Geometry of Quantum Correlations

Correlations are everything we can measure

Outcomes are random.

Expectation values are the only measurable quantities. From experiment: frequencies of measurement, $$\frac{n_k}{\sum_j n_j}\rightarrow \lvert \langle \psi \vert \psi_k\rangle \rvert^2=:P_k.$$ expectation value: weighted average of numbers $\{\lambda_k\}$, \[\langle A \rangle := \sum \lambda_k P_k.\]

What are the relations between observables?

What are the possible regions of…

• joint expectation values? of $A_1, A_2,\ldots$ $$ \left\{\langle A_1\rangle,\langle A_2\rangle,\ldots)_\rho : \Tr \rho=1, \rho\succeq 0\right\}$$
• covariance matrices? $$ \left\{ \begin{pmatrix} \Cov(A,A) &\Cov(A,B)\\ \Cov(B,A) &\Cov(B,B)\end{pmatrix}_\rho \right\},$$

  $\Cov_\rho(A,B)=\frac12\langle AB+BA\rangle_\rho-\langle A\rangle_\rho\langle B\rangle_\rho$
  (+other quadratic expressions, e.g. sector lengths))

Relations between observables are important!

Goal: Precise rotation angle measurements – $\SUT$

(State evolved by $\exp(-\theta Z)$. $X$ and $Y$ measured.)

Precision of $\theta$ bound by variances, $$ \Delta^2(\theta) \le \text{const.}\times({\operatorname{Var}( X) +\operatorname{Var}( Y)}),$$ for spin operators $X$ and $ Y$.

[KS, Karol Życzkowski, J Phys A 2019]

Observation: Geometry $\implies$ $H$ is gapless

`Cusp height' $\ge$ spectral gap

[KS, Karol Życzkowski, EPL 2022]

Goal: Entanglement detection

• $ W$: $\langle { W}\rangle_\rho~ $$<0 \implies \rho~$ entangled
  

  (ent. witness)

• Generalization: $ (\langle A_1\rangle, \langle A_2\rangle, \langle A_3\rangle)_\rho$ outside separable region
  $\implies $ $\rho~$ entangled.

[J. Czartowski et al, PRA 2019]

$$ \operatorname{Var}(X)+\operatorname{Var}(Y)=\langle X^2+Y^2\rangle-\langle X\rangle^2-\langle Y\rangle^2$$

$$ H = \sum_{n=1}^{N} \sigma^{(x)}_n \sigma^{(x)}_{n+1} + \sigma^{(y)}_n\sigma^{(y)}_{n+1}$$ $$V = \sum_{n=1}^{N} \sigma^{(x)}_{n-1} \sigma^{(z)}_{n} \sigma^{(y)}_{n+1} - \sigma^{(y)}_{n-1} \sigma^{(z)}_{n} \sigma^{(x)}_{n+1} $$ `Cusp height'=0!

Examples of mathematical treatment

Bloch ball:

$X, Y, Z$: Pauli matrices

$$\{{\cblue(\langle X\rangle, \langle Y\rangle, \langle Z\rangle)_\rho} : \Tr \rho=1, \rho\succeq 0\}\\=\left\{{\cblue \vec r}\in\mathbb{R}^3 : \lvert {\cblue \vec r}\rvert\le 1\right\}.$$

$\cblue (\Tr \sigma X, \Tr \sigma Y, \Tr \sigma Z)$ outside $\implies$ negative eigenvalues of $\sigma$.

Higher-dimensional analogues much more complex.

Numerical ranges: projections of state space

(like a photo is a 2D projection of 3D space)

Numerical range: \[W( A_1, \ldots, A_n) \\= \{ (\langle A_1\rangle_{ \rho} ,\ldots, \langle A_n \rangle_{ \rho}) \}\]

$W(X)$: all possible expectation values

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

$\lambda_{\max}$ – maximal eigenvalue

$W( X_1, X_2)$: pairs 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.]
[KS, S. Weis, K. Życzkowski, LAA 2018]

Numerical ranges geometry is defined by polynomials

Practically: $(x_1, x_2)\in W( X_1, X_2)$ boundary $\leftrightarrow f(x_1,x_2)=0$ for some polynomial $f$.

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

$\sim$ max. root of characteristic polynomial $\det( X-\lambda \I)$

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

$\sim$ max. root of characteristic polynomial $\det(n_1 X_1+n_2 X_2-\lambda \I)$

convex duality and polynomial theory!

[search for Kippenhahn theorem, e.g. in K.S, 2303.07390]

Quadratic surfaces: partial descriptions

Tailored calculations

[from S. Morelli et al, PRA 109, 012423 (2024)]

$$\lVert T\rVert = \sqrt{\sum \langle \sigma_\alpha \otimes \sigma_\beta \rangle^2},$$ $$\lVert a \rVert = \sqrt{\sum \langle \sigma_\alpha \otimes 1 \rangle^2}$$

Linear estimators

$$ X=\operatorname{diag}(1,0,-1),~~ Y=\frac{1}{\sqrt2}\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}$$

$$\Var A = \min_{a\in\mathbb{R} } \langle A^2-2a A+a^2\rangle$$

[KS, Karol Życzkowski, J Phys A 2019]

Entangled qudits in photonic systems

[G. F. Borges et al, JOSA B 2021]

Summary

• Frequent problem: characterize geometry of correlations

  (e.g. uncertainty relations, entanglement detection)

• Nontrivial mathematics!

  (see e.g. Kippenhahn theorem)

• Application to experimental entangled qudits

Thank you for your attention!

References:

1. K.S., Numerical ranges and geometry in quantum information: Entanglement, uncertainty relations, phase transitions, and state interconversion (PhD thesis), arXiv: 2303.07390,
2. K.S., S. Weis, K. Życzkowski, Classification of joint numerical ranges of three hermitian matrices of size three, LAA 2018,
3. T. Simnacher, J. Czartowski, K.S., K. Życzkowski, Confident entanglement detection via the separable numerical range, PRA 2021,
4. K.S., K. Życzkowski, Geometric and algebraic origins of additive uncertainty relations, J Phys A 2019,
5. J. Xie et al., Observing geometry of quantum states in a three-level system, PRL 2020
6. S. Morelli et al., Correlation constraints and the Bloch geometry of two qubits, PRA 2024
7. G. F. Borges et al., Angular spectrum influence and entanglement characterization of Gaussian-path encoded photonic qudits, JOSA B 2021