Fisher symmetry

and the geometry of quantum states

Jonathan A. Gross, Ninnat Dangniam, Carlton M. Caves

Center for Quantum Information and Control, University of New Mexico

Overview

Goal: Measure deviations from a reference quantum state with equal precision in all directions
Background: How do deviations in different directions relate?

Different directions

You've never heard of the Millennium Falcon?…It's the ship that made the Kessel Run in less than twelve parsecs.

We will use the concept of statistical distance to naturally relate all the directions in quantum state space

Statistical distance

Quantify distance between two quantum states by how statistically distinguishable they are

More distinguishable: further apart

Less distinguishable: closer together

Example: coin bias

Count the number of intermediary distinguishable states

W. K. Wootters, Phys. Rev. D 23, 357 (1981)

Example: coin bias

Count the number of intermediary distinguishable states

W. K. Wootters, Phys. Rev. D 23, 357 (1981)

Example: coin bias

\begin{align} ds\sim\frac{dp}{\Delta \hat{p}} \end{align}

Statistical fluctuations

The fluctuations depend on the estimator used

The Cramér–Rao bound gives us an achievable lower bound

\begin{align} \mathbb{E}\left[(\Delta\hat{x})^2\right]&\geq\frac{1}{F(x)} \end{align}

This allows us to generally define our statistical distance in terms of the Fisher information (FI): $ds\sim\sqrt{F(x)}dx$

Fisher information

\begin{align} F(x)&=\sum_\xi\operatorname{Pr}(\Xi=\xi|x)\left(\frac{d}{dx}\ln\,\operatorname{Pr}(\Xi=\xi|x)\right)^2 \\ &=\sum_\xi\frac{\left(\frac{d}{dx}\operatorname{Pr}(\Xi=\xi|x)\right)^2}{\operatorname{Pr}(\Xi=\xi|x)} \end{align}

The FI quantifies the information $\Xi$ has about $x$

Multiple parameters

So far we've only discussed fluctuations in 1 "direction"

If you're trying to measure multiple parameters, the FI becomes a matrix

\begin{align} F_{\alpha\beta}(\mathbf{x})&=\sum_\xi\frac{\frac{\partial}{\partial x^\alpha}\operatorname{Pr}(\Xi=\xi|\mathbf{x}) \frac{\partial}{\partial x^\beta}\operatorname{Pr}(\Xi=\xi|\mathbf{x})}{\operatorname{Pr}(\Xi=\xi|\mathbf{x})} \end{align}

Diagonal elements are the FIs for individual parameters

This matrix can be used as a metric: $ds\sim\sqrt{d\mathbf{x}^\mathsf{T}F(\mathbf{x})d\mathbf{x}}$

Quantum mechanics

Random variable defined by a positive-operator-valued measure (POVM), $\{E^\xi\}$

\begin{align} \operatorname{Pr}(\Xi=\xi|\mathbf{x})&=\operatorname{tr}(E^\xi\rho(\mathbf{x})) & \sum_\xi E^\xi=I \end{align}

We write the FI generated by a particular POVM as

\begin{align} C_{\alpha\beta}(\mathbf{x})&=\sum_\xi\frac{ \operatorname{tr}\left(E^\xi\left.\frac{\partial\rho}{\partial x^\alpha}\right\vert_\mathbf{x}\right) \operatorname{tr}\left(E^\xi\left.\frac{\partial\rho}{\partial x^\beta}\right\vert_\mathbf{x}\right)} {\operatorname{tr}\left(E^\xi\rho(\mathbf{x})\right)} \end{align}

Quantum Fisher information

\begin{align} ds_\mathbf{v}&\sim\max_{\{E^\xi\}}\sqrt{\mathbf{v}^\mathsf{T}C(\mathbf{x})\mathbf{v}} \hphantom{=\sqrt{\mathbf{v}^\mathsf{T}Q(\mathbf{x})\mathbf{v}}\,\,\,\,} \end{align}

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994)

Quantum Fisher information

\begin{align} ds_\mathbf{v}&\sim\max_{\{E^\xi\}}\sqrt{\mathbf{v}^\mathsf{T}C(\mathbf{x})\mathbf{v}} \hphantom{=\sqrt{\mathbf{v}^\mathsf{T}Q(\mathbf{x})\mathbf{v}}\,\,\,\,} \end{align}

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994)

Quantum Fisher information

\begin{align} ds_\mathbf{v}&\sim\max_{\{E^\xi\}}\sqrt{\mathbf{v}^\mathsf{T}C(\mathbf{x})\mathbf{v}} \hphantom{=\sqrt{\mathbf{v}^\mathsf{T}Q(\mathbf{x})\mathbf{v}}\,\,\,\,} \end{align}

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994)

Quantum Fisher information

\begin{align} ds_\mathbf{v}&\sim\max_{\{E^\xi\}}\sqrt{\mathbf{v}^\mathsf{T}C(\mathbf{x})\mathbf{v}} \hphantom{=\sqrt{\mathbf{v}^\mathsf{T}Q(\mathbf{x})\mathbf{v}}\,\,\,\,} \end{align}

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994)

Quantum Fisher information

\begin{align} ds_\mathbf{v}&\sim\max_{\{E^\xi\}}\sqrt{\mathbf{v}^\mathsf{T}C(\mathbf{x})\mathbf{v}} =\sqrt{\mathbf{v}^\mathsf{T}Q(\mathbf{v})d\mathbf{x}} \end{align}

S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994)

State-space geometry

Consider equatorial plane of the Bloch ball

As in the simplex case, Euclidean distance doesn't reflect statistical distance

State-space geometry

Just like the simplex is naturally a semicircle, the Bloch ball is naturally the upper hemisphere of $S^3$

Simultaneous measurement

QFI gives bounds for individual parameter estimation

The Gill–Massar bound gives us simultaneous bounds on multiple parameters for a $d$ -dimensional system

\begin{align} \operatorname{tr}\left(Q^{-1}C\right)&\leq d-1 \\ Q^{\alpha\beta}C_{\alpha\beta}&\leq d-1 \end{align}

Saturated iff $\{E^\xi\}$ is rank-1

R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000)

Gill–Massar bound

Through change of parameters we can set $Q=I$

In this natural parametrization

\begin{align} \operatorname{tr}C&\leq d-1 \end{align}

If we measured all parameters with quantum-limited accuracy

\begin{align} \operatorname{tr}C&=\begin{cases} 2d-2 & \mathrm{pure} \\ d^2-1 & \mathrm{general} \end{cases} \end{align}

Gill–Massar bound

Fisher symmetry

Can we saturate bound and locally measure all parameters with the same fraction of the quantum-limited accuracy?

\begin{align} C(\mathbf{x})&=\begin{cases} \frac{1}{2}Q(\mathbf{x}) & \mathrm{pure} \\ \frac{1}{d+1}Q(\mathbf{x}) & \mathrm{general} \end{cases} \end{align}

We call a measurement with this property a Fisher-symmetric measurement (FSM)

Fisher symmetry

Pure states

Li et al. have shown FSMs to exist for pure states in all dimensions and given an explicit construction

N. Li, C. Ferrie, and C. M. Caves, arXiv:1507.06904

Mixed states

A symmetric informationally complete POVM (SIC-POVM) is a FSM at the maximally mixed state, provided it exists

\begin{align} E^\xi&\propto\Pi^\xi \\ \operatorname{tr}\left(\Pi^\xi\Pi^\eta\right)&=\mathrm{const.} & \xi&\neq\eta \end{align}

Mixed states

For qubits, can explicitly calculate the FSM for an arbitrary mixed state

\begin{align} E^\xi&=\left\vert\phi_\xi\middle\rangle\middle\langle\phi_\xi\right\vert \\ \left\vert\phi_0\right\rangle&=\frac{1}{\sqrt{2-z}}\left\vert0\right\rangle \\ \left\vert\phi_j\right\rangle&=\frac{1}{\sqrt{3}}\left( \sqrt{\frac{1-z}{2-z}}\left\vert0\right\rangle+ e^{i\varphi_j}\left\vert1\right\rangle \right) \end{align}

POVM

| ϕ_{j} ⟩ = \frac{1}{\sqrt{3}} (\sqrt{\frac{1 - z}{2 - z}} | 0 ⟩ + e^{i φ_{j}} | 1 ⟩)

POVM

| ϕ_{j} ⟩ = \frac{1}{\sqrt{3}} (\sqrt{\frac{1 - z}{2 - z}} | 0 ⟩ + e^{i φ_{j}} | 1 ⟩)

POVM

| ϕ_{j} ⟩ = \frac{1}{\sqrt{3}} (\sqrt{\frac{1 - z}{2 - z}} | 0 ⟩ + e^{i φ_{j}} | 1 ⟩)

Back to the coin

I need more coin!

"Walken-Cowbell". Licensed under Fair use via Wikipedia.

Back to the coin

POVM

| ϕ_{j} ⟩ = \frac{1}{\sqrt{3}} (\sqrt{\frac{1 - z}{2 - z}} | 0 ⟩ + e^{i φ_{j}} | 1 ⟩)

POVM

| ϕ_{j} ⟩ = \frac{1}{\sqrt{3}} (\sqrt{\frac{1 - z}{2 - z}} | 0 ⟩ + e^{i φ_{j}} | 1 ⟩)

POVM

| ϕ_{j} ⟩ = \frac{1}{\sqrt{3}} (\sqrt{\frac{1 - z}{2 - z}} | 0 ⟩ + e^{i φ_{j}} | 1 ⟩)

Higher dimensions

SIC-POVMs are hard to find, so it looks like FSMs are going to be hard to find (if they exist at all)

Easier problem (?): given a SIC-POVM, can one transform it to be a FSM at an arbitrary state?

Since FSMs are intimately related to the metric, parallel transport recommends itself as a metric-dependent means of transforming objects on state space