Introduction

Parameter Estimation in Quantum Systems

In the probe state \(\rho\), the parameter \(\theta\) is encoded through a physical evolution \(\mathcal{E}_\theta\).

Quantum statistical model:

\[\left\{ \rho_\theta = \mathcal{E}_\theta(\rho): \theta \in \Theta \subset \mathbb{R} \right\}\]

Parametric set: \(\Theta\) is the set of possible values of the parameter \(\theta\).

The goal is to infer the value of \(\theta\) by performing measurements on \(\rho_\theta\).

Optical Phase estimation:

  • \(\Theta \subset [0, 2\pi)\).

  • \(\rho\) is a quantum state of light.

Positive Operator-Valued Measure (POVM)

Sample space:

\[\mathcal{X} \subset \mathbb{R}^n\]

Set of events:

\[\text{Borel } \sigma\text{-algebra } \mathcal{B}(\mathcal{X}) \text{ of } \mathcal{X}\]

A POVM is a \(\sigma\)-additive map

\[\Pi: \mathcal{B}(\mathcal{X}) \to \mathcal{L}(\mathcal{H})\]

  • For any \(B_1, B_2, \ldots \in \mathcal{B}(\mathcal{X})\) such that \(B_i \cap B_j = \emptyset\) for \(i \ne j\), we have that \(\Pi(\cup B_i ) = \sum_{i} \Pi(B_i)\).
  • \(\Pi(\mathcal{X}) = 1_{\mathcal{H}}.\)

When \(\mathcal{X} = \left\{x_1,x_2, \ldots, x_n \right\}\) is a discrete set:

\[\Pi = \left\{ \Pi(x); \, x \in \mathcal{X} \right\}; \quad \sum_{x \in \mathcal{X}} \Pi(X) = 1_{\mathcal{H}}.\]

Examples of POVMs

Projective measurements

For any self-adjoint operator \(A\) with spectral decomposition

\[A = \int_{-\infty}^{\infty}\lambda E(d\lambda).\]

The spectral measure \(E_A(d\lambda)\) defines a (continuous) POVM \(\Pi\), such that \(\Pi(d\lambda) = E_A(d\lambda)\), where

  • \(E_A(B)^2 = E_A(B)\) for any Borel set \(B\)
  • \(E(B_1) E(B_2) = 0\) for \(B_1 \cap B_2 = \emptyset\).

Quadrature operator

The operator \(\hat{X}\) define a POVM \[\Pi(dx) = \Pi(dx) = \lvert x \rangle \langle x \rvert dx.\] This POVM can be implemented using homodyne measurements.

Number operator

The operator, defined as \[ \hat{n} = \sum_{n=0}^{\infty} \lvert n \rangle \langle n \rvert, \] defines the photon-counting POVM with elements \[ \Pi(n) = \lvert n \rangle \langle n \rvert. \]

Example of a POVM that is not projective:

The double homodyne (heterodyne) define the POVM \[\Pi(x_1,x_2) = \frac{1}{\pi} \lvert z \rangle \langle z \rvert dz, \quad z = x_1+ix_2.\]

Strategies of Estimation

Any estimation strategy can be characterized by a POVM and an estimator: \[\left( \Pi, \hat{\theta}(X) \right).\]

Estimator

The estimator is a function \(\hat{\theta}(X): \mathcal{X} \to \Theta\) that maps the outcomes of the measurements to the parameter space.

The outcomes of the POVM define a random variable \(X\) with probability distribution \[f(x \mid \theta) = \mathrm{Tr}\left[ \rho_\theta \Pi(x) \right].\]

Mean Squared Error:

\[C(\hat{\theta}(X), \theta) = \mathrm{E}_\theta\left[ \left( \hat{\theta}(X) - \theta \right)^2 \right].\]

Goal:

Find a measurement (POVM) and estimator that minimizes the Mean Squared Error \(\forall \theta \in \Theta\).

🎯 Limits in precision

Unbiased Estimators

\[\mathrm{E}_\theta\left[ \left( \hat{\theta}(X) - \theta \right)^2 \right] = \mathrm{Var}_{\theta}(\hat{\theta}(X)) + b_{\theta}(\hat{\theta}(X))^2\]

\[ \require{cancel} \mathrm{E}_\theta\left[ \left( \hat{\theta}(X) - \theta \right)^2 \right] = \mathrm{Var}_{\theta}(\hat{\theta}(X)) + \cancelto{0}{\color{red}{b_{\theta}(\hat{\theta}(X))^2}}\]

Quantum Cramér-Rao Bound & Fisher Information\(^1\)

\(\mathrm{Var}_{\theta}(\hat{\theta}(X)) \geq \left[ n F_Q(\theta) \right]^{-1}\)

\[\, F_X(\theta) = \mathrm{E}\left[ \left( \frac{\partial}{\partial \theta} \ln\left( f(X \mid \theta) \right) \right)^2 \right], \quad F_Q(\theta) = \underset{POVMs}{max} F_X(\theta)\]

The QFI quantifies the amount of information that the probe state \(\rho_\theta\) carries about the parameter \(\theta\).

A POVM \(\Pi\) with outcome space \(\mathcal{X}\) is optimal if \(F_X(\theta) = F_Q(\theta)\).

The QCRB can be saturated with the maximum likelihood estimator in the asymptotic limit.

Maximum Likelihood Estimator

\(\left\{ \Pi(x) \right\}_{x\in \mathcal{X}} \implies \hat{\theta}_{MLE}(X) = \mathrm{argmax}_{\theta \in \Theta} f(X|\theta)\)

\(^1\)Barndorff-Nielsen, O. E., & Gill, R. D. (2000).

🔦 Optical phase estimation

Ligth to probe systems…

Probe State


Dynamical Evolution

Phase Change


Measurement \(\,\)

Applications: Quantums Sensing, Quantum Imaging, Quantum communications, …

Holevo Variance

In the task of phase estimation, the probability distributions are periodic functions.

  • The linear variance fails into capture this periodicity.

The Holevo variance is a measure of the uncertainty in the estimation of a periodic parameter\(^1\).

\[\mathrm{Var}^{H}_\theta(\hat{\theta}(X)) = \dfrac{\color{blue}{T}^2}{4 \pi^2} \Big\lvert \mathrm{E}_{\theta}\left[ e^{i \frac{2\pi}{\color{blue}{T}} \left( \hat{\theta}(X) \right)} \right] \Big\rvert^{-2} - 1\]

\(^1\)Holevo, A. S. (2011).

Adaptive estimation with coherent states

Phase estimation with coherent states

Shot Noise Limit (classical correlations)

\[\mathrm{Var}^{H}_\theta(\hat{\theta}(X)) \geq \dfrac{1}{4\left(\mathrm{E}\left[ \hat{n} \right] \right)} = \dfrac{1}{4 |\alpha|^2}\]

\[\mathrm{Var}^{H}_\theta(\hat{\theta}(X_1,\ldots,X_{\color{red}{\nu}})) \geq \dfrac{1}{ \color{red}{\nu} \cdot 4 |\alpha|^2} \]

Canonical Phase Measurement\(^1\)

It is the POVM that achieves QCRB saturation with the highest rate of convergence.

\[\Pi_{\text{CPM}}(d\phi) = \frac{1}{2\pi} \lvert \phi \rangle \langle \phi \rvert d\phi, \quad \lvert \phi \rangle = \sum_{n=0}^{\infty} e^{i n \phi} \lvert n \rangle.\]

This POVM cannot be implemented using liner optics, but can be approximated using adaptive methods.

\(^1\)Wiseman, H. M., & Killip, R. B. (1998).

Adaptive estimation strategies

Canonical Phase Measurement

\[\mathrm{Var}^{H}_\theta(\hat{\theta}^{\text{can}}(X)) = \frac{1}{4 |\alpha|^2} + \frac{5}{32 |\alpha|^4} + \mathcal{O}(|\alpha|^{-6})\]

There is no effective way to physically implement the canonical phase measurement.

Mark II (Adaptive Homodyne)

\[\mathrm{Var}^{H}_\theta(\hat{\theta}^{\text{MK II}}(X)) = \frac{1}{4 |\alpha|^2} + \frac{1}{8 |\alpha|^3} + \mathcal{O}(| \alpha|^{-4})\]

The strategy Mark II is the best as far as we know based in gaussian measurements.

\(\quad\) Can we get an improvement using non-gaussian measurements?

\(^1\)Wiseman, H. M., & Killip, R. B. (1998).

Dolinar receiver

Dolinar receiver for phase estimation

\[\Pi_m = \left\{ \Pi(0; \beta), \ldots, \Pi(m; \beta), 1_{\mathcal{H}} - \sum_{i=0}^{m} \Pi(i; \beta) \right\}\]

\[\Pi(n; \beta) = D(\beta)\lvert n \rangle \langle n \rvert D^{\dagger}(\beta)\]

It employs feedback based on photon detection results to dynamically adjust \(\beta\) in the measurement process.

Bayesian experimental design

  • \(\theta \sim \pi(\theta), \,\) \(X \sim \text{Pois}( \lvert \alpha - \beta \rvert^2)\)
  • \(p(\theta \mid x; \beta) \propto f(x \mid \theta; \beta) \pi(\theta)\)
  • \(\hat{\theta}_{\text{MAP}}(X; \beta) = \arg\max_{\theta \in \Theta} p(\theta \mid X; \beta)\)

Mutual information

\[\beta_{\text{opt}} = \arg \max_{\beta \in \mathbb{C}} I_{\beta}(\theta; X) \\ = \arg \max_{\beta \in \mathbb{C}} \left[ H_{\beta}(\theta) - H_{\beta}(\theta \mid X) \right]\]

As the number of adaptive steps increases, this optimization problem becomes computationally intractable due to its high complexity

DiMario, M. T., & Becerra, F. E. (2020).

Reduction of optimization complexity

Asymptotic behavior

Under some regularity conditions on the posterior distribution \(p(\theta; X)\), as the number of adaptive steps \(\nu\) increases:

  • \(\hat{\theta}_{\text{MAP}}(X_1, \ldots, X_\nu) \to \theta\).
  • Its distribution becomes asymptotically normal with variance:

\[ \sigma^2 = \arg \max_{ F \in \mathrm{hull} \{ F_X(\theta; \beta) \} } |F|^{-1} \]

Most informative design

Given that \(\beta = |\beta| \exp(i \phi)\), the Fisher information is given by:

\[F_X(\theta; \beta) = \dfrac{4 |\alpha|^2 | \beta |^2\sin^2(\theta - \phi) }{ \lvert \alpha - \beta \rvert^2 }\]

Thus, the optimal choice of \(|\beta|\) is:

\[|\beta|_{\text{opt}} = \frac{|\alpha|^2}{\cos(\theta - \phi)} \implies F_X(\theta; \beta_{\text{opt}}) = F_{Q}(\theta).\]

Since the MAP estimator is asymptotically consistent, the corresponding estimated optimal amplitude is:

\[|\hat{\beta}|_{\text{opt}} = \frac{|\alpha|^2}{\cos(\hat{\theta}_{\text{MAP}} - \phi)}.\]

Rodríguez-García, M. A., DiMario, M. T., Barberis-Blostein, P., & Becerra, F. E. (2022).

Performance of the MAP estimator

\[\color{blue}{\mathrm{Var}^{H}_\theta(\hat{\theta}^{\text{can}}(X)) = \frac{1}{4 |\alpha|^2} + \frac{5}{32 |\alpha|^4} + \mathcal{O}(|\alpha|^{-6})}\\ \leq \color{darkgreen}{\mathrm{Var}^{H}_\theta(\hat{\theta}^{\text{NG}}(X)) \approx \frac{1}{4 |\alpha|^2} + \frac{0.53}{|\alpha|^4}} \\ \leq \color{darkred}{\mathrm{Var}^{H}_\theta(\hat{\theta}^{\text{MK II}}(X)) = \frac{1}{4 |\alpha|^2} + \frac{1}{8 |\alpha|^3} + \mathcal{O}(| \alpha|^{-4})}\]

Rodríguez-García, M. A., DiMario, M. T., Barberis-Blostein, P., & Becerra, F. E. (2022).

Adaptive estimation with squeezed vaccum states

Phase estimation with squeezed vaccum states

Heisenberg scaling (quantum correlations)

\[\, F_Q = 8\left(\mathrm{E}\left[ \hat n \right]^2 + \mathrm{E}\left[ \hat n \right] \right) \, \]

Canonical phase measurement

There is no effective way to physically implement it.

Non-gaussian measurements

Requires a PNR\(\sim 12-20\).

Gaussian measurements

Heterodyne Measurement

Homodyne Measurement

POVM

\[\Pi_Z(dz) = \frac{1}{\pi} \lvert z \rangle \langle z \rvert d\alpha,\]

\[f(z \mid \theta) = \mathrm{Tr}\left[ \Pi_Z(z) \rho(\theta) \right]\] \[ f(z \mid \theta) \propto \exp(-|z|^2 - \tanh(-r) \mathrm{Re}(z^2 e^{2i\theta} )),\] \[r = |\xi|\]

\(F_Z = 4 \sinh^2(r) < F_Q\)

POVM

\[ \Pi_X(dx) = \lvert x \rangle \langle x \rvert dx, \]

\[f(x \mid \theta) = \mathrm{Tr}\left[ \Pi_X(x) \rho(\theta) \right],\] \[x \sim \mathcal{N}(0, \sigma^2(\theta))\] \[\sigma^2(\theta) = e^{-2r}\cos^2(\theta) + e^{2r}\sin^2(\theta)\]

\(\, \theta_{\mathrm{opt}} = \text{arccos}\left(\tanh(2r)\right)/2\)

Nagaoka’s two-step method1

  • We split the total number of probe states \(N\) into two chuncks

    • \(N = \sqrt{N} + (N- \sqrt{N})\)

  • We perform \(\sqrt{N}\) homodyne measurements on the state \(\rho(\theta)\)

    • \(\vec{X}_{\sqrt{N}} = X_1, \ldots, X_{\sqrt{N}}\)

  • We use the Maximum Likelihood Estimator (MLE)

    • \(\hat{\theta}(\vec{X}_{\sqrt{N}}) = \arg\max_{\theta \in \Theta} \prod_{i=1}^{N} f(X_i \mid \theta)\)

  • We rotate the remaining \(N-\sqrt{N}\) probe states by \(\hat{\theta}(X_1, \ldots, X_{\sqrt{N}}) + \theta_{\text{opt}}\) angles and measure them

    • \(\rho(\theta) \rightarrow \rho(\theta - \hat{\theta}(X_1, \ldots, X_{\sqrt{N}}) + \theta_{\text{opt}})\)

  • We rotate the remaining \(N-\sqrt{N}\) probe states by \(\hat{\theta}(X_1, \ldots, X_{\sqrt{N}}) + \theta_{\text{opt}}\) angles and measure them

    • \(\rho(\theta) \rightarrow \rho(\theta - \hat{\theta}(X_1, \ldots, X_{\sqrt{N}}) + \theta_{\text{opt}}) \approx \rho(\theta_{\text{opt}}), \quad N \to \infty\)

Phase Estimation with Squeezed States1

Parameters: (\(r = 1, N = 3705\) )


  • \(r\): Squeezing strength,
  • \(N\): Total number of probe states.

Set of locally optimal POVMs

\(\,\)

Rotations + Homodyne

\[\Pi_x = \lvert x \rangle \langle x \rvert\]

\[\Pi_{x}(\varphi) = \lvert x_\varphi \rangle \langle x_\varphi \rvert = U(\varphi)\Pi_xU^{\dagger}(\varphi)\]

Locally optimal measurement

\[\varphi = \check{\theta} - \theta_{\text{opt}}\]

\[f(x \mid \theta, \Pi_{x}(\varphi)) = f(x \mid \varphi) = f(x \mid \theta + \theta_{\text{opt}} - \check{\theta})\] \[\varphi_{\text{opt}} = \theta + \theta_{opt}, \, \theta \in [0, \pi/2)\]

Adaptive Phase Estimation

Proposal

  • For a set of \(N\) input squeezed probe states \({\lvert 0, \xi\rangle \langle0, \xi\rvert}\), the dynamical evolution \(U(\theta)\) encodes the parameter \(\theta\) into the probes.
  • We implement the measurement \(\left \{ \Pi_X(\varphi_1) \right\}\) with \(\varphi_1 \sim [0, \pi/2)\) over \(\nu= N/m\) probe states to get \(\vec{X}_{\nu}(\varphi_1)\).
  • The MLE is applied to obtain an estimate, and we use the estimate to adjust the phase of the homodyne measurement \(\varphi_2 = \hat{\theta}(\vec{X}_{\nu}(\varphi_1)) + \theta_{\text{opt}}\).
  • The process is repeated \(m\) times iteratively during the strategy.

Rodríguez-García, M. A., & Becerra, F. E. (2024).

📊 Asymptotically consistent estimator

Almost Sure Convergence

Theorem

Let \(\theta_0 \in \tilde{\Theta}\), with \(\tilde{\Theta}\) a compact set, and suppose that the regularity conditions are satisfied in the statistical model of each adaptive step.

\[ \widehat{\theta}(\vec{X}_\nu(\varphi_1) ,\ldots, \vec{X}_\nu(\varphi_m)) \xrightarrow{a.s} \theta_0\]

\[\lvert \widehat{\theta}(\vec{x}_\nu(\varphi_1) ,\ldots,\vec{x}_\nu(\varphi_m) ) - \theta_0 \rvert \geq a.\]

  • To bound the probability of observing “far” estimates of the real parameter \(\theta_0\) after \(m\) adaptive steps.
  • Moreover, this bound decays exponentially with \(m\).

\[P_{\theta_0}\left( \lvert \widehat{\theta}(\check{x}_{\nu}(\varphi_1) ,\ldots, \check{x}_{\nu}(\varphi_m) ) - \theta_0 \rvert \geq a \right) \leq j \exp\left[ - \min_{1 \leq i \leq j}[b_i] \, m \right], \, m \geq 1. \]

Rodríguez-García, M. A., & Becerra, F. E. (2024).

Asymptotic normality

Limit distribution

\[ \hat{\theta}(\vec{X}_\nu(\varphi_1), \ldots, \vec{X}_{\nu}(\varphi_{m}) ) \xrightarrow{d} \mathcal{N}\left( \theta_0, \frac{1}{N F_Q(\theta_0)} \right)\]

Convergence rate

💻 Numerical Experiments (\(r = 1, N = 3705, \nu = 1235, 741, 247\) )[^3]



  • \(r\): Squeezing Strength,
  • \(N\): Total number of probe states,
  • \(m:\) Number of adaptive steps,
  • \(\nu:\) Sample size per adaptive step.

\([1]\) Berni, A., Gehring, T., Nielsen, B., & Andersen, U. L. (2015).

Extension to the full range of phases

Complete phase range: \([0, \pi)\)


Heterodyne preliminary sampling

\[\Pi_{z} = \frac{1}{\pi}\lvert z \rangle \langle z \rvert, \, z \in \mathbb{C}.\]

\[\varphi_2 = \hat{\theta}\left( Z_1, \ldots, Z_\nu \right) + \theta_{\text{opt}} \rightarrow \Pi_x\left( \varphi_2 \right)\]

For full period \([0, 2\pi)\) phase estimation, it is necessary to use displaced states.

📃 Numerical Results

Monte Carlo Simulations

🧭 Summary & Takeaway Message

Adaptive estimation strategies are essential tools for extracting phase information in quantum systems with high precision using feasible measurements.

  • By using locally optimal POVMs it is possible to construct estimators that are: asymptotically consistent and statistically efficient.
  • Phase estimation with coherent states allow significant gains using non-Gaussian adaptive schemes, while squeezed states unlock quantum-enhanced scaling even with realistic homodyne measurements.

The proposed strategies offer a scalable path to quantum-limited phase estimation over a broad range of parameters using physically accessible resources.

Thanks