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Error Probability Analysis of Quantum Communication with Phase-squeezed M-PSK

This paper presents a theoretical and numerical analysis of the symbol error probability for phase-squeezed M-PSK communication using an adaptive Mark-II receiver, demonstrating that phase squeezing significantly enhances photon efficiency and proposing computationally efficient approximation models that closely match rigorous operator-based calculations.

Original authors: Nikos A. Mitsiou, Ioannis Krikidis

Published 2026-06-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikos A. Mitsiou, Ioannis Krikidis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a secret message using a flashlight in the dark. In the world of "classical" communication, you might just wiggle the light's brightness or color. But in the quantum world, the paper you provided suggests a smarter way: instead of just turning the light on and off, you wiggle the direction the light is pointing. This is called M-PSK (M-ary Phase-Shift Keying). Think of it like a clock face where you send a message by pointing the hand at 12 o'clock, 3 o'clock, 6 o'clock, etc.

The problem? Quantum physics has a rule called the "Uncertainty Principle." It's like trying to hold a spinning top perfectly still; if you try to pin down exactly where the hand is pointing (the phase), it starts to wobble randomly. This "wobble" is noise, and if the wobble is too big, the receiver might think you pointed at 12 o'clock when you actually meant 1 o'clock. This causes errors.

The Big Idea: Squeezing the Noise

The authors propose a clever trick called Phase Squeezing.

Imagine your signal is a balloon filled with air (the noise). Usually, this balloon is perfectly round. If you squeeze it, it stops being round; it gets thinner in one direction and fatter in the other.

The Trick: The authors squeeze the balloon so that it becomes very thin in the direction of the "clock hand" (the phase) but very fat in the direction of the "radius" (how far out the hand is).
Why? Because for this specific type of message, we only care about the angle of the hand, not how far out it is. By making the "angle wobble" tiny (squeezing it), we make the signal much clearer, even if the "distance wobble" gets bigger.

The Receiver: The "Mark-II" Eye

To read these messages, you need a special detector. The paper uses a device called the Adaptive Mark-II Receiver.

Analogy: Imagine trying to guess the direction of a spinning top while it's wobbling. A standard camera just takes a blurry snapshot. The Mark-II receiver is like a smart camera that constantly adjusts its focus and angle in real-time to track the top as it spins. It's not perfect, but it's the best physically possible way to measure the angle without breaking the laws of physics.

What They Found

The researchers did the math to see if this "squeezed balloon" idea actually works better than the standard "round balloon" (coherent states).

It depends on how many "clock positions" you use:
- If you only have 4 positions (like a compass: N, E, S, W), squeezing doesn't help. The math shows the extra "fatness" in the radius cancels out the benefit of the thinness in the angle.
- But, if you have many positions (like a clock with 8, 16, or 32 numbers), squeezing works wonders. The more crowded the clock face is, the more you need to stop the "angle wobble."
The Results:
- For complex messages (high-order M-PSK), squeezing the noise can almost double the efficiency. This means you can send the same amount of information using half the number of photons (light particles), or send much more information with the same amount of light.
- They developed two new "shortcuts" (mathematical approximations) to predict how well this works. These shortcuts are much faster to calculate than the full, complex quantum math, but they are still very accurate (within a few photons of the exact answer).

The Bottom Line

The paper proves that by "squeezing" the quantum noise in the right direction, we can make quantum communication much more reliable, especially when we are trying to send complex messages using very few particles of light. It's like taking a shaky, blurry photo and using a special filter to sharpen the edges exactly where you need them, making the picture clear enough to read even in the dark.

Technical Summary: Error Probability Analysis of Quantum Communication with Phase-squeezed M-PSK

Problem Statement
This paper investigates the Symbol Error Probability (SEP) of $M$ -ary Phase-Shift Keying (M-PSK) in quantum optical communication, specifically focusing on the use of phase-squeezed states. While coherent states are the standard for quantum optical communication, their isotropic noise distribution in phase space limits distinguishability under strict photon budgets. Squeezed states offer a mechanism to redistribute quantum uncertainty, reducing noise in one quadrature at the expense of the conjugate. However, the specific impact of phase squeezing on M-PSK constellations with $M > 4$ remains unexplored. Previous literature suggested that for low-order constellations (BPSK and QPSK), the benefits of angular noise reduction are offset by increased radial noise. This paper addresses whether symbol-dependent phase squeezing can improve performance for higher-order M-PSK constellations when detected by a physically realizable phase receiver.

Methodology
The authors analyze the system under a fixed total photon budget, where the mean photon number per symbol is the sum of displacement photons and squeezing photons. The proposed constellation utilizes displaced-squeezed states where the squeezing angle is dependent on the symbol index, aligning the low-variance (squeezed) quadrature with the local phase-sensitive direction of each PSK symbol.

To evaluate performance, the paper employs the adaptive Mark-II receiver, a physically realizable phase measurement scheme that approximates an ideal phase measurement. The analysis proceeds through three distinct analytical layers:

Exact Quantum Analysis (Fock Basis): The authors derive an exact SEP expression using the Phase Probability Operator Measure (POM) of the Mark-II scheme. The measured phase distribution is expressed as a weighted Fourier series in the Fock basis, determined by the Mark-II matrix elements and the Fock coefficients of the displaced-squeezed states. The exact SEP is calculated by integrating this distribution over the decision wedges defined by the M-PSK constellation.
Polar-Domain Statistical Analysis: To avoid the computational complexity of the Fock-basis truncation, a statistical approach is developed in the polar domain. This method models the state-induced noise as independent Gaussian radial and tangential components. The phase distribution of the received state is derived in closed form, and the total measured phase distribution is obtained by convolving this state-induced distribution with the Gaussian phase error distribution inherent to the Mark-II receiver.
Closed-Form Approximation (Effective Tangential Variance): A further simplification is introduced by modeling the Mark-II phase uncertainty as an additional tangential noise component. This allows the derivation of a closed-form SEP expression involving Owen's T-function, which accounts for the joint effects of displacement, radial anti-squeezing, tangential noise reduction, and receiver phase uncertainty.

Key Contributions

Novel Constellation Design: The paper proposes and analyzes a phase-squeezed M-PSK constellation where the squeezing angle is symbol-dependent, specifically optimized for higher-order constellations ( $M > 4$ ) where the error mechanism is dominated by phase boundary crossings rather than radial separation.
Exact SEP Formulation: The authors provide the first exact SEP expression for phase-squeezed M-PSK under the adaptive Mark-II measurement, utilizing the Mark-II POM and Fock-basis coefficients.
Tractable Analytical Tools: Two approximation methods are developed: a polar-domain analysis involving angular convolution and a closed-form approximation using Owen's T-function. These tools offer computational efficiency while maintaining high accuracy.
Performance Characterization: The study quantifies the trade-offs between squeezing amplitude, photon budget, and constellation order, demonstrating that optimal squeezing exists and is non-monotonic with respect to the squeezing parameter.

Results
Numerical results comparing the exact POM benchmark, the polar-domain analysis, the Owen's T-function approximation, and standard coherent state transmission yield the following findings:

Higher-Order Gains: Phase squeezing substantially reduces the SEP for higher-order M-PSK constellations (e.g., $M=16, 32$ ) compared to coherent states. The gains increase with the constellation order.
Low-Order Behavior: Consistent with prior findings for low-order PSK, the analysis confirms that QPSK ( $M=4$ ) yields an optimal squeezing value of zero ( $r^*=0$ ), as the circular symmetry of the noise is optimal for this geometry.
Photon Efficiency: For the investigated scenarios, phase squeezing can nearly double the photon efficiency of M-PSK as the mean number of transmitted photons increases. Specifically, the required photon number to achieve a target SEP is reduced by approximately 20% for $M=8$ , 50% for $M=16$ , and 66% for $M=32$ .
Accuracy of Approximations: Both proposed approximation methods closely track the exact POM analysis. The polar-domain method is slightly more accurate (within 2 photons), while the Owen's T-function approximation is typically within 2–4 photons of the exact benchmark.
Optimal Squeezing: The optimal squeezing amplitude $r^*$ increases with the constellation order but saturates at higher total photon numbers, indicating a balance between noise reduction and the energy cost of squeezing.

Significance and Claims
The paper claims that its primary significance lies in demonstrating that phase squeezing is a viable strategy for enhancing the distinguishability of higher-order M-PSK constellations in photon-limited regimes, provided a symbol-dependent squeezing strategy is employed. The authors assert that their proposed approximations provide accurate and computationally efficient tools for analyzing phase-squeezed quantum M-PSK communication, bridging the gap between complex exact quantum calculations and practical system design. The work validates the Mark-II phase measurement model for squeezed states and highlights the geometric intuition that aligning the squeezed quadrature with the phase-sensitive direction of the symbol is crucial for performance gains in high-order modulations.

The Big Idea: Squeezing the Noise

The Receiver: The "Mark-II" Eye

What They Found

The Bottom Line

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