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

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}

Fisher information

The Fisher information (FI) quantifies fluctuations

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

‌

Measurement dependence

‌

Measurement dependence

‌

Measurement dependence

‌

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 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}

The bound is saturable

tr⁡[ Q−1 C]= d−1E ξ= | ϕξ ⟩⟨ ϕξ | \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 =IQ=I .

Saturate GM bound: ∑jC jj= d−1 \sum_jC_{jj}=d-1

Fisher symmetry

Can choose parameters so Q =IQ=I .

Saturate GM bound: ∑jC jj= d−1 \sum_jC_{jj}=d-1

Fisher symmetry

Fisher symmetry

CFS= { 12Qpure 1d+ 1Qfull rank \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}

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: ✘

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 tr⁡[ ρ2 ]> tr⁡[ σ2 ] \begin{align} \operatorname{tr}[\rho^2]&>\operatorname{tr}[\sigma^2] \end{align}

New idea of symmetry

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}

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}

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