Derivation and experimental test of fidelity benchmarks for remote preparation of arbitrary qubit states

Remote state preparation (RSP) is the act of preparing a quantum state at a remote location without actually transmitting the state itself. Using at most two classical bits and a single shared maximally entangled state, one can in theory remotely prepare any qubit state with certainty and with perfect fidelity. However, in any experimental implementation the average fidelity between the target and output states cannot be perfect. In order for an RSP experiment to demonstrate genuine quantum advantages, it must surpass the optimal threshold of a comparable classical protocol. Here we study the fidelity achievable by RSP protocols lacking shared entanglement and determine the optimal value for the average fidelity in several different cases. We implement an experimental scheme for deterministic remote preparation of arbitrary photon polarization qubits, preparing $178$ different pure and mixed qubit states with an average fidelity of $0.995$. Our experimentally achieved average fidelities surpass our derived classical thresholds whenever the classical protocol does not trivially allow for perfect RSP.

Figure 1

Evaluating RSP protocols. Alice samples a state ${\rho }^{\mathrm{tar}}$ from a given distribution of target states and Bob aims to prepare a closely matching state. In classical RSP protocols, Alice may send only a limited number of classical bits to Bob. In quantum RSP protocols, the parties also share some predistributed entanglement. Their goal is to maximize $\langle F({\rho }^{\mathrm{tar}},{\rho }^{\mathrm{out}})\rangle$, the RSP fidelity averaged over the entire target distribution.

Figure 2

Example of a possible classical RSP strategy. The target ensemble consists of the six pure states $\{|0\rangle ,|1\rangle ,\frac{|0\rangle \pm |1\rangle }{\sqrt{2}},\frac{|0\rangle \pm i|1\rangle }{\sqrt{2}}\}$ (represented here as the vertices of an octahedron inscribed within the Bloch sphere) with equal probabilities. A possible partitioning strategy is given for the case where two cbits of classical communication are allowed, and the optimal output state for partition 11 is detailed.

Figure 3

Five examples of pure-state target ensembles, given by the vertices of the Platonic solids inscribed within the Bloch sphere, with uniform probability distributions. For both two and three cbits message capacity, optimal partitioning strategies are shown, along with the corresponding optimal average fidelity benchmarks. States labeled with the same symbol (e.g., red circles) are in the same partition. The tetrahedron (two or three cbits), octahedron (three cbits), and cube (three cbits) examples can in principle be remotely prepared with perfect fidelity using only classical communication, whereas the remaining ensembles cannot. Experimentally achieved mean fidelities for these ensembles are given in the bottom row; the reported uncertainty is the standard error of the mean. For all nonunity benchmarks, the experimental values surpass the benchmarks for two (three) transmitted cbits by at least 96 (46) times the standard error of the mean. The two-cbit octahedron bound also appears in Ref. [13].

Figure 4

RSP experiment. (a) Entangled photon pairs are produced via parametric downconversion in a polarization-based Sagnac interferometer [41, 42] (see text for details) and distributed to Alice and Bob in single-mode fibers (SMFs). (b) The apparatus used by Alice to perform a POVM on her photon. It is a double interferometer based on calcite beam displacers (BDs) which couple the polarization qubit to a “path” qubit for the generalized measurement. Further details are in the text. (c) Schematic of the entire experimental RSP protocol. The POVM $\{E_m\}$ is performed on Photon A. Based on the outcome $m$, a message is encoded in two classical electronic signals and then sent to Bob with probability $r$ which is controled by a veto signal. Dependent on the message, up to two Pockels cells (PCs) fire to perform the necessary unitary correction on Photon B, which has been delayed in a 50-m fiber to allow time to trigger the PCs. Bob’s output is analyzed using quantum state tomography. DM, dichroic mirror; IF, blocking and interference filter; PBS, polarizing beamsplitter; H(Q)WP, half (quarter) wave plate; SPCM, fiber-coupled single-photon counting module.

Figure 5

Experimentally reconstructed density matrices of our two-photon entangled state: real part (left panels) and imaginary part (right panels). The top row [(a) and (b)] represents the source state, as aligned for the remote preparation of the pure states (Fig. 6); the bottom row [(c) and (d)] represents the source state as aligned (on a subsequent day) for the preparation of mixed states (states with $r<1$ in Fig. 7 and Table .) The state in the top (bottom) row has fidelity $F=0.9807\pm 0.0004$ ($0.9813\pm 0.0003$) with the ideal state $|{\Phi }^{-}\rangle$, tangle $T=0.935\pm 0.002$ ($0.932\pm 0.001$), and purity $P=\mathrm{Tr}({\rho }^2)=0.9676\pm 0.0009$ ($0.9659\pm 0.0007$) [48].

Figure 6

Experimentally achieved mean fidelities $\langle F({\rho }^{\mathrm{tar}},{\rho }_m^{\mathrm{out}})\rangle$ and optimal classical benchmarks for target ensembles of pure states based on the five Platonic solids shown in Fig. 3. The error bars shown are the standard error of the mean. Any experimental data point above the green diamonds (blue squares) represents results that are not possible with only two (three) cbits communication and no preshared entanglement. In all cases where the classical benchmark is less than unity, the experimental results surpass the benchmarks conclusively. Lines are included only to guide the eye and do not represent calculated thresholds.

Figure 7

Experimentally achieved mean fidelities and optimal classical thresholds versus Bloch vector radius. The target ensembles consist of uniform distributions of states of Bloch radius $r$ which form the vertices of either an icosahedron or dodecahedron. The lower (upper) pair of lines are bounds on the classical average fidelity arising from specific two (three) cbit classical strategies. Within each pair, the higher bound is for the icosahedron. Experimental data must lie above and to the right of the bounds to be in the quantum regime, but even points in this region may be possible to achieve without preshared entanglement by some nondeterministic classical strategy. Experimental data points for the icosahedron ensembles are similar but were not plotted because at this scale they are not distinguishable from those of the dodecahedron.