Fisher symmetry

and the geometry of quantum states

Jonathan A. Gross, Amir Kalev, Howard Barnum, Carlton M. Caves

Center for Quantum Information and Control, University of New Mexico

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

How far apart are coins with different biases?

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

Example: coin bias

How far apart are coins with different biases?

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{p})^2\right]&\geq\frac{1}{F(p)} \end{align}

This allows us to generally define our statistical distance in terms of the Fisher information (FI)

\begin{align} ds&\sim\frac{dp}{\Delta\hat{p}}\to\sqrt{F(p)}\,dp \end{align}

Fisher information

\begin{align} F(p)&=\sum_{\text{Outcomes}}\operatorname{Pr}(\text{outcome}|p) \left\{\frac{d}{dp}\ln\Big[\operatorname{Pr}(\text{outcome}|p) \Big]\right\}^2 \\ &=\sum_{\text{Outcomes}}\frac{\left[\frac{d}{dp} \operatorname{Pr}(\text{outcome}|p)\right]^2} {\operatorname{Pr}(\text{outcome}|p)} \end{align}

The FI quantifies the information a random variable provides about a parameter

Coin FI

\begin{align} \operatorname{Pr}(H|p)&=p & F(p)&=\frac{1^2}{p}+\frac{(-1)^2}{1-p} \\ \operatorname{Pr}(T|p)&=1-p & &=\frac{1}{p(1-p)} \end{align}

Kullback–Leibler divergence

\begin{align} \mathcal{D}(f\Vert g)&:=\int dx\,f(x)\, \ln\left(\frac{f(x)}{g(x)}\right) \end{align}

Quantifies how likely you are to mistake one distribution for another (not symmetric)

Locally it defines the same distance as the FI

\begin{align} \mathcal{D}(f_p\Vert f_{p+\delta p})&=\frac{1}{2}\delta p^2\,F(p) \end{align}

Multiple parameters

\begin{align} F_{jk}(\mathbf{p})&=\sum_\text{Outcomes} \frac{\partial_j\operatorname{Pr}(\text{outcome}|\mathbf{p}) \,\partial_k\operatorname{Pr}(\text{outcome}|\mathbf{p})} {\operatorname{Pr}(\text{outcome}|\mathbf{p})} \end{align}

This matrix can be used as a metric

\begin{align} ds&\sim\sqrt{d\mathbf{p}^\mathsf{T}F(\mathbf{p})\,d\mathbf{p}} \end{align}

Including 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}[E^\xi\partial_\alpha\rho(\mathbf{x})] \operatorname{tr}[E^\xi\partial_\beta\rho(\mathbf{x})]} {\operatorname{tr}[E^\xi\rho(\mathbf{x})]} \end{align}

Measurement dependence

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Measurement dependence

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Measurement dependence

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Measurement dependence

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Quantum Fisher information

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

Gill–Massar bound

Assume Hilbert space of finite dimension $d$

\begin{align} \operatorname{tr}[Q^{-1}C]&\leq d-1 & Q^{\alpha\beta}C_{\alpha\beta}&\leq d-1 \end{align}

The bound is saturable

\begin{align} \operatorname{tr}[Q^{-1}C]&=d-1 & E^\xi&=|\phi^\xi\rangle\langle\phi^\xi| \end{align}

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

Fisher symmetry

Can choose parameters so $Q=I$ .

Saturate GM bound: $\sum_jC_{jj}=d-1$

Fisher symmetry

Can choose parameters so $Q=I$ .

Saturate GM bound: $\sum_jC_{jj}=d-1$

Fisher symmetry

\begin{align} C_\text{FS}&=\begin{cases} \frac{1}{2}Q & \text{pure} \\ \frac{1}{d+1}Q & \text{full rank} \end{cases} \end{align}

Local tomography

Have device to prepare a target state

Local tomography

Want to measure deviations

Pure states: ✔

FSMs have been constructed for pure states in all dimensions

N. Li, C. Ferrie, J. A. Gross, A. Kalev, and C. M. Caves, Phys. Rev. Lett. 116, 180402 (2016)

Maximally mixed state: ✔

POVM is an FSM iff it is a 2-design

\begin{align} \sum_\xi\frac{E^\xi\otimes E^\xi}{\operatorname{tr}[E^\xi]} &=\frac{1}{d+1}\sum_{j,k}\Big( |e_j\rangle\langle e_j|\otimes|e_k\rangle\langle e_k| \\ &\hphantom{=\frac{1}{d+1}\sum_{j,k}\Big(} +|e_j\rangle\langle e_k|\otimes|e_k\rangle\langle e_j|\Big) \end{align}

Examples: SIC-POVMs, MUBs, uniformly random basis

Qubits: ✔

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 $S^3$

Qubits: ✔

Symmetry of CFI

Consider tensor form of CFI

\begin{align} \sum_\xi\frac{E^\xi\otimes E^\xi}{\operatorname{tr}[E^\xi\rho]} \end{align}

This is invariant under the Choi isomorphism

\begin{align} |e_j\rangle\langle e_k|\otimes|e_m\rangle\langle e_n| &\leftrightarrow|e_j\rangle\langle e_n|\otimes|e_m\rangle\langle e_k| \end{align}

Asymmetry of QFI

\begin{align} \rho&=\sum_j\lambda^j|e_j\rangle\langle e_j| \end{align}

For the QFI to be invariant under the Choi isomorphism, it must be that

\begin{align} \lambda^j+\lambda^k&=\frac{2}{d} & j&\neq k \end{align}

Only true for qubits and at the maximally mixed state

Higher dimensions: ✘

Need to find a new geometry in the higher-dimensional multiparameter setting

Purity for CFI

Want as close to uniform accuracies as possible

Purity measures uniformity of eigenvalues $\begin{align} \operatorname{tr}[\rho^2]&>\operatorname{tr}[\sigma^2] \end{align}$

New idea of symmetry

Analogous quantity for the CFI

\begin{align} \operatorname{tr}[Q^{-1}CQ^{-1}C]&=C^{\alpha\beta}C_{\alpha\beta} \end{align}

Minimize the purity of the CFI

\begin{align} \{E^\xi_\text{FS}\}&=\underset{\{E^\xi\}}{\operatorname{arg\,min}} \,C^{\alpha\beta}C_{\alpha\beta} \end{align}

What have we learned?

Limitations of strict symmetry requirements (or: why one should never trust qubits)

Pure states ✔

Maximally mixed state ✔

Qubits ✔

Full-rank higher dimension ✘
Promising preliminary perturbative results

Pure states	✔
Maximally mixed state	✔
Qubits	✔
Full-rank higher dimension	✘