Tomography on Continuous Variable Quantum Systems

Ludmila Augusta Soares Botelho

Departamento de Física - ICEx - Universidade Federal de Minas Gerais

Representations

Discrete

Continuous

Experiment

Data

Quantum Mechanics

Data

Quantum State

Representations

Discrete

Continuous

Quantum States

Discrete

Continuous

\vert \uparrow \rangle = \begin{bmatrix} 1\\ 0\\ \end{bmatrix}
\vert \downarrow \rangle = \begin{bmatrix} 0\\ 1\\ \end{bmatrix}
\vert 0 \rangle = \begin{bmatrix} 1\\ 0\\ \end{bmatrix}
\vert 1 \rangle = \begin{bmatrix} 0\\ 1\\ \end{bmatrix}

Phase Space

  • Wigner Function

$$W(q,p) = \frac{1}{2\pi\hbar} \int \left\langle q-\frac{v}{2}\right\vert \rho\left\vert q+\frac{v}{2}\right\rangle e^{ipv/\hbar}\mathrm{d}v$$

 

  • Allows Negative Values
  • Quasi-probability

Homodyne Detection

2-1

$$\vert \alpha \rangle$$

signal

50/50

Intensity     Photon number

$$I_{21} = I_1 - I_2$$

$$ \propto$$

\hat{n}_1=\hat{a}_1'^\dagger\hat{a}_1' \qquad \hat{n}_2=\hat{a}_2'^\dagger\hat{a}_2'\\ \hat{a}_1'=2^{1/2}(\hat{a}'-\alpha_{LO}) \qquad \hat{a}_2'=2^{1/2}(\hat{a}'+\alpha_{LO}) \\ \hat{n}_{21} = \hat{n}_{2} - \hat{n}_{1} = \alpha_{LO}^*\hat{a}+\alpha_{LO}\hat{a}^{\dagger} \\ \alpha = \vert\alpha_{LO}\vert(\cos{\theta} +i\sin{\theta}) \\ \hat{n}_{21} = \frac{1}{\sqrt{2}} \vert{\alpha_{LO}}\vert \left[ (\cos{\theta} +i\sin{\theta})(\hat{q} - i\hat{p}) + (\cos{\theta} +i\sin{\theta})(\hat{q} - i\hat{p}) \right]\\
\hat{n}_{21} = \frac{2}{\sqrt{2}}\vert{\alpha_{LO}}\vert\hat{q}_{\theta}
\langle{q_{\theta}}\vert\rho \vert{q_{\theta}}\rangle = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} W(q_{\theta}\cos{\theta} - p_{\theta}\sin{\theta},q_\theta \sin{\theta} + p_{\theta}\cos{\theta}) \mathrm{d}p_{\theta}

Radom Transform

$$\langle{q_{\theta}}\vert\rho \vert{q_{\theta}}\rangle \rightarrow W(q,p)$$

$$W(q,p) \rightarrow \langle{q_{\theta}}\vert\rho \vert{q_{\theta}}\rangle  $$

?

W(q,p) = \frac{1}{2 \pi^{2}} \int_{-\infty}^{+\infty} \int_{0}^{\pi} pr(x,\theta) K(q\cos{\theta} + p\sin{}\theta -x) \mathrm{d}x \mathrm{d}\theta

Inverse Radom

K(x)= \frac{1}{2} \int_{-\infty}^{+\infty} \vert{\xi}\vert \exp{(i\xi x)} \mathrm{d}\xi

What about multi-mode state?

Convex Optmization

min

s.t

$$\rho \succeq 0$$

$$\mathrm{Tr}(\rho) = 1$$

  • Semi-definite programming

  • Fock Basis

$$\sum_{i \in \mathcal{I}} \Delta_i + \delta$$

$${\rho, \Delta, \delta}$$

$$\left\vert \mathrm{Tr}({E_i \rho) -f_i} \right\vert  \leqslant \Delta_i f_i$$

$$i \in \mathcal{I}$$

$$\mathrm{Tr}({ E_i\rho} )\leqslant \delta$$

$$i \notin \mathcal{I}$$

$$\vert q_{\theta}\rangle\!\langle{q_{\theta}}\vert= \sum \psi_n^*(q) \psi_m(q) \exp[i(m-n)]\vert{n}\rangle\!\langle{m}\vert$$

Ludmila A. S. Botelho, Reinaldo O. Vianna, Eur. Phys. J. D 74, 42 (2020)

10.1140/epjd/e2020-100649-3

And now for something completely different

Separability

  • Partial Transposition

$$\rho^{T_B} = \sum_i p_i \left(\rho_i^A \otimes (\rho_i^B)^T\right)$$

Huge Dimension

Separability

  • Covariance Matrix
\gamma \geq \begin{pmatrix} \gamma_A & 0 \\ 0 & \gamma_B \end{pmatrix}
  • SDP

Positive Maps

  • Density Operator vs Moment Matrix
  • Entanglement Witness

(But Not Completely Positive)

Sep

$$\rho$$

$$\mathit{W}$$

Ent

$$M$$

$$\mathit{W}'$$

Thank You!

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