Passive interferometric symmetries of multimode Gaussian pure states

As large-scale multimode Gaussian states begin to become accessible in the laboratory, their representation and analysis become a useful topic of research in their own right. The graphical calculus for Gaussian pure states provides powerful tools for their representation, while this work presents a useful tool for their analysis: passive interferometric (i.e., number-conserving) symmetries. Here we show that these symmetries of multimode Gaussian states simplify calculations in measurement-based quantum computing and provide constructive tools for engineering large-scale harmonic systems with specific physical properties, and we provide a general mathematical framework for deriving them. Such symmetries are generated by linear combinations of operators expressed in the Schwinger representation of $\mathrm{U}\left(2\right)$, called nullifiers because the Gaussian state in question is a zero eigenstate of them. This general framework is shown to have applications in the noise analysis of continuous-various cluster states and is expected to have additional applications in future work with large-scale multimode Gaussian states.

Figure 1

Precise graphical representation (described in Sec. 2c) of a two-mode squeezed (TMS) state with adjacency matrix $\mathrm{K}$ [given in Eq. (2.13)], where $\alpha >0$ is an overall squeezing parameter.

Figure 2

The graph of the adjacency matrix $\mathrm{K}$ for the dual-rail quantum wire with periodic boundary conditions. The weights of the links are $\pm \frac{1}{2}tanh\alpha$, differentiated by the blue and yellow links, respectively. Schwinger spins are formed by pairing qumodes together vertically, where each Schwinger spin is labeled by integer values 1–5 and contains two qumodes, labeled by $a$ and $b$.

Figure 3

The graph of the adjacency matrix for the dual-rail quantum wire in the distributed qumodes, with the qumode-operator relabeling (in terms of $\pm$ along the vertical axis) as indicated in Eq. (5.1). The weights of the links are all $tanh\alpha$ in this qumode decomposition.

Figure 4

A portion of the dual-rail quantum wire on periodic boundaries interpreted as a ring of Schwinger spins. The qumodes are paired vertically into Schwinger spins, labeled 1–5. The colored ellipses are an illustration of the chainlike nullifier that results from taking linear positive combinations of three of the $n$ independent nullifiers from Eq. (5.3) (where the green and red ellipses correspond to ${\widehat{S}}_{1a,1b}^0-{\widehat{S}}_{1a,1b}^x$ and $-{\widehat{S}}_{4a,4b}^0-{\widehat{S}}_{4a,4b}^x$, respectively), and the $2{\widehat{S}}^x$ operator acts on spins 2 and 3, indicated by the blue ellipses.

Figure 5

Schwinger spin structure for the dual-rail quantum wire with periodic boundary conditions, where the qumodes (labeled by $a$ and $b$) are paired together vertically into Schwinger spins, labeled 1–4. The blue ellipses are an illustration of the global nullifier for this state, i.e., an ${\widehat{S}}^x$ operator acting on every Schwinger spin of the system.

Figure 6

Alternate Schwinger spin structure for the dual-rail quantum wire with periodic boundary conditions (where the number of Schwinger spins is even). Note that the qumodes are now paired horizontally into spins. A global nullifier for this state is illustrated by the purple ellipses, which denote the ${\widehat{S}}^z$ operator acting on all of the Schwinger spins in the system.

Figure 7

Illustration of the local symmetry from (6.1). Applying $e^{-i2\varphi ({\widehat{S}}_{2a,2b}^0-{\widehat{S}}_{2a,2b}^x)}$ (illustrated by the green ellipse) is equivalent to applying $e^{-i2\varphi ({\widehat{S}}_{3a,3b}^0+{\widehat{S}}_{3a,3b}^x)}$ (illustrated by the red ellipse).