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Jonathan A. Gross, Ninnat Dangniam, Carlton M. Caves
Center for Quantum Information and Control, University of New Mexico
equal precisionin all directions
You've never heard of the Millennium Falcon?…It's the ship that made the Kessel Run in less than twelve parsecs.
We will use the concept of statistical distance to naturally relate all the directions in quantum state space
Quantify distance between two quantum states by how statistically distinguishable they are
More distinguishable: further apart
Less distinguishable: closer together
Count the number of intermediary distinguishable states
W. K. Wootters, Phys. Rev. D 23, 357 (1981)Count the number of intermediary distinguishable states
W. K. Wootters, Phys. Rev. D 23, 357 (1981)The fluctuations depend on the estimator used
The Cramér–Rao bound gives us an achievable lower bound
E[(Δˆx)2]≥1F(x)This allows us to generally define our statistical distance in terms of the Fisher information (FI): ds∼√F(x)dx
The FI quantifies the information Ξ has about x
So far we've only discussed fluctuations in 1 "direction"
If you're trying to measure multiple parameters, the FI becomes a matrix
Fαβ(x)=∑ξ∂∂xαPr(Ξ=ξ|x)∂∂xβPr(Ξ=ξ|x)Pr(Ξ=ξ|x)Diagonal elements are the FIs for individual parameters
This matrix can be used as a metric: ds∼√dxTF(x)dx
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∂ρ∂xα|x)tr(Eξ∂ρ∂xβ|x)tr(Eξρ(x))Consider equatorial plane of the Bloch ball
As in the simplex case, Euclidean distance doesn't reflect statistical distance
Just like the simplex is naturally a semicircle, the Bloch ball is naturally the upper hemisphere of S3
QFI gives bounds for individual parameter estimation
The Gill–Massar bound gives us simultaneous bounds on multiple parameters for a d-dimensional system
tr(Q−1C)≤d−1QαβCαβ≤d−1Saturated iff {Eξ} is rank-1
R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000)Through change of parameters we can set Q=I
In this natural parametrization
trC≤d−1If we measured all parameters with quantum-limited accuracy
trC={2d−2pured2−1generalCan we saturate bound and locally measure all parameters with the same fraction of the quantum-limited accuracy?
C(x)={12Q(x)pure1d+1Q(x)generalWe call a measurement with this property a Fisher-symmetric measurement (FSM)
Li et al. have shown FSMs to exist for pure states in all dimensions and given an explicit construction
N. Li, C. Ferrie, and C. M. Caves, arXiv:1507.06904A symmetric informationally complete POVM (SIC-POVM) is a FSM at the maximally mixed state, provided it exists
Eξ∝Πξtr(ΠξΠη)=const.ξ≠ηFor qubits, can explicitly calculate the FSM for an arbitrary mixed state
Eξ=|ϕξ⟩⟨ϕξ||ϕ0⟩=1√2−z|0⟩|ϕj⟩=1√3(√1−z2−z|0⟩+eiφj|1⟩)I need more coin!"Walken-Cowbell". Licensed under Fair use via Wikipedia.
SIC-POVMs are hard to find, so it looks like FSMs are going to be hard to find (if they exist at all)
Easier problem (?): given a SIC-POVM, can one transform it to be a FSM at an arbitrary state?
Since FSMs are intimately related to the metric, parallel transport recommends itself as a metric-dependent means of transforming objects on state space
"Parallel Transport" by Fred the Oyster - . Licensed under CC BY-SA 4.0 via Commons
Need to associate operators with vectors/covectors
ρ(x)=I/d+xαXα∂∂xα→∂ρ∂xα=Xαδαβ=:tr(˜XαXβ)dxα→˜XαPOVM elements naturally correspond to covectors
Cαβ(x)=∑ξtr(Eξ∂ρ∂xα|x)tr(Eξ∂ρ∂xβ|x)tr(Eξρ(x))Naïve parallel transport does not work (doesn't preserve positivity or rank of POVM elements)
˜Xz=σz−zIdirections
Several possible things to try moving forward
Special thanks to Adrian, Jim, Qi, Rick, Matt, Travis, and Ninnat