Introduction
Phase estimation is a fundamental problem in quantum metrology, where we aim to estimate an unknown phase parameter $\phi$ with the highest possible precision. In interferometry, the choice of input quantum state dramatically affects the measurement sensitivity. This article explores how to optimize quantum states for phase estimation, comparing classical and quantum strategies.
Theoretical Background
Fisher Information and Quantum Cramér-Rao Bound
The precision of any unbiased estimator is bounded by the Cramér-Rao bound:
$$\Delta\phi \geq \frac{1}{\sqrt{M F(\phi)}}$$
where $M$ is the number of measurements and $F(\phi)$ is the Fisher information. For quantum states, the quantum Fisher information (QFI) provides the ultimate bound:
$$F_Q(\phi) = 4(\langle\partial_\phi\psi|\partial_\phi\psi\rangle - |\langle\psi|\partial_\phi\psi\rangle|^2)$$
Phase Encoding in Interferometry
Consider a Mach-Zehnder interferometer where the phase $\phi$ is encoded via the unitary operator:
$$U(\phi) = e^{-i\phi \hat{n}}$$
where $\hat{n}$ is the photon number operator. For $N$ photons, we compare:
- Classical strategy (coherent state): $|\alpha\rangle$ with $F_Q = N$
- Quantum strategy (NOON state): $\frac{1}{\sqrt{2}}(|N,0\rangle + |0,N\rangle)$ with $F_Q = N^2$
The NOON state achieves the Heisenberg limit, providing quadratic improvement over the shot-noise limit.
Problem Setup
We investigate phase estimation with $N=10$ photons, comparing:
- Coherent state input
- NOON state input
- Squeezed state input
We calculate the quantum Fisher information and phase sensitivity for each state across different phase values.
Python Implementation
1 | import numpy as np |
Code Explanation
Class Structure
The QuantumPhaseEstimation class encapsulates all functionality for comparing different quantum states in interferometric phase estimation:
Initialization: Sets the number of photons $N$, which determines the quantum resources available.
QFI Calculation Methods: Each quantum state has its own QFI formula:
coherent_state_qfi(): Returns $F_Q = N$, representing the shot-noise limitnoon_state_qfi(): Returns $F_Q = N^2$, achieving the Heisenberg limitsqueezed_state_qfi(): Returns $F_Q = N e^{2r}$, where $r$ is the squeezing parameter
Phase Sensitivity: The phase_sensitivity() method implements the quantum Cramér-Rao bound:
$$\Delta\phi = \frac{1}{\sqrt{M \cdot F_Q(\phi)}}$$
Measurement Simulation: The simulate_measurement() method generates realistic measurement outcomes by:
- Computing the expected signal based on the interference pattern
- Adding appropriate quantum noise (different for each state type)
- Returning $M$ simulated measurements
For NOON states, the signal oscillates at frequency $N\phi$ instead of $\phi$, providing enhanced sensitivity but also introducing phase ambiguity.
Visualization Components
Plot 1 - QFI Comparison: Shows that QFI is phase-independent for these states. The NOON state provides $N^2 = 100$ times more Fisher information than the coherent state.
Plot 2 - Phase Sensitivity: Displays the achievable precision on a logarithmic scale. Lower values indicate better precision. NOON states achieve $N = 10$ times better sensitivity.
Plot 3 - Scaling Analysis: Demonstrates the fundamental difference in scaling:
- Coherent states: $\Delta\phi \propto N^{-1/2}$ (classical scaling)
- NOON states: $\Delta\phi \propto N^{-1}$ (quantum scaling)
This log-log plot clearly shows the different slopes, confirming theoretical predictions.
Plot 4-5 - Measurement Histograms: Visualize the distribution of actual measurement outcomes. The narrower distribution for NOON states reflects reduced uncertainty.
Plot 6 - 3D Surface: Shows how NOON state sensitivity improves with both increasing photon number and varies across phase space. The surface demonstrates that precision improves dramatically with more photons.
Performance Optimization
The code is optimized for Google Colab execution:
- Vectorized NumPy operations instead of loops where possible
- Pre-allocated arrays for 3D mesh calculations
- Efficient histogram binning for measurement simulations
- Moderate resolution (30×30) for 3D plots to balance quality and speed
Physical Interpretation
The results demonstrate the quantum advantage in metrology:
Shot-noise vs Heisenberg limit: Classical coherent states are limited by statistical fluctuations scaling as $\sqrt{N}$. Quantum NOON states exploit entanglement to achieve the fundamental Heisenberg limit scaling as $N$.
Practical considerations: While NOON states offer superior sensitivity, they are also more fragile and difficult to prepare experimentally. Squeezed states provide an intermediate option with moderate enhancement and better practical feasibility.
Measurement tradeoff: The NOON state’s $N$-fold enhanced oscillation frequency provides better sensitivity but creates $N$-fold phase ambiguity, requiring additional measurements or prior knowledge to resolve.
Execution Results
====================================================================== QUANTUM PHASE ESTIMATION: SENSITIVITY OPTIMIZATION ====================================================================== System Parameters: Number of photons (N): 10 Number of measurements (M): 1000 Quantum Fisher Information (constant across phase): Coherent state: F_Q = 10.00 NOON state: F_Q = 100.00 Squeezed state: F_Q = 27.18 Phase Sensitivity (Δφ): Coherent state: 0.010000 rad NOON state: 0.003162 rad Squeezed state: 0.006065 rad Quantum Advantage: NOON vs Coherent: 3.16x improvement Squeezed vs Coherent: 1.65x improvement Scaling Analysis (at φ = π/4): Coherent state: Δφ ∝ 1/√N (shot-noise limit) NOON state: Δφ ∝ 1/N (Heisenberg limit) With N=10, NOON state achieves 10x better sensitivity ====================================================================== Measurement Statistics: Coherent State: Mean: 0.5018 Std Dev: 0.3108 Estimated phase precision: 0.310840 rad Noon State: Mean: -0.5019 Std Dev: 0.1027 Estimated phase precision: 0.102662 rad Squeezed State: Mean: 0.4905 Std Dev: 0.1902 Estimated phase precision: 0.190245 rad