Anonymous broadcasting of classical information with a continuous-variable topological quantum code

Broadcasting information anonymously becomes more difficult as surveillance technology improves, but remarkably, quantum protocols exist that enable provably traceless broadcasting. The difficulty is making scalable entangled resource states that are robust to errors. We propose an anonymous broadcasting protocol that uses a continuous-variable surface-code state that can be produced using current technology. High squeezing enables large transmission bandwidth and strong anonymity, and the topological nature of the state enables local error mitigation.

Figure 1

Sketch of the protocol. (a) A CV surface-code ground state (with squeezed logical modes) is prepared on a torus. The players decide beforehand to perform measurements along a loop around the torus (shown in blue). (b) The state is distributed to the players, one wedge to each. (c) Closeup of the lattice on the broadcaster's wedge. Physical bosonic modes are assigned to each edge, and each edge is assigned an orientation. Similarly, the faces are given a uniform orientation (one face is shown for reference). For the indicated edge $e, f(e,\mathcal{P})=+1$ with respect to path $\mathcal{P}$ and $f(e,\tilde{\mathcal{P}})=+1$ with respect to path $\tilde{\mathcal{P}}$ (see Sec. 3a). The broadcaster performs the unitary displacement, Eq. (17), on a loop $\tilde{\mathcal{P}}$ around her wedge (shown in red), which encodes a message $r\in \mathbb{R}$. Next, each party $j$ measures an operator, Eq. (18), along an arc ${\mathcal{P}}_j$ of the loop $\mathcal{P}$ (shown in blue). The players publicly announce their measurement outcomes ${\left\{m_j\right\}}_{j=1}^n$, and the broadcast message is computed as their (noisy) weighted sum. (d) Error mitigation strategy. Additional blue (green) ancillae are quasilocally coupled to modes surrounding faces (vertices), which perform error mitigation by dissipative cooling (see Sec. 7).

Figure 2

Contour plot of the geometric-mean probability $p$ that the broadcaster is correctly identified during the protocol as a function of the number of players $n$ and the channel capacity $C$ [Eq. (22)] in the limit of a large number of independent broadcast events. More precisely, we plot an upper bound on $p$, which we calculate using Eq. (44) and the upper bound for $I(\mathbf{M},A)$ from Eq. (42). Contours corresponding to $p=0.01,0.02,0.03$ are labeled, and subsequent contours increase by 0.01 each. The white region corresponds to $p>0.25$. The squeezing is 20 dB ($s={10}^{\text{(\#dB)}/20}=10$), and each player's wedge width is $w=6$ (see Appendix pp2). The inset shows (i) a solid blue curve corresponding to a cross section of the main plot along the green $C=1$ line and (ii) a dashed red curve corresponding to $p=1/n$. The latter corresponds to perfect tracelessness (no more risk than guessing randomly), which is only achieved in the trivial limit of no broadcast ($C=0$) or, for any $C>0$, in the asymptotic limit of infinite squeezing.

Figure 3

Average probability that the broadcaster is correctly identified in a single-shot broadcast of exactly one bit of data (i.e., only the sign of the broadcast message is recorded). The dots are plots of the probability to guess correctly, Eq. (66), for various values of $n$ and amounts of squeezing (in dB), with a wedge width $w=6$. The orange plane is $1/n$, which corresponds to uniformly random guessing. The broadcast magnitude $r_0$ is scaled according to Eq. (48) to correspond to (a) 1% probability of bit-flip error and (b) 0.0001% probability of bit-flip error.

Figure 4

Example of state maintenance via dissipative error mitigation. The initial state is the CV toric-code vacuum $|{\text{GS}}_{\mathrm{vac}}\rangle$, Eq. (12), defined by the symmetric nullifiers, Eqs. (6), on a $6\times 24$ lattice with 10 dB of squeezing ($s=\sqrt{10}$). (a) Infidelity with $|{\text{GS}}_{\mathrm{vac}}\rangle , 1-\mathcal{F}(\widehat{\rho },\widehat{\sigma })$, where fidelity is given by Eq. (70). (b) Excitation number, $\langle {\widehat{\eta }}^{†}\widehat{\eta }\rangle$, for a single nullifier (identical for face or vertex). In both (a) and (b), the curves are ordered ${\gamma }_{\mathrm{cool}}/{\gamma }_{\mathrm{loss}}=\{{10}^3,{10}^4,{10}^5,{10}^6\}$ from top to bottom.

Figure 5

Basic graph $\mathbf{G}$ for temporal-mode CV cluster states [38]; the full graph [58] is given in Eq. (71). $\mathbf{G}$, as shown, also represents frequency-mode CV cluster states [36, 59, 60] up to trivial $\pi$ phase shifts that merely flip the sign of some of the edges. Notice that $\mathbf{G}$ has the overall structure of a square lattice, but the individual nodes of that lattice are now collections of four nodes called macrocodes. Each macronode is identified by its surrounding red oval. In the temporal-mode case [38], each of the four nodes within a macronode is a synchronous temporal mode in four spatially separate laser beams. In the cylindrical frequency-mode case [36], each of the four nodes within a macronode share a common frequency but differ in spatial beam and polarization. The toroidal frequency-mode case [59, 60] is more complicated in structure and offers no advantages over the cylindrical one, so we do not consider it further.

Figure 6

Basic graph $\mathbf{G}$ for the temporal-mode linear CV cluster state reported in Ref. [39]; the full graph $\mathbf{Z}$ [58] is obtained from this through Eq. (71). $\mathbf{G}$, as shown, also represents frequency-mode CV cluster states reported in Ref. [37] up to trivial $\pi$ phase shifts that merely flip the sign of some of the edges. Notice that $\mathbf{G}$ has the overall structure of a line graph, but the individual nodes of that lattice are now collections of two nodes called macrocodes. Each macronode is identified by its surrounding red oval. In the temporal-mode experiment [38, 39], each node within a macronode is a synchronous temporal mode in spatially separate laser beams. In the frequency-mode experiment [36, 37], each node is one of two polarizations with the same frequency.

Figure 7

Toroidal CV cluster state and toroidal CV surface-code state $|{\text{GS}}_{\mathrm{sq}}\rangle$ [16]. (a) Portion of a CV cluster state with toroidal boundary conditions. Red edges have weight 1, and cyan self-loops have weight $i{s}^{-2}$ [58]. (b) Portion of a CV surface-code state with toroidal boundary conditions (CV toric-code state). Unlabeled edges all have weight $i{s}^2$. This state is generated by measuring $\widehat{p}$ and $\widehat{q}$ on the odd nodes of panel (a) in a diagonally alternating pattern. The $\widehat{p}$ measurements delete the node and produce a criss-cross pattern in panel (b) where the node used to be. The $\widehat{q}$ measurements just delete the node. (In this case, $\widehat{q}$ was measured on nodes $71,73,97,99$; $\widehat{p}$ was measured on the other visible odd-numbered nodes; and so on.)

Figure 8

Graphs for the (a) linear CV cluster state and (b) CV GHZ state, with all edge weights labeled explicitly. Measuring $\widehat{p}$ on the even nodes in panel (a) produces panel (b). Notice that the self-loops at the ends of the GHZ state have a different weight from the ones in the middle.

Figure 9

Open-boundary CV cluster state and CV surface-code state. (a) CV cluster state with open boundaries. Red edges have weight 1, and cyan self-loops have weight $i{s}^{-2}$ [58]. (b) CV surface-code state with smooth boundaries on the top and bottom and with rough boundaries on the left and right. Unlabeled edges all have weight $i{s}^2$. Starting from panel (a), the smooth boundaries are generated by measuring $\widehat{p}$ on nodes $6,16,26,10,20,30$. The rough boundaries are generated by measuring $\widehat{q}$ on nodes $2,4,32,34$. An alternating pattern of $\widehat{p}$ and $\widehat{q}$ measurements on all remaining even nodes completes the transition to the surface-code state. (The terms “smooth” and “rough” are chosen by convention to visually match the boundaries of the resulting graph.) Also notice that the three horizontal lines extending the full width of panel (b) have the same weights as the CV GHZ state from Fig. 8.