Fisher symmetry – and the geometry of quantum states

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Fisher symmetry – and the geometry of quantum states

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apsmarch2016

Slides for my reveal.js presentation at the 2016 APS March Meeting: "Fisher symmetry and the geometry of quantum states"

On Github jarthurgross / apsmarch2016

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

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∼dpΔˆp

Fisher information

The Fisher information (FI) quantifies fluctuations

Fjk(p)=∑Outcomes∂jPr(outcome|p)∂kPr(outcome|p)Pr(outcome|p)

This matrix can be used as a metric: ds∼√dxTF(x)dx

Including quantum mechanics

Random variable defined by a positive-operator-valued measure (POVM), {Eξ}

Pr(Ξ=ξ|x)=tr[Eξρ(x)]∑ξEξ=I

We write the FI generated by a particular POVM as

Cαβ(x)=∑ξtr[Eξ∂αρ(x)]tr[Eξ∂βρ(x)]tr[Eξρ(x)]

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

tr[Q−1C]≤d−1QαβCαβ≤d−1

The bound is saturable

tr[Q−1C]=d−1Eξ=|ϕξ⟩⟨ϕξ|R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000)

Fisher symmetry

Can choose parameters so Q=I.

Saturate GM bound: ∑jCjj=d−1

Fisher symmetry

Can choose parameters so Q=I.

Saturate GM bound: ∑jCjj=d−1

Fisher symmetry

Fisher symmetry

CFS={12Qpure1d+1Qfull rank

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

∑ξEξ⊗Eξtr[Eξ]=1d+1∑j,k(|ej⟩⟨ej|⊗|ek⟩⟨ek|=1d+1∑j,k(+|ej⟩⟨ek|⊗|ek⟩⟨ej|)

Examples: SIC-POVMs, MUBs, uniformly random basis

Qubits: ✔

Qubits: ✔

Symmetry of CFI

Consider tensor form of CFI

∑ξEξ⊗Eξtr[Eξρ]

This is invariant under the Choi isomorphism

|ej⟩⟨ek|⊗|em⟩⟨en|↔|ej⟩⟨en|⊗|em⟩⟨ek|

Asymmetry of QFI

ρ=∑jλj|ej⟩⟨ej|

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

λj+λk=2dj≠k

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]

New idea of symmetry

Analogous quantity for the CFI

tr[Q−1CQ−1C]=CαβCαβ

Minimize the purity of the CFI

{EξFS}=argmin{Eξ}CαβCαβ

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

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