Geometric Methods in Quantum Information

Konrad Szymański

Universität Siegen

Numerical ranges are sets of expectation values.

Given observables $\color{blue}X_1, \ldots, X_k$, what $({\color{navy}\braket{X_1}}, \ldots, {\color{navy}\braket{X_k}})$ can be realized?
\[{\color{navy}W({\color{blue}X_1}, \ldots, {\color{blue}X_k})} := \left\{({\color{navy}\langle \psi \vert {\color{blue}X_1} \vert \psi \rangle},\ldots,{\color{navy}\langle \psi \vert {\color{blue}X_k} \vert \psi \rangle}): {\color{navy}\lVert \ket\psi \rVert} =1\right \}\] (possible extensions to other sets of states)

Numerical range $W(X_1, \ldots, X_k)$ provides the description of correlations. (PhD thesis in 2022)
  • Ground states of defining operators lie at the boundary (phase transitions, (K.S., K. Życzkowski, Universal witnesses of vanishing energy gap , EPL 2022).
  • Metrological performance: related to the geometry of $\color{navy}W(X,Y,X^2+Y^2)$. (K.S., K.Ż., Geometric and algebraic origins of additive uncertainty relations, J Phys A 2019)
  • Entanglement characterization: qubit-qudit entanglement related to 4D numerical range. (J. Czartowski, K.S. et al., Separability gap and large-deviation entanglement criterion, PRA 2019)
  • Dynamical correlation constraints: reducible to numerical range-like problem. (organized conference in 2024, articles in preparation)
\[\Delta^2X+\Delta^2Y=\braket{X^2+Y^2}-\braket{X}^2-\braket{Y}^2\]


all states $\color{navy}W(X,Y)$: blue, orbit $\color{darkred}\exp(-iH t)\ket{0}$: red

What combinations of spin components are possible?

With total spin of $j$, \[\color{navy}W({\color{blue}J_x, J_y, J_z}) \color{black}= \{{\color{navy}\vec r}\in\mathbb{R}^3 : \vert{\color{navy}\vec r}\vert \le j\}\]

Qubit: modelled by $j=\frac12$.\[\ket{1}=\ket{\uparrow}, ~\ket{0}=\ket\downarrow.\]

Any qubit numerical range: linear image of the Bloch sphere.
(observables expressible with $J_x, J_y, J_z$)

Much more complex shapes for the qutrit case

Main result of K.S., S. Weis, K. Życzkowski,
Classification of joint numerical ranges of three hermitian matrices of size three, LAA 2018

  • Each ellipse is an image of a qubit.
  • Each flat part is an ellipse.
  • Two qubit subspaces in a qutrit intersect.
  • Hence, the ellipses pairwise intersect.
  • 3D geometry $\implies$ up to four ellipses.

Qutrit classification has been used in experiments

J. Xie et al, Observing Geometry of Quantum States in a Three-Level System, PRL 2020.

Structure of high-dimensional quantum states: future plans.

Numerical range $W(X_1, \ldots, X_k)$ provides the description of correlations. (PhD thesis in 2022)
  • Ground states of defining operators lie at the boundary (phase transitions, (K.S., K. Życzkowski, Universal witnesses of vanishing energy gap , EPL 2022).
  • Metrological performance: related to the geometry of $\color{navy}W(X,Y,X^2+Y^2)$. (K.S., K.Ż., Geometric and algebraic origins of additive uncertainty relations, J Phys A 2019)
  • Entanglement characterization: qubit-qudit entanglement related to 4D numerical range. (J. Czartowski, K.S. et al., Separability gap and large-deviation entanglement criterion, PRA 2019)
  • Dynamical correlation constraints: reducible to numerical range-like problem. (organized conference in 2024, articles in preparation)
\[\Delta^2X+\Delta^2Y=\braket{X^2+Y^2}-\braket{X}^2-\braket{Y}^2\]


all states $\color{navy}W(X,Y)$: blue, orbit $\color{darkred}\exp(-iH t)\ket{0}$: red

  • Investigate the possible correlations in realistic high (and $\infty$)-dimensional systems.
  • Determine what can be reached using restricted dynamics.
  • Rigorously study the next unsolved case of mathematical classification (2 qubits).

The plan: use the machinery in realistic systems

Analysis of possible correlations, dynamically reachable ones and mathematical structure of the problem.
  • Information processing in optical cavities
    (non-reciprocal interactions)
  • Strongly coupled qubits$\otimes$cavity systems
    (what possible with the intermediate?)
Thank you!

Dynamically accessible expectation values: how does that work?

What is the minimal value of $\color{navy}\braket{n_x X+n_y Y}_{\color{#a00}\psi(t)}$ for \[{\color{#a00}\ket{\psi(t)}}=\exp(-i {\color{#d70}H} t) \color{#090}\ket{\psi}?\] Relaxation (only moments have to agree): \[\left\{\begin{matrix}\min_{\rho} \operatorname{Tr} {\color{darkred}\rho} {\color{navy}(n_X X+n_Y Y)},\\\operatorname{Tr} {\color{#d70}H}^n {\color{darkred}\rho} = \braket{{\color{#d70}H}^n}_{\color{#090}\psi},\\\color{darkred}\rho\succcurlyeq0 \end{matrix}\right.\]
\[\text{SDP} \leftrightarrow \min_{\vec r\in\text{Spec}} \vec r \cdot \vec n\]

Geometry implies uncertainty relations

$W(X,Y,X^2+Y^2)$ determines $\min \Delta^2 X + \Delta^2 Y$.

Given observables $X$ and $Y$, how do we determine the minimal sum of variances:

\[ \min_{\ket{\psi}} \Delta^2_\psi X + \Delta^2_\psi Y? \]
  • $\Delta^2 X = \langle X^2 \rangle - \langle X \rangle^2.$
  • Therefore, $\Delta^2 X + \Delta^2 Y = \langle X^2 + Y^2 \rangle - \langle X \rangle^2 - \langle Y \rangle^2.$
  • $\Delta^2 X + \Delta^2 Y$ is constant on a paraboloid; minimal uncertainty $\Leftrightarrow$ tangent paraboloid.
  • Solvable with polynomial equations! (e.g., spin squeezing inequalities)

Vanishing energy gap can be detected using geometry

The following are equivalent (if the ground state of $H$ is an eigenstate of $V$):

  • Hamiltonian $H$ has energy gap $\Delta(H)=E_1(H)-E_0(H)$ at most $\varepsilon$.
  • $\langle H \rangle$ on the ground state of $H+\lambda V$ has a jump of size $\varepsilon \ge \Delta(H)$.
  • Flat parts at the bottom of $W(V,H)$ (cusp) have height $\varepsilon \ge \Delta(H)$.

Geometric criterion for estimating $\Delta(H)$! Works for XY model:

\[ \begin{aligned} H &= \sum_{n=1}^{N} \sigma^{(x)}_n \sigma^{(x)}_{n+1} + \sigma^{(y)}_n \sigma^{(y)}_{n+1}, \\ V &= \sum_{n=1}^{N} \sigma^{(x)}_{n-1} \sigma^{(z)}_{n} \sigma^{(y)}_{n+1} - \sigma^{(y)}_{n-1} \sigma^{(z)}_{n} \sigma^{(x)}_{n+1}. \end{aligned} \]

Numerical results (MPS): $N \approx 150$.

Entanglement witnesses can be generated from arbitrary observables

\[(*)~~~\lambda^\otimes_{\min}(X)=\min \langle \alpha \otimes \beta |X| \alpha \otimes \beta \rangle.\]

$\langle \psi | X | \psi \rangle < \lambda^\otimes_{\min} \implies \ket{\psi}$ is entangled.

Idea: in $(*)$, for fixed $\ket{\beta}$, optimal $\ket{\alpha}$ $\approx$ min. eigenvalue of $X_\beta = \operatorname{Tr}_B X (\mathbb{1}_A \otimes \ket{\beta}\bra{\beta})$.

Also a geometric problem!

\[ X_i = \operatorname{Tr}_A X (\sigma_i \otimes \mathbb{1}_B). \]