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 the covariance matrix of the 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.

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