Remotely Preparing Many Qubits with a Single Photon

Abstract

A single photon in a superposition of $d$ modes naturally encode a $d$-dimensional quantum system, a so-called qudit. We show that such superpositions can be leveraged to achieve a quantum speed-up of remote remote state preparation (RSP): a primitive for several quantum network protocols. For a superposition over $d\geq 2$ modes, the photon state can encode up to ${\rm Log}_2(d)$ qubits, which we exploit in a proposed reflection based RSP protocol with multiple variations. For single qubit RSP, we achieve a performance comparable to the best known existing schemes but with reduced requirements for phase stabilization. For many qubit RSP the achievable success rates remain high despite needing exponentially many temporal modes, since only one photon needs to be transmitted and detected to prepare multiple qubits. By simultaneously preparing many qubits at once, we bypass limited qubit lifetimes limited qubit lifetimes and improve fidelities beyond what is achievable with existing RSP protocols.

Figures (5)

• Figure 1: Sketch of the setup for generalized remote state preparation of a multi-qubit register of length $n$. The steps are (a) preparation of a train of $2^n$ time-bins (purple), (b) the qubit-photon interaction, and (c) the removal of the which time-bin information (e.g. using a quantum Fourier transform or time-lens Barak2007Donohue2016) followed by single click heralding. The server does two of the steps including (b) and the remaining step, either (a) or (c), is performed by the client. The client additionally imprints individual phases on the time-bins, either after splitting in (a) or before removing the which-time-bin information in (c). The qubit photon interaction (b) is engineered by optical switches (small gray boxes) controling the path of the time-bins and thus with which qubits a time-bin interacts. The protocols requires that time-bin $x$ with binary representation $\vec{x} \in \{0,1\}^n$ interacts with qubit $l$ implementing a conditional phase flip iff $x_l = 1$.
• Figure 2: Tradeoff between fidelity $F$ and success probability $P$ when using a weak coherent pulse and photon number resolving detection. We compare the reflection based (R), single-click (SC), and double-click (DC) remote state preparation protocols, see legend. (a) a scenario limited by photon routing and detection where we take $\eta_s = \eta_0 \eta_d = \eta_1 \eta_d$ and (b) a scenario limited by the matter-photon interaction where we consider $\eta_0,\eta_d=1$, $\eta_s = C/(1+C)$, and $\eta_1 = [C/(2+C)]^2$. Note that we do not account for the contribution of phase fluctuations which are strongest for the SC protocol.
• Figure 3: Rate to prepare $n=k q =8$ qubits within a sliding window of $w=2000/2^{k-1}$ attempts using single-photon sources as a function of distance. Different distribution in $q$ batches (photon detections) and $k$ qubits prepared using a single photon are encoded in the linestyle, see legend. We also include the performance of DC-RSP. We assume $\eta_t$ corresponding to transmission over a distance $L$ in a fiber with a loss rate of $0.2\,\text{db}/\text{km}$ and the remaining quantum efficiencies during emission are in $\eta_{t,\text{intrinsic}} = 0.9$. Additionally we assume efficient spin-photon interaction with $\eta_0 = 0.9$ and $\eta_1 = \eta_0 [C/(C+2)]^2$ with $C=38$ as well as a detection efficiency of $\eta_d = 0.9$. We take the pulse duration to be $T_{\text{TB}} = 30\,$ns.
• Figure 4: Circuit illustrating an alternative implementation of the protocol based on photon absorption.
• Figure 5: Sketch of the input-output model for a single photon pulse interaction with a trapped ion mediated by a cavity.