Modular Bosonic Subsystem Codes

We introduce a framework to decompose a bosonic mode into two virtual subsystems—a logical qubit and a gauge mode. This framework allows the entire toolkit of qubit-based quantum information to be applied in the continuous-variable setting. We give a detailed example based on a modular decomposition of the position basis and apply it in two situations. First, we decompose Gottesman-Kitaev-Preskill grid states and find that the encoded logical state can be damaged due to entanglement with the gauge mode. Second, we identify and disentangle qubit cluster states hidden inside of Gaussian continuous-variable cluster states.

Figure 1

Subsystem decomposition of a single mode. (a) Modular decomposition of the position spectrum (the real line) into bins of size $\alpha$. Each bin is centered at $m\alpha$, with $m$ labeling the bin and $u\in [-\alpha /2,\alpha /2)$ giving the fractional remainder. Blue (red) indicates even (odd) values of $m$. (b) The even- and odd-$m$ subspaces are each isomorphic to the original Hilbert space. (c) Decomposition into a tensor product of a qubit subsystem and a gauge mode, where the parity of $m$ defines a logical qubit.

Figure 2

Encoding quality of approximate square-lattice GKP states ($\alpha =\sqrt{\pi }$) intended to encode the qubit state $|+⟩$. (a) Logical Bloch vector in the $xz$ plane for $|{+}_{\mathrm{GKP}}⟩$ states with fixed spike width $\mathrm{\Delta }=0.1$ and varying envelope parameter $\kappa$. Wave functions (solid blue) and envelopes (dashed black) are shown for $\kappa \in \{0.05,0.3,0.75,2\}$ to illustrate various approximate GKP states (A–D). For large envelopes, $\kappa \ll 1$, the logical qubit faithfully encodes $|+⟩$. For small envelopes, $\kappa \gg 1$, the logical state approaches $|0{⟩}_{\mathcal{L}}$, because the envelope damps all spikes except the one centered at zero. (b) Encoding quality captured by the logical fidelity, Eq. (4), and logical purity $\mathrm{Tr}[({\widehat{\rho }}_{\mathcal{L}}{)}^2]$.

Figure 3

Graphical representations of states in the subsystem decomposition. Different line styles indicate edge weights from Eq. (7). (a) Logical-gauge entanglement structure for an infinitely squeezed two-mode squeezed state. The decomposition of a single-mode state $|0{⟩}_p$ is grouped inside the dotted lines. (b) A modular-position measurement on mode 1 disconnects the middle node, indicated by dashed lines. (c) Then, a similar measurement on mode 2 fully disentangles the logical two-qubit cluster state. (d) Entanglement structure in a linear CVCS.