Fisher symmetry – and the geometry of quantum states
Fisher symmetry – and the geometry of quantum states
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sydney2016
Talk I gave at the University of Sydney on 2016-05-12.On Github jarthurgross / sydney2016
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
Quantify distance between two quantum states by how statistically distinguishable they are
More distinguishable: further apart
Less distinguishable: closer together
Example: coin bias
How far apart are coins with different biases?
W. K. Wootters, Phys. Rev. D 23, 357 (1981)Example: coin bias
How far apart are coins with different biases?
W. K. Wootters, Phys. Rev. D 23, 357 (1981)Example: coin bias
ds∼d pΔp^ \begin{align} ds\sim\frac{dp}{\Delta \hat{p}} \end{align}Statistical fluctuations
The fluctuations depend on the estimator used
The Cramér–Rao bound gives us an achievable lower bound
E [(Δp^)2]≥1F(p) \begin{align} \mathbb{E}\left[(\Delta\hat{p})^2\right]&\geq\frac{1}{F(p)} \end{align}This allows us to generally define our statistical distance in terms of the Fisher information (FI)
ds∼dpΔp^→F(p )dp \begin{align} ds&\sim\frac{dp}{\Delta\hat{p}}\to\sqrt{F(p)}\,dp \end{align}Fisher information
F(p)=∑Outcomes Pr(outcome|p){ ddpln [Pr( outcome|p) ]}2=∑ Outcomes[ddpPr(outcome|p)]2Pr(outcome|p) \begin{align} F(p)&=\sum_{\text{Outcomes}}\operatorname{Pr}(\text{outcome}|p) \left\{\frac{d}{dp}\ln\Big[\operatorname{Pr}(\text{outcome}|p) \Big]\right\}^2 \\ &=\sum_{\text{Outcomes}}\frac{\left[\frac{d}{dp} \operatorname{Pr}(\text{outcome}|p)\right]^2} {\operatorname{Pr}(\text{outcome}|p)} \end{align}The FI quantifies the information a random variable provides about a parameter
Coin FI
Pr(H|p)=pF(p)=1 2p+(−1) 21−pPr(T |p )=1−p=1p(1−p) \begin{align} \operatorname{Pr}(H|p)&=p & F(p)&=\frac{1^2}{p}+\frac{(-1)^2}{1-p} \\ \operatorname{Pr}(T|p)&=1-p & &=\frac{1}{p(1-p)} \end{align}Kullback–Leibler divergence
D(f∥ g):=∫dxf (x)ln( f(x)g(x)) \begin{align} \mathcal{D}(f\Vert g)&:=\int dx\,f(x)\, \ln\left(\frac{f(x)}{g(x)}\right) \end{align}Quantifies how likely you are to mistake one distribution for another (not symmetric)
Locally it defines the same distance as the FI
D(fp∥fp+δ p)= 12δp2F(p) \begin{align} \mathcal{D}(f_p\Vert f_{p+\delta p})&=\frac{1}{2}\delta p^2\,F(p) \end{align}Multiple parameters
Fjk(p)=∑Outcomes∂jPr( 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
ds∼dp TF(p)dp \begin{align} ds&\sim\sqrt{d\mathbf{p}^\mathsf{T}F(\mathbf{p})\,d\mathbf{p}} \end{align}Including quantum mechanics
Random variable defined by a positive-operator-valued measure (POVM), {Eξ} \{E^\xi\}
Pr (Ξ=ξ|x)= tr[Eξρ(x)]∑ξEξ=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
Cαβ(x)=∑ ξtr[Eξ∂αρ(x)]tr[Eξ∂βρ(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 dd
tr [Q−1C]≤d−1 QαβCαβ≤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−1C]=d−1Eξ=|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: ∑jCjj=d−1\sum_jC_{jj}=d-1
Fisher symmetry
Can choose parameters so Q=IQ=I .
Saturate GM bound: ∑jCjj=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}Local tomography
Have device to prepare a target state
Local tomography
Want to measure deviations
Pure states: ✔
FSMs have been constructed for pure states in all dimensions
N. Li, C. Ferrie, J. A. Gross, A. Kalev, and C. M. Caves, Phys. Rev. Lett. 116, 180402 (2016)Maximally mixed state: ✔
POVM is an FSM iff it is a 2-design
∑ξEξ⊗Eξ tr[Eξ]=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: ✔
Qubits: ✔
Qubits: ✔
State-space geometry
Consider equatorial plane of the Bloch ball
As in the simplex case, Euclidean distance doesn't reflect statistical distance
State-space geometry
Just like the simplex is naturally a semicircle, the Bloch ball is naturally the upper hemisphere of S 3S^3
Qubits: ✔
Symmetry of CFI
Consider tensor form of CFI
∑ξEξ⊗Eξ tr[Eξρ] \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−1CQ −1C]=CαβCαβ \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}=argmin{Eξ }CαβCαβ \begin{align} \{E^\xi_\text{FS}\}&=\underset{\{E^\xi\}}{\operatorname{arg\,min}} \,C^{\alpha\beta}C_{\alpha\beta} \end{align}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