Simply Symplectic

Gerardo Adesso changed the gears of the school, and gave a more theoretical kind of talk. One could feel the enthusiasm among the more mathematically inclined participants.

Quantum information with continuous variables — lecture I, Gerardo Adesso

Gerardo started by defining continuous variables (CV) systems. As the name already gives away, CV systems are systems in which we exploit a degree-of-freedom with a continuous spectrum. An atom, for example, has discrete energy levels, but its position can, of course, assume any value on the real line. The Hilbert space is thus infinite-dimensional.

I guess you’re already probably thinking, “there comes again the quantum harmonic oscillator”, and you’re right. For bosons, the best way to describe CV systems is to define a (symmetric) Fock space, creation $a^\dagger$ and annihilation $a$ operators… and all that jazz.

For $N$ harmonic oscillator modes, it’s convenient to define a vector with the canonical operators $\hat{R}=(\hat{q}_1,\hat{p}_1,\hat{q}_2,\hat{p}_2,\ldots, \hat{q}_N,\hat{p}_N)$, and the symplectic matrix $\Omega_N=\Omega^{\oplus^N}$, with $\Omega=\begin{pmatrix} 0&1\\-1&0\end{pmatrix}$. In this way the commutation relations are easily written as $[R_j,R_k]= i (\Omega_N)_{jk}$. To explain, in part, this post’s title, a $2N \times 2N$ matrix $M$ is called symplectic ($\in Sp(2N,\mathbb{R})$) iff $M^T\Omega_N M = \Omega_N$.

Why do we care about this symplectic structure? Because it will be very handy to describe properties of bosonic Gaussian states. Before we do that, however, we must be able to write down the state of these $N$ bosonic modes. The first thing that might came to your mind is to write down its density matrix. Remember, though, that the Hilbert space is infinite-dimensional and thus also the density matrix. To cope with that Adesso employed the Wigner function of the state.

Without going in too much detail (you should go to the lectures!), a bosonic state is said Gaussian whenever its Wigner function is Gaussian. Defining a vector $\xi \in \mathbb{R}_{2N}$, the Wigner function of a Gaussian state $\rho$ can be written as:

$W_\rho(\xi) = \frac{\exp[-(\xi -\overline{R})^T\sigma^{-1}(\xi -\overline{R})]}{\pi^N \sqrt{\det \sigma}}$,

where $\overline{R}=\langle R \rangle_\rho$ is the vector containing the first moments, and $\sigma$ is the correlation matrix, with $(\sigma)_{jk} = \langle R_j R_k + R_k R_j\rangle_\rho - 2 \langle R_j\rangle_\rho \langle R_k\rangle_\rho$. A necessary condition for an operator in the space of $N$ bosonic modes to be positive semi-definite is given by $\sigma + i\Omega_N\ge 0$, and this condition becomes also sufficient for Gaussian operators.

Now that we have a description of bosonic Gaussian states, we can talk about Gaussian operations. These are operations generated by Hamiltonians that are at most quadratic in the creation and annihilation operators. These operations, as expected, map Gaussian states onto Gaussian states. The displacement operator, the squeezing operator, rotations and homodyne detection are all examples of Gaussian operations, and they can be associated with a symplectic matrix. The nice thing is that to update a Gaussian state after a Gaussian operation with associated symplectic martrix $S$ all one has to do is to map $\sigma \mapsto S^T \sigma S$. This can be efficiently done, despite the fact that you are dealing with a infinite-dimensional state!

Let’s see what Gerardo is planning to do with all this machinery! Next lecture tomorrow (08.08) at 14h.