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Rank deficiency

and the Euclidean geometry of quantum states

Jonathan A. Gross, Carlton M. Caves

Center for Quantum Information and Control, University of New Mexico

Quantum state tomography

\begin{aligned} ρ & = [\begin{matrix} ρ_{11} & ρ_{12} & \dots & ρ_{1 d} \\ ρ_{12}^{*} & ρ_{22} & ρ_{2 d} \\ ⋮ & ⋱ & ⋮ \\ ρ_{1 d}^{*} & ρ_{2 d}^{*} & \dots & 1 - \sum_{j} ρ_{j j} \end{matrix}] \end{aligned}

Need $d^{2} - 1$ parameters

Maximum-likelihood estimation

\begin{aligned} \hat{ρ} & = \underset{ρ \geq 0, tr [ρ] = 1}{a r g m a x} L (ρ | d) & \hat{A} & = \underset{A = A^{†}, tr [A] = 1}{a r g m a x} L (A | d) \end{aligned}

$\begin{aligned} L (ρ | d_{1}, \dots, d_{N}) & = tr [ρ E_{d_{1}}] \dots tr [ρ E_{d_{N}}] \end{aligned}$

Solid angle

\begin{aligned} ∠ ψ & = lim_{ε \to 0} \frac{vol [B_{ε}^{ψ} \cap M_{d}]}{vol [B_{ε}^{ψ}]} \\ B_{ε}^{ψ} & = {A | A = A^{†}, tr [A] = 1, ∥ A - | ψ ⟩ ⟨ ψ | ∥ \leq ε} \\ M_{d} & = {ρ | ρ \in C^{d \times d}, ρ \geq 0, tr [ρ] = 1} \end{aligned}

Polar coördinates

\begin{aligned} x & = r \cos φ \sin θ \\ y & = r \sin φ \sin θ \\ z & = r \cos θ \\ d s^{2} & = d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d φ^{2}) \\ = d r^{2} + r^{2} d Ω^{2} \end{aligned}

Spectral coördinates

$\begin{aligned} ρ & = U Λ U^{†} \end{aligned}$

$\begin{aligned} U & = [\begin{matrix} \cos θ & \sin θ \\ e^{i ϕ} \sin θ & - e^{i ϕ} \cos θ \end{matrix}] & Λ & = [\begin{matrix} λ_{1} & 0 \\ 0 & λ_{2} \end{matrix}] \end{aligned}$

Qutrit

\begin{aligned} d V & = d A_{d - 1} \prod_{j < k} (λ_{j} - λ_{k})^{2} d Υ_{d} \\ d V^{'} & = d A_{d - 1} \prod_{j < k} (λ_{j} - λ_{k})^{2} d Υ_{d - 1} \end{aligned}

Qutrit

\begin{aligned} \Pr [rank (\hat{ρ}) = 1 | ρ = | ψ ⟩ ⟨ ψ |] & = \frac{1}{3} - \frac{\sqrt{3}}{4 π} \\ \Pr [rank (\hat{ρ}) = 2 | ρ = | ψ ⟩ ⟨ ψ |] & = \frac{1}{2} + \frac{\sqrt{3}}{2 π} \end{aligned}