Physical significance of Williamson parameters

Question

I am trying to read some of the quantum mechanical problems from a mathematical point of view, and came to the following problem. Let us consider a $n$ mode quantum Gaussian state (which is in $L^2(\mathbb{R}^n)$) in which every real linear combinations of canonical momentum and position observables $p_1,p_2,\cdots, p_n;~q_1,q_2,\cdots,q_n$ has a normal distribution on the real line. Such a state is uniquely determined by expectation values of $p_1,p_2,\cdots, p_n;~q_1,q_2,\cdots,q_n$ and their covariance matrix of order $2n$.

A real strictly positive definite matrix $A$ of order $2n$ is a valid covarience matrix of observables $p_1,p_2,\cdots, p_n;~q_1,q_2,\cdots,q_n$ if and only if the matrix inequality \begin{equation} 2A-\imath J_{2n}\geq 0, \end{equation} where $J_{2n}=\begin{pmatrix} 0 & I_n\\ -I_n & 0\end{pmatrix}$.

Such a matrix has a symplectic diagonalisation. There exists a (real) symplectic matrix $S\in Sp(2n)$ such that $SAS^T=\kappa_1 I_2 \oplus \kappa_2 I_2 \oplus \cdots \oplus \kappa_n I_2$, where $\kappa_1 \geq \kappa_2 \geq \cdots \geq \kappa_n \geq \frac{1}{2}$. Such a form is also known as Williamson normal form.

My question is, what is the physical significance (process, phenomena, temperature of system, & etc.) of these $\kappa$'s. Advanced thanks for any help or suggestion.

Answer 1

DISCLAIMER: I usually use a different definition of the covariance matrix: It should only fulfill $\gamma\geq iJ_{2n}$. The factor 1/2 is irrelevant. I tried to adjust everything, but there might be some problems with factors of 1/2.

I don't have a complete answer to this question (still learning myself), but here are some ideas. Let's call the $\kappa_i$ the "symplectic spectrum":

1. The most basic equation $2A-iJ_{2n}\geq 0$, which implies that the $\kappa_i$ must be bigger or equal to $1/2$ is nothing but a basis independent formulation of Heisenberg's uncertainty relation. I guess you already know this, because $A\geq \frac{i}{2}J_{2n}$ is exactly Heisenberg's uncertainty relation with $\hbar=1$.

2. Another interesting feature of the $\kappa_i$ is that it provides us with some measure of mixedness:

   • For a Gaussian state, if all the $\kappa_i$ are 1/2 (or 1, if you get rid of your factor 1/2), then the state is pure. In other words, a pure state is $1/2$ of a symplectic matrix. In this sense, the symplectic spectrum relates to the "mixedness" of the Gaussian state.
   • Via this, it becomes rather clear that the states where at least one $\kappa_i$ is 1/2 lie on the boundary of the set of Gaussian states, so, again, this leads us to the $\kappa_i$ being some measure of mixedness.
   • By construction, the $\kappa_i$ are invariant under symplectic transformations. If we have a look at the most basic Gaussian operations, we find that they either do not change the spectrum (symplectic transformations such as one-mode squeezers, beam-splitters or phase-shifters in quantum optics) or only increase the spectrum (adding noise, which is represented by some positive definite covariance matrix; maybe homodyne/Gaussian measurements? I don't know about those). Again, this shows that the Williamson spectrum is some measure for mixedness.
   • If I'm not mistaken, any symplectic matrix can be constructed with beam-splitters, one-mode-squeezers and phase-shifters, i.e. the symplectic spectrum characterizes the states that can be obtained by just these operations.
   • I don't know, however, how well this works as a measure for mixedness in general. In particular, I don't know enough about possible state transitions between states with different symplectic spectra.

3. On a different level, there is a relation between the symplectic eigenvalues and temperature:

Let's consider thermal Gaussian states. A thermal state $$\rho_{th}=e^{-\beta \hat{n}}/\operatorname{tr}(e^{-\beta \hat{n}})$$ with temperature $\beta$ and number expectation value $N:=\operatorname{tr}(\rho_{th}\hat{n})$ has a characteristic function

$$ \chi(q,p)=\operatorname{exp}(-(q^2+p^2)(2N+1)/4) $$ hence its covariance matrix is $1/2I_2 (2N+1)$.

This means that Williamson's theorem provides us with a decomposition of a Gaussian state into a tensor product of thermal states with temperatures $\beta_k$ such that $$N_k=(e^{\beta_k}-1)^{-1}=1/2(2\kappa_k-1)$$

This is also sometimes known as "normal mode decomposition".