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

Creative Commons License

Local tomography

Have device to prepare a target state

Local tomography

Want to measure deviations

Distances between states

Is the blue or green state closer to the target?

Want an operational notion of distance

Statistical distance

How far apart are coins with different biases?

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

Statistical distance

How far apart are coins with different biases?

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

Statistical distance

ds∼d pΔ p ^ \begin{align} ds\sim\frac{dp}{\Delta \hat{p}} \end{align}

Statistical distance

Δ p^ ≥1 F(p ) \begin{align} \Delta\hat{p}&\geq\frac{1}{\sqrt{F(p)}} \end{align}

Fisher information

The Cramér–Rao bound is achievable

Δ p^ →1 F(p ) \begin{align} \Delta\hat{p}&\to\frac{1}{\sqrt{F(p)}} \end{align}

The Fisher information (FI) F (p) F(p) is given by

F(p )=∑ Outcomes∂ pPr⁡ (outcome |p ) ∂p Pr⁡(outcome |p )Pr ⁡(outcome |p ) \begin{align} F(p)&=\sum_\text{Outcomes} \frac{\partial_p\operatorname{Pr}(\text{outcome}|p) \,\partial_p\operatorname{Pr}(\text{outcome}|p)} {\operatorname{Pr}(\text{outcome}|p)} \end{align}

Multiple parameters

For multiple parameters, the FI becomes a matrix

Fjk ( p) =∑Outcomes ∂j Pr⁡( outcome| p ) ∂k Pr⁡(outcome | p )Pr ⁡(outcome | p ) \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: d s∼d x T F( x)d x ds\sim\sqrt{d\mathbf{x}^\mathsf{T}F(\mathbf{x})d\mathbf{x}}

Including quantum mechanics

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

Pr⁡ (Ξ=ξ | x )= tr⁡[E ξρ ( x) ]∑ ξE ξ=I \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

C αβ ( x) =∑ ξtr⁡ [Eξ ∂α ρ( x )] tr⁡[ Eξ ∂β ρ( x )] tr⁡[ Eξ ρ( x) ] \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)

Uncertainty relations

Would like to have quantum-limited accuracy in all directions

Complementarity forbids this. A useful expression of complementarity is the Gill–Massar (GM) bound.

Gill–Massar bound

Assume Hilbert space H \mathcal{H} of finite dimension d d

tr⁡ [Q−1 C] ≤d−1 Qα βC α β ≤d−1 \begin{align} \operatorname{tr}[Q^{-1}C]&\leq d-1 & Q^{\alpha\beta}C_{\alpha\beta}&\leq d-1 \end{align}

Can always choose parameters so Q =IQ=I .

tr⁡ [C] ≤d−1 \begin{align} \operatorname{tr}[C]&\leq d-1 \end{align}

C =Q=I C=Q=I is unphysical because

tr⁡ [I]= { 2(d−1 )pure (d+1 )(d− 1)full rank \begin{align} \operatorname{tr}[I]&=\begin{cases}2(d-1) & \text{pure} \\ (d+1)(d-1) & \text{full rank}\end{cases} \end{align} R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000)

Fisher symmetry

Extract all information: tr⁡[ Q−1 C]=d−1 \operatorname{tr}[Q^{-1}C]=d-1

Measure with equal precision in all directions

C= { 12Q pure1d+ 1Qfull rank \begin{align} C&=\begin{cases}\frac{1}{2}Q & \text{pure} \\ \frac{1}{d+1}Q & \text{full rank}\end{cases} \end{align}

Fisher symmetry

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Fisher symmetry

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Fisher symmetry

Can we do this?

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, J. A. Gross, A. Kalev, and C. M. Caves, arXiv:1507.06904

Maximally mixed state: ✔

POVM is an FSM iff it is a 2-design

∑ξ Eξ ⊗E ξ tr⁡[E ξ] =1d+ 1∑ j,k (| ej⟩ ⟨ej | ⊗ |ek ⟩ ⟨ek | =1d+1 ∑j ,k (+ | ej⟩ ⟨ek | ⊗ |ek ⟩ ⟨ej | ) \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: ✔

Qubits: ✔

Symmetry of CFI

Consider tensor form of CFI

∑ξ Eξ ⊗E ξ tr⁡[E ξρ ] \begin{align} \sum_\xi\frac{E^\xi\otimes E^\xi}{\operatorname{tr}[E^\xi\rho]} \end{align}

This is invariant under the Choi isomorphism

| ej⟩ ⟨ek | ⊗ |em ⟩ ⟨en |↔ | ej⟩ ⟨en | ⊗ |em ⟩ ⟨ek | \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

ρ= ∑j λj |ej ⟩ ⟨ej | \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

λj+ λk =2dj ≠k \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: ✘

What is the next-best thing?

Purity for CFI

Want accuracy spread out over all parameters as much as possible

Purity measures the uniformity of the eigenvalues of ρ \rho

purity= tr⁡[ ρ2] \begin{align} \text{purity}&=\operatorname{tr}[\rho^2] \end{align}

Analogous quantity for the CFI

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

New idea of symmetry

Minimize the purity of the CFI

{E FSξ }= a rg m in {E ξ }C α βC α β \begin{align} \{E^\xi_\text{FS}\}&=\underset{\{E^\xi\}}{\operatorname{arg\,min}} \,C^{\alpha\beta}C_{\alpha\beta} \end{align}

Will this exist?

Since we are minimizing a quadratic function over a convex set, the minima exist, so now we just have to find them.

The hunt

Still not trivial to find

Strategy: perturb SIC-POVM as we leave maximally mixed state such that the CFI purity is minimized

What have we learned?

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

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

  2. Promising preliminary perturbative results

What's next?

  1. Extend perturbative results
  2. Understand structure of CFI
  3. Employ tools to understand other problems in quantum information