Matrix properties¶

Here I provide a detailed justification that the matrix \(B = \Re(A)\) introduced in equation (36) during the derivation of the coincidence probability is real symmetric and positive definite under physically relevant conditions. This property is essential for the convergence of the Gaussian integral and for the existence of the factorization of \(A\).

Structure of B¶

As we see in the derivation of the coincidence probability, the matrix \(A\) can be decomposed into its real and imaginary parts as

\[\begin{equation} A = B + iC. \end{equation}\]

The real part is explicitly given by

\[\begin{equation} B = \begin{pmatrix} I - \frac{\lambda}{2}(L + M) & 0 \\ 0 & I + \frac{\lambda}{2}(L + M) \end{pmatrix}. \end{equation}\]

Link to the definitions of:

The matrix \(L = DMD\) is used as an abbreviation. First, observe that both \(M\) and \(D\) are real symmetric matrices. Since the product of symmetric matrices of the form \(DMD\) preserves symmetry, it follows that \(L\) is also real symmetric. Consequently, \(B\) is real symmetric as it consists of symmetric blocks on the diagonal.

Spectral properties of M¶

The matrix \(M\) has the explicit form

\[\begin{equation} M = \begin{pmatrix} 2cs & s^2 - c^2 \\ s^2 - c^2 & -2cs \end{pmatrix}, \end{equation}\]

where \(c = \cos(2\vartheta)\) and \(s = \sin(2\vartheta)\). A direct computation shows that

\[\begin{equation} (2cs)^2 + (s^2 - c^2)^2 = 1, \end{equation}\]

which implies

\[\begin{equation} M^2 = \mathbb{1}. \end{equation}\]

This relation has an important consequence for the spectrum of \(M\). If \(v\) is an eigenvector of \(M\) with eigenvalue \(\lambda\), then

\[\begin{equation} M^2 v = \lambda^2 v. \end{equation}\]

Since \(M^2 = \mathbb{1}\), it follows that \(\lambda^2 = 1\), and therefore \(\lambda = \pm 1\). Thus, all eigenvalues of \(M\) lie in the set \(\{-1,1\}\).

Because \(M\) is real symmetric, its operator norm is equal to the largest absolute value of its eigenvalues (see [1]). Hence,

\[\begin{equation} \|M\|_2 = 1. \end{equation}\]

Bound on L¶

Next, we analyze the matrix \(L = DMD\). The matrix \(D = \mathrm{diag}(t_H, t_V)\) encodes detector losses, where \(t_{H,V} = 1 - \eta_{H,V}\) and \(0 \le \eta_{H,V} \le 1\). Therefore,

\[\begin{equation} 0 \le t_H, t_V \le 1. \end{equation}\]

For diagonal matrices, the operator norm is given by the largest absolute diagonal entry, and hence

\[\begin{equation} \|D\|_2 = \max(t_H, t_V) \le 1. \end{equation}\]

Using the submultiplicativity of the operator norm,

\[\begin{equation} \|AB\|_2 \le \|A\|_2 \|B\|_2, \end{equation}\]

we obtain

\[\begin{equation} \|L\|_2 = \|DMD\|_2 \le \|D\|_2^2 \|M\|_2 \le 1. \end{equation}\]

We now consider the sum \(L + M\). Using the triangle inequality,

\[\begin{equation} \|L + M\|_2 \le \|L\|_2 + \|M\|_2 \le 2. \end{equation}\]

Since \(L+M\) is real symmetric, this bound implies that all of its eigenvalues lie in the interval \([-2,2]\).

Positivity of B¶

We now determine the eigenvalues of \(B\). Due to its block-diagonal structure, the eigenvalues of \(B\) are given by those of the two blocks

\[\begin{equation} \mathbb{1} \pm \frac{\lambda}{2}(L+M). \end{equation}\]

Let \(v\) be an eigenvector of \(L+M\) with eigenvalue \(\mu\). Then

\[\begin{equation} (L+M)v = \mu v. \end{equation}\]

Applying the block matrices to \(v\) yields

\[\begin{equation} \left(\mathbb{1} \pm \frac{\lambda}{2}(L+M)\right)v = \left(1 \pm \frac{\lambda}{2}\mu\right)v. \end{equation}\]

Thus, the eigenvalues of \(B\) are

\[\begin{equation} \lambda_\pm = 1 \pm \frac{\lambda}{2}\mu. \end{equation}\]

Since \(\mu \in [-2,2]\), the smallest possible value of \(\lambda_\pm\) occurs for \(\mu = \pm 2\), yielding

\[\begin{equation} \lambda_\pm \ge 1 - \lambda. \end{equation}\]

For finite squeezing parameter \(r\), we have \(\lambda = \tanh(r)\) with \(0 \le \lambda < 1\), and therefore

\[\begin{equation} \lambda_\pm > 0. \end{equation}\]

Hence, all eigenvalues of \(B\) are strictly positive, and we conclude that \(B\) is positive definite.

Existence and uniqueness of the matrix square root of B¶

Since \(B\) is real symmetric, the spectral theorem guarantees an orthogonal diagonalization

\[\begin{equation} B = Q \Lambda Q^T, \end{equation}\]

where \(Q\) is orthogonal and \(\Lambda = \mathrm{diag}(\lambda_1,\dots,\lambda_n)\) contains the eigenvalues of \(B\) (see [1]). Since \(B\) is positive definite, all eigenvalues satisfy \(\lambda_i > 0\).

We define

\[\begin{equation} B^{1/2} := Q \Lambda^{1/2} Q^T, \end{equation}\]

where \(\Lambda^{1/2} = \mathrm{diag}(\sqrt{\lambda_1},\dots,\sqrt{\lambda_n})\). By construction,

\[\begin{equation} B^{1/2} B^{1/2} = B. \end{equation}\]

Since all eigenvalues \(\lambda_i\) are strictly positive, their square roots are well-defined and nonzero, implying that \(B^{1/2}\) is invertible. The inverse is given by

\[\begin{equation} B^{-1/2} = Q \Lambda^{-1/2} Q^T. \end{equation}\]

The square root constructed in this way is the unique real symmetric positive definite square root of \(B\) (see [1]).

Conclusion¶

We have shown that \(B\) is real symmetric and positive definite for all physically relevant parameters (\(0 \le \eta_{H,V} \le 1\), finite squeezing \(r\)). This ensures both the convergence of the Gaussian integral and the validity of the factorization used in equation (43) of Derivation of the coincidence probability.

References¶

[1] Gilbert Strang, “Introduction to Linear Algebra,” Wellesley - Cambridge Press, ed. 5, isbn 978-0-9802327-7-6, 2016.