# Decompositions¶

Note

In Strawberry Fields we use the convention $$\hbar=2$$ by default, but other conventions can also be chosen by setting the global variable sf.hbar at the beginning of a session. In this document we keep $$\hbar$$ explicit.

This page contains some common continuous-variable (CV) decompositions. However, before beginning, it is useful to state some important properties of Gaussian unitaries and the symplectic group.

## Gaussian unitaries¶

Gaussian states are those with positive Wigner distributions, and are fully characterised by their first and second moments; the means vector $$\hat{\mathbf{r}}=(\x_1,\dots,\x_N,\p_1,\dots,\p_N)$$, and the covariance matrix $$V_{ij}=\frac{1}{2}\langle\Delta r_i\Delta r_j + \Delta r_i\Delta r_j\rangle$$ respectively (for more details, see Gaussian States).

Gaussian unitaries, it then follows, are quantum operations that retain the Gaussian character of the state; i.e., the set of unitary transformations $$U$$ that transform Gaussian states into Gaussian states:

$\rho\rightarrow U\rho U^\dagger$

In the Hilbert space, these are represented by unitaries of the form $$U=e^{-iH/2}$$ with Hamiltonians $$H$$ that are at most second-order polynomials in the quadrature operators $$\x$$ and $$\p$$. In order to preserve the commutation relations $$[r_i,r_j]=i\hbar\Omega$$, where

$\begin{split}\Omega = \begin{bmatrix}0 & \I_N \\-\I_N & 0 \end{bmatrix}\end{split}$

is the symplectic matrix, the set of Gaussian unitary transformations $$U$$ in the Hilbert space is represented by real symplectic transformations on the first and second moments of the Gaussian state:

$\begin{split}\rho\rightarrow U\rho U^\dagger ~~~\Leftrightarrow ~~~ \begin{cases}\mathbf{r}\rightarrow S\mathbf{r}+\mathbf{d}\\ V\rightarrow S V S^T\end{cases}\end{split}$

Here, $$\mathbf{d}\in\mathbb{R}^{2N}$$, and $$S\in\mathbb{R}^{2N\times 2N}$$ is a symplectic matrix satisfying the condition $$S\Omega S^T=S$$.

## Williamson decomposition¶

Definition

For every positive definite real matrix $$V\in\mathbb{R}^{2N\times 2N}$$, there exists a symplectic matrix $$S$$ and diagonal matrix $$D$$ such that

$V = S D S^T$

where $$D=\text{diag}(\nu_1,\dots,\nu_N,\nu_1,\dots,\nu_N)$$, and $$\{\nu_i\}$$ are the eigenvalues of $$|i\Omega V|$$, where $$||$$ represents the element-wise absolute value.

Tip

Implemented in Strawberry Fields as a state preparation decomposition by strawberryfields.ops.Gaussian

The Williamson decomposition allows an arbitrary Gaussian covariance matrix to be decomposed into a symplectic transformation acting on the state described by the diagonal matrix $$D$$.

The matrix $$D$$ can always be decomposed further into a set of thermal states with mean photon number given by

$\bar{n}_i = \frac{1}{\hbar}\nu_i - \frac{1}{2}, ~~i=1,\dots,N$

### Pure states¶

In the case where $$V$$ represents a pure state ($$|V|-(\hbar/2)^{2N}=0$$), the Williamson decomposition outputs $$D=\frac{1}{2}\hbar I_{2N}$$; that is, a symplectic transformation $$S$$ acting on the vacuum. It follows that the original covariance matrix can therefore be recovered simply via $$V=\frac{\hbar}{2}SS^T$$.

Note

$$V$$ must be a valid quantum state satisfying the uncertainty principle: $$V+\frac{1}{2}i\hbar\Omega\geq 0$$. If this is not the case, the Williamson decomposition will return non-physical thermal states with $$\bar{n}_i<0$$.

## Bloch-Messiah (or Euler) decomposition¶

Definition

For every symplectic matrix $$S\in\mathbb{R}^{2N\times 2N}$$, there exists orthogonal symplectic matrices $$O_1$$ and $$O_2$$, and diagonal matrix $$Z$$, such that

$S = O_1 Z O_2$

where $$Z=\text{diag}(e^{-r_1},\dots,e^{-r_N},e^{r_1},\dots,e^{r_N})$$ represents a set of one mode squeezing operations with parameters $$(r_1,\dots,r_N)$$.

Tip

Implemented in Strawberry Fields as a gate decomposition by strawberryfields.ops.GaussianTransform

Gaussian symplectic transforms can be grouped into two main types; passive transformations (those which preserve photon number) and active transformations (those which do not). Compared to active transformation, passive transformations have an additional constraint - they must preserve the trace of the covariance matrix, $$\text{Tr}(SVS^T)=\text{Tr}(V)$$; this only occurs when the symplectic matrix $$S$$ is also orthogonal ($$SS^T=\I$$).

The Bloch-Messiah decomposition therefore allows any active symplectic transformation to be decomposed into two passive Gaussian transformations $$O_1$$ and $$O_2$$, sandwiching a set of one-mode squeezers, an active transformation.

### Acting on the vacuum¶

In the case where the symplectic matrix $$S$$ is applied to a vacuum state $$V=\frac{\hbar}{2}\I$$, the action of $$O_2$$ cancels out due to its orthogonality:

$SVS^T = (O_1 Z O_2)\left(\frac{\hbar}{2}\I\right)(O_1 Z O_2)^T = \frac{\hbar}{2} O_1 Z O_2 O_2^T Z O_1^T = \frac{\hbar}{2}O_1 Z^2 O_1^T$

As such, a symplectic transformation acting on the vacuum is sufficiently characterised by single mode squeezers followed by a passive Gaussian transformation ($$S = O_1 Z$$).

## Rectangular decomposition¶

The rectangular decomposition allows any passive Gaussian transformation to be decomposed into a series of beamsplitters and rotation gates.

Definition

For every real orthogonal symplectic matrix

$\begin{split}O=\begin{bmatrix}X&-Y\\ Y&X\end{bmatrix}\in\mathbb{R}^{2N\times 2N},\end{split}$

the corresponding unitary matrix $$U=X+iY\in\mathbb{C}^{N\times N}$$ representing a multiport interferometer can be decomposed into a set of $$N(N-1)/2$$ beamsplitters and single mode rotations with circuit depth of $$N$$.

For more details, see [1].

Tip

Implemented in Strawberry Fields as a gate decomposition by strawberryfields.ops.Interferometer

Note

The rectangular decomposition as formulated by Clements [1] uses a different beamsplitter convention to Strawberry Fields:

$BS_{clements}(\theta, \phi) = BS(\theta, 0) R(\phi)$

## References¶

1(1,2)

William R Clements, Peter C Humphreys, Benjamin J Metcalf, W Steven Kolthammer, and Ian A Walsmley. Optimal design for universal multiport interferometers. Optica, 3(12):1460–1465, 2016. doi:10.1364/OPTICA.3.001460.