Fisher symmetry – and the geometry of quantum states
Fisher symmetry – and the geometry of quantum states
0 0
squint2016
Talk I gave at SQuInT 2016On Github jarthurgross / squint2016
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
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∼dpΔˆpStatistical distance
Δˆp≥1√F(p)Fisher information
The Cramér–Rao bound is achievable
Δˆp→1√F(p)The Fisher information (FI) F(p) is given by
F(p)=∑Outcomes∂pPr(outcome|p)∂pPr(outcome|p)Pr(outcome|p)Multiple parameters
For multiple parameters, the FI becomes a matrix
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=IWe write the FI generated by a particular POVM as
Cαβ(x)=∑ξtr[Eξ∂αρ(x)]tr[Eξ∂βρ(x)]tr[Eξρ(x)]Measurement dependence
Measurement dependence
Measurement dependence
Measurement dependence
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 of finite dimension d
tr[Q−1C]≤d−1QαβCαβ≤d−1Can always choose parameters so Q=I.
tr[C]≤d−1C=Q=I is unphysical because
tr[I]={2(d−1)pure(d+1)(d−1)full rankR. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000)Fisher symmetry
Extract all information: tr[Q−1C]=d−1
Measure with equal precision in all directions
C={12Qpure1d+1Qfull rankFisher symmetry
Fisher symmetry
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:1507.06904Maximally 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≠kOnly 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
purity=tr[ρ2]Analogous quantity for the CFI
tr[Q−1CQ−1C]=CαβCαβNew idea of symmetry
Minimize the purity of the CFI
{EξFS}=argmin{Eξ}CαβCαβWill 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?
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