm and quantum squeezed states in communication systems, inIEEE INFOCOM 2022 - IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)(2022) pp. 1–5

2022
[20]

E. Bai, J. Peng, T. Wu, K. Wen, F. Sun, C. Zhou, Y. Li, Z. Zhang, and C. Dong, Near-optimal discrimination of displaced squeezed binary signals using displacement, inverse-squeezing, and photon-number-resolving detection (2026), arXiv:2601.09073 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[21]

M. G. Genoni, S. Olivares, and M. G. A. Paris, Optical phase estimation in the presence of phase diffusion, Phys. Rev. Lett.106, 153603 (2011)

2011
[22]

M. N. Notarnicola and S. Olivares, A robust hybrid receiver for binary phase-shift keying discrimination in the presence of phase noise, Int. J. Quantum Inf.22, 2450008 (2024)

2024
[23]

Lambert, E

N. Lambert, E. Gigu‘ere, P. Menczel, B. Li, P. Hopf, G. Su’arez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, Qutip 5: The quantum toolbox in Python, Phys. Rep.1153, 1 (2026)

2026
[24]

Cariolaro and G

G. Cariolaro and G. Pierobon, Theory of quantum pulse position modulation and related numerical problems, IEEE Trans. Commun58, 1213 (2010)

2010
[25]

Izumi, M

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, Quantum receivers with squeezing and photon-number-resolving detectors form-ary coherent state discrimination, Phys. Rev. A87, 042328 (2013)

2013
[26]

Humer, M

G. Humer, M. Peev, C. Schaeff, S. Ramelow, M. Stipˇ cevi´ c, and R. Ursin, A simple and robust method for estimating afterpulsing in single photon detectors, J. Lightwave Technol.33, 3098 (2015)

2015
[27]

J. Chen, J. Habif, Z. Dutton, R. Lazarus, and S. Guha, Optical codeword demodulation with error rates below standard quantum limit using a conditional nulling receiver, Nat. Photon.6(2012)

2012

[1] [1]

=|⟨0|0, r⟩| 2 = 1 coshr , v0 =p(n̸= 0|H ′

[2] [2]

= 1−u 0, u1 =p(n= 0|H ′

[3] [3]

=|⟨0|α, r⟩| 2 = 1 coshr exp −α2 (1 + tanhr) , v1 =p(n̸= 0|H ′

[4] [4]

(13) Sinceu 1 < u0 for anyα >0, one hasv 1 > v0

= 1−u 1. (13) Sinceu 1 < u0 for anyα >0, one hasv 1 > v0. Therefore, under equal priors, the MAP rule reduces to the following intuitive decision strategy: if at least one slot clicks, choose uniformly among the clicked slots; if no slot clicks, choose uniformly among allMhypotheses. Without loss of generality, suppose the true hypothesis isH 1, so that t...

[5] [5]

= 1√2πVx exp − x2 2Vx , p(x|H ′

[6] [6]

no-click

= 1√2πVx exp −(x−α) 2 2Vx ,(17) whereV x = 1 4 e−2r. For equal priors, the optimal threshold isx=α/2. The false-alarm and miss probabilities are then identical and are given by ϵ=Q α/2√Vx =Q(αe r) =Q p Neff = 1 2 erfc r Neff 2 ! (18) whereQ(·) is the standard Gaussian Q-function. Then we adopt the following classical post-processing rule: if exactly one s...

[7] [7]

= Z R pσϕ(ϕ)p(x|H ′ 0, ϕ) dϕ, ¯p(x|H′

[8] [8]

(45) Letx th be the single-slot decision threshold

= Z R pσϕ(ϕ)p(x|H ′ 1, ϕ) dϕ. (45) Letx th be the single-slot decision threshold. The receiver declares the slot to be a pulse slot whenx > x th, and declares it to be an empty slot otherwise. The corresponding false-alarm and miss probabilities are pF (xth) = Z ∞ xth ¯p(x|H′

[9] [9]

dx= Z R pσϕ(ϕ)Q xthp Vϕ ! dϕ, pM(xth) = Z xth −∞ ¯p(x|H′

[10] [10]

(46) Unlike the ideal phase-matched case, these two probabilities are generally not equal

dx= Z R pσϕ(ϕ)Q αcosϕ−x thp Vϕ ! dϕ. (46) Unlike the ideal phase-matched case, these two probabilities are generally not equal. Therefore, the optimal homodyne threshold must be found numerically. TheM-ary decision is obtained using the same post-processing rule as in Sec. II. If exactly one slot is declared as the pulse slot, that slot is selected. If mo...

[11] [11]

Formally, the stateρ i+1 is obtained fromρ i through a symmetry operatorTthat pushes each Kronecker factor by one position (modulo M) to the left

GUS reduction of the phase-diffused ensemble The geometrically uniform symmetry (GUS) structure of the PPM, whether C-PPM or S-PPM, constellation is apparent [11]. Formally, the stateρ i+1 is obtained fromρ i through a symmetry operatorTthat pushes each Kronecker factor by one position (modulo M) to the left. ρm ∼ |Ψ m⟩=T m−1 |Ψ1⟩, m= 1,· · ·, M.(A1) Unde...

[12] [12]

SRM benchmark for high-order phase-diffused PPM To obtain a tractable quantum benchmark for high-order phase-diffused S-PPM, we employ the square-root mea- surement associated with the mixed-state ensemble{ρ pd m }M m=1. This measurement is not assumed to be the exact Helstrom measurement for general mixed-state ensembles; rather, it provides a physically...

[13] [13]

curse of dimensionality

Efficient Numerical Construction of the Gram Matrix Although the SRM error probability has been analytically reduced to operations on the Gram matrixG, constructing Gexplicitly still suffers from the “curse of dimensionality” because the inner productsζ † i ζj reside in an exponentially large tensor-product space (N cut)M. Here we introduce an efficient n...

[14] [14]

I. A. Burenkov, M. V. Jabir, and S. V. Polyakov, Practical quantum-enhanced receivers for classical communication, AVS Quantum Sci.3, 025301 (2021)

2021

[15] [15]

C. W. Helstrom, Quantum detection and estimation theory, J. Stat. Phys.1, 231 (1969)

1969

[16] [16]

S. Dolinar Jr, A near-optimum receiver structure for the detection of m-ary optical ppm signals, The Telecommunications and Data Acquisition Progress Report 42-72, October–December 1982 , 30 (1982)

1982

[17] [17]

M. G. A. Paris, Nearly ideal binary communication in squeezed channels, Phys. Rev. A64, 014304 (2001)

2001

[18] [18]

Chesi, S

G. Chesi, S. Olivares, and M. G. A. Paris, Squeezing-enhanced phase-shift-keyed binary communication in noisy channels, Phys. Rev. A97, 032315 (2018)

2018

[19] [19]

Wang and X

Y. Wang and X. Chen, Joint modulation of 3-ppm and quantum squeezed states in communication systems, inIEEE INFOCOM 2022 - IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)(2022) pp. 1–5

2022

[20] [20]

E. Bai, J. Peng, T. Wu, K. Wen, F. Sun, C. Zhou, Y. Li, Z. Zhang, and C. Dong, Near-optimal discrimination of displaced squeezed binary signals using displacement, inverse-squeezing, and photon-number-resolving detection (2026), arXiv:2601.09073 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2026

[21] [21]

M. G. Genoni, S. Olivares, and M. G. A. Paris, Optical phase estimation in the presence of phase diffusion, Phys. Rev. Lett.106, 153603 (2011)

2011

[22] [22]

M. N. Notarnicola and S. Olivares, A robust hybrid receiver for binary phase-shift keying discrimination in the presence of phase noise, Int. J. Quantum Inf.22, 2450008 (2024)

2024

[23] [23]

Lambert, E

N. Lambert, E. Gigu‘ere, P. Menczel, B. Li, P. Hopf, G. Su’arez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, Qutip 5: The quantum toolbox in Python, Phys. Rep.1153, 1 (2026)

2026

[24] [24]

Cariolaro and G

G. Cariolaro and G. Pierobon, Theory of quantum pulse position modulation and related numerical problems, IEEE Trans. Commun58, 1213 (2010)

2010

[25] [25]

Izumi, M

S. Izumi, M. Takeoka, K. Ema, and M. Sasaki, Quantum receivers with squeezing and photon-number-resolving detectors form-ary coherent state discrimination, Phys. Rev. A87, 042328 (2013)

2013

[26] [26]

Humer, M

G. Humer, M. Peev, C. Schaeff, S. Ramelow, M. Stipˇ cevi´ c, and R. Ursin, A simple and robust method for estimating afterpulsing in single photon detectors, J. Lightwave Technol.33, 3098 (2015)

2015

[27] [27]

J. Chen, J. Habif, Z. Dutton, R. Lazarus, and S. Guha, Optical codeword demodulation with error rates below standard quantum limit using a conditional nulling receiver, Nat. Photon.6(2012)

2012