Quantum Information Geometry

Konrad Szymański

RCQI Bratislava

Quantum mechanics primer

Lee, G. J. et al., Micromachines 2020.

Einstein: light is photons — 'wave components', each with definite energy and wavelength.
de Broglie: … maybe electrons are similar? (yes!)
Schrödinger: insight from the limit
wave optics $\rightarrow$ ray optics, `electron wave' described by $\ket{\psi(t)}=\psi(x,y,z,t)$, \[i \frac{\partial \ket{\psi}}{\partial t} \!=\! \overbrace{\left(\!-(\partial^2_x\!+\!\partial^2_y\!+\!\partial^2_z)\!+\!\frac{k}{\sqrt{x^2\!+\!y^2\!+\!z^2}}\!\right)}^{H} \ket{\psi}\]
Quantum information: restricted linear subspaces of $\ket{\psi}$:\[\ket{\psi} = c_1 \ket{\psi_1} +\ldots+c_d \ket{\psi_d}.\]

What is measurement?

Coupling between the system and the outside world: \[i\frac{\partial \ket{\psi}}{\partial t} = (H_{\text{sys}}\otimes\operatorname{Id} + \operatorname{Id}\otimes H_{{world}} + H_{\text{int}} ) \ket{\psi}\]
Parts separate in space: clear interpretation!
Result (in the simplest case): $\exists$ orthonormal bases $\color{blue}\{\ket{e_i}\}$ and $\color{green}\{\ket{f_j}\}$ such that \[\left.\ket{\psi}\right|_{t=0}=\left(\sum {\color{red}c_k}{\color{blue}\ket{e_k}} \right)\otimes \ket{w} \mapsto\sum {\color{red}c_k} {\color{blue}\ket{e_k}} \otimes {\color{green}\ket{f_k}} =: \left. \ket{\psi}\right|_{t\rightarrow \infty}\]
Born interpretation: probability $\propto~$norm$^2=\lvert {\color{red}c_k}\rvert^2$ .
We don't know the details!

What is measurement?

Observations are discrete, single observation $\approx$ element of the basis $\{\ket{e_k}\}$. Often an eigenbasis of some physically relevant operator $A$: \[A=\sum \lambda_k \ket{e_k}\bra{e_k}.\]

($A$ Hermitian, real eigenvalues)
`Moments of $A$' inferred from observation statistics. Expectation value: \[E(A) = \braket{\psi\vert A\vert\psi} \color{gray} =\sum_k \frac{n_k}{\sum_j n_j} \lambda_k.\]
(with $\ket\psi=(\psi_1,\ldots, \psi_d)^T$, $\langle \psi \vert A \vert \psi \rangle = \sum_{i,j=1}^d \psi_i^* A_{i,j} \psi_j$)

State vectors are not the entire story

What if the black box generates $\ket{\psi_k}$ according to the ensemble $\{(p_k,\ket{\psi_k})\}$?

Qubit example:

$$ X = \begin{pmatrix} 0 & 1 \\ 1&0 \end{pmatrix},~ Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, ~Z=\begin{pmatrix} 1 & 0 \\ 0&-1\end{pmatrix}.$$

With $\ket\psi = \begin{pmatrix} \cos\frac\theta2\\e^{i\phi} \sin\frac\theta2\end{pmatrix},$ $$ \vec r=\begin{pmatrix} \langle X\rangle\\\langle Y\rangle\\\langle Z\rangle\end{pmatrix}=\begin{pmatrix}\cos\phi\sin\theta\\\sin\phi\sin\theta\\\cos\theta\end{pmatrix}, \Vert \vec r \Vert =1 $$ But if $\ket0, \ket1$ are taken at random, $\Vert \vec r\Vert =0$!

Mixed states are statistical averages

Expectation value linear: \[\sum p_k \braket{\psi_k \vert A\vert \psi_k}=\operatorname{Tr} A \overbrace{\sum p_k \ket{\psi_k} \bra{\psi_k}}^{\rho}\]
Definition (mixed states of dimension $d$): \[\mathcal{M_d} = \{ \rho \in \mathbb{C}^{d\times d} : \rho=\rho^\dagger, \operatorname{Tr} \rho=1, \rho \succeq 0\}.\]
$X\succeq 0$: positive semidefinite matrix
$\Leftrightarrow X=\sum x_k \ket{\psi_k}\bra{\psi_k}$ with positive $x_k$.
All properties defined by $\rho$. Details of the ensemble do not matter!
Tomography: reconstruction of $\rho$ from measurement statistics.

Numerical ranges: projections of state space

Definition (joint numerical range): \[W(A_1, \ldots, A_n) \\= \{ (\langle A_1\rangle_\rho ,\ldots, \langle A_n \rangle_\rho) : \rho \in \mathcal{M} \}\]

$A_i$ Hermitian: observations related to orthogonal eigenbases.
Expectation values $\langle A_i\rangle$ real.

$W(X)$: all possible expectation values

$$W(X)=[\lambda_{\min}(X),\lambda_{\max}(X)]$$

$W(X_1, X_2)$: all possible tuples of expectation values

$W(X_1, X_2, X_3)$: triples

J. Xie et al., Observing geometry of quantum states in a three-level system, PRL 2020.

Numerical ranges geometry: cuts through state space

$$W(X)=[\lambda_{\min}(X),\lambda_{\max}(X)],\color{gray}~~\max(W(X))= \lambda_{\max}(X)$$

$$\max_{(x_1, x_2)\in W(X_1, X_2)} (n_1 X_1+n_2 X_2) = \lambda_{\max} (n_1 X_1+n_2 X_2)$$

$\vec n \in S^\circ$: $\max_{\vec x \in S}\vec n\cdot\vec x \le 1$ – convex duality

Polynomial theory!

Uncertainty relations

Variance: also inferred from observation statistics, \[\operatorname{Var}(A)=E(A^2)-E(A)=\braket{\psi\vert A^2\vert \psi}-(\braket{\psi\vert A\vert \psi})^2.\]
Given two measurements $A$ and $B$, what is the relation between variances?
The Heisenberg uncertainty relation: \[\operatorname{Var}(X) \operatorname{Var}(P) \ge \frac14,\] for \[(X {\psi})(x)=x {\psi}(x), (P {\psi})(x) = -i\left.\frac{\partial}{\partial x}{\psi}\right|_{x}\]

Variance can be encoded in 2D numerical range

With $(x,x')~\in~W(X,X^2),$ $\operatorname{Var}(X)=x'-x^2.$

Sum of variances: 3D joint numerical range

($SU(2)$ metrology)

Precise measurements of magn. field strength $B$.

Measurement precision bound by variances, $$ \Delta^2(B) \le C\times({\operatorname{Var}(X) +\operatorname{Var}(Y)}),$$ for specific $X$ and $Y$.

\[(x,y,z) \in W(X,Y,X^2+Y^2):\] \[\operatorname{Var}(X)+\operatorname{Var}(Y)=z-x^2-y^2.\]

Observation: minimal sum on variances attained on the surface of $W(X,Y,X^2+Y^2)$, tangent to the paraboloid of revolution.

With \[f:=\det((X-x \operatorname{Id})^2+(Y-y\operatorname{Id})^2-\lambda \operatorname{Id}),\] points tangent to some paraboloid defined by \[f=0,\] \[\partial_x f=0,\] \[\partial_y f=0.\]

Summary

Different definition of sets of quantum states help in some cases.
Numerical ranges (possible tuples of expectation values) are their projections.
Interesting polynomial theory connection through Kippenhahn theorem.

Thank you for your attention!

References:

K.S., Numerical ranges and geometry in quantum information: Entanglement, uncertainty relations, phase transitions, and state interconversion (PhD thesis), arXiv: 2303.07390,
K.S., S. Weis, K. Życzkowski, Classification of joint numerical ranges of three hermitian matrices of size three, LAA 2018,
T. Simnacher, J. Czartowski, K.S., K. Życzkowski, Confident entanglement detection via the separable numerical range, PRA 2021,
K.S., K. Życzkowski, Geometric and algebraic origins of additive uncertainty relations, J Phys A 2019,
Lee, G. J. et al., Micromachines 2020.
J. Xie et al., PRL 2020

Correlation between different parts of the system

State of two systems: $\rho\in \mathcal{M}_{d_1\times d_2}$
($\rho=$ convex combination of projectors onto subspaces in $\mathcal{H}_{d_1}\otimes\mathcal{H}_{d_2} $)
No correlation at all: $\rho = \rho' \otimes \rho''$
Any non-tensor $\rho$: correlation when measuring two subsystems!
Definition (separable states of bipartite $\mathcal{H}_{d_1}\otimes\mathcal{H}_{d_2}$): \[ \mathcal{M}^{\text{sep}}_{d_1,d_2} = \operatorname{conv}\{ \Pi_{\ket{v_k'}}\otimes \Pi_{\ket{v_k''}}: \ket{v'}\in\mathcal{H}_{d_1}, \ket{v''}\in\mathcal{H}_{d_2}\} \]
$\mathcal{M}^{\text{sep}}_{d_1,d_2}$ allows for correlations… but not all of them!

$\mathcal{M}^{\text{sep}}_{d_1,d_2}$ is a proper convex subset of $\mathcal{M}_{d_1\times d_2}$.

Entanglement and separable numerical ranges

Joint numerical range vs separable numerical range

$W_{\text{sep}}=\{\vec a \in \mathbb{R}^n | a_i = \operatorname{Tr} \rho A_i, \rho \in \mathcal{M}^{\text{sep}}\}$
Expectation values outside $W_{\text{sep}}~\implies$ entanglement!
$W_{\text{sep}}$ convex $\implies$ defined by linear constraints.
Approximation by half-space intersection in general NP-hard!
S. Friedland, L-H. Lim, Nuclear Norm of Higher-Order Tensors, Math. Comp. 2014

Entanglement and separable numerical ranges

Goal: maximize $\operatorname{Tr} X(\rho'\otimes \rho'')$ over $ \rho'\in\mathcal{M}_2, \rho''\in\mathcal{M}_d $.

Lemma: Optimization reduces to finding a point on the surface of 4D joint numerical range, tangent to a hypercone.

Main points:

$ \operatorname{Tr} X (\rho'\otimes \rho'')=\operatorname{Tr} \rho' X''$, where \[X''=\operatorname{Tr}_2 X (\operatorname{Id} \otimes \rho'')\]
For $\rho''$ fixed, \[\max_{\rho'} \operatorname{Tr} \rho' X'' = \lambda_{\max}(X'').\]
$X'' \in M_{2\times 2} \implies $ for $X_i=\operatorname{Tr}_1 X(\sigma_i\otimes \operatorname{Id})$, \[\lambda_{\max}(X'')=x_0+\sqrt{x_1^2+x_2^2+x_3^2}\]
Linearization possible: then semidefinite programming.

$\operatorname{Tr}_2$: linear extension of \[\operatorname{Tr}_2 A \otimes B = A \operatorname{Tr} B.\] Similarly, \[\operatorname{Tr}_1 A \otimes B=B \operatorname{Tr} A.\]