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.


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".

  • $\begingroup$ Thank you for your answer, in particular for the last point. Of course, the entropy of the state comes in terms of the Williamson parameters. It is also suggested in your point 2 c. I am still looking, whether there is some more interesting physics hiding behind those parameters. $\endgroup$ – RSG Jan 19 '15 at 16:58
  • 2
    $\begingroup$ @RSG As Martin wrote, the $\kappa_i$ are the mixedness (or, if you wish, the effective temperatures) associated to each of the normal modes of the system. What would you consider "more interesting physics"? -- Note that this decomposition also allows you to write a quadratic Hamiltonian which has this state as a thermal state (and where the terms in the Hamiltonian are determined by the $\kappa_i$ and $S$). $\endgroup$ – Norbert Schuch Jan 19 '15 at 17:03

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