Optimal quantum-programmable projective measurements with coherent states

We consider a device which can be programed using coherent states of light to approximate a given projective measurement on an input coherent state. We provide and discuss three practical implementations of this programmable projective measurement device with linear optics, involving only balanced beam splitters and single photon threshold detectors. The three schemes optimally approximate any projective measurement onto a program coherent state. We further extend these to the case where there are no assumptions on the input state. In this setting, we show that our scheme enables an efficient verification of an unbounded untrusted source with only local coherent states, balanced beam splitters, and threshold detectors. Exploiting the link between programmable measurements and generalized swap test, we show as a direct application that our schemes provide an asymptotically quadratic improvement in existing quantum fingerprinting protocol to approximate the Euclidean distance between two unit vectors.

Figure 1

Controlled-swap circuit employed to compare the incoming qubit states. The ancilla qubit is measured in the computational basis and this relates to the probability of telling the two qubit states apart.

Figure 2

Balanced beam splitter (BS) operation acting on input coherent states $|\alpha \rangle$ and $|\beta \rangle$. The lower output mode of the BS is measured with a single-photon threshold detector $D_0$. The probability of obtaining a click in $D_0$ relates to the projective measurement test of distinguishing the two coherent states.

Figure 3

Hadamard interferometer with four input modes.The input states are one input register state $|\alpha \rangle$ and three local states $|\beta \rangle$, one in each mode. The detectors $D_i$ are single-photon threshold detectors, $\forall i\in \{0,2\}$.

Figure 4

Amplifier scheme with four input modes. The input states are one tested state $|\alpha \rangle$ and three local states $|\beta \rangle$, one in each mode. The detectors $D_i$ are single-photon threshold detectors. This scheme may be contrasted with the one in Fig. 3

Figure 5

General amplifier scheme of size $M$, with one copy of $|\alpha \rangle$ and $M-1$ copies of $|\beta \rangle$: The first output modes of two interferometers described by $U_{n-1}$ are mixed on a balanced beam splitter. The input states are one tested state $|\alpha \rangle$ and $M-1$ local states $|\beta \rangle$, one in each mode. Indexing the spatial modes from 0 to $M-1$, the $2^k$ output modes are measured with single-photon threshold detectors labeled $D_k$, for $k=0\cdots n-1$.

Figure 6

The looped amplifier scheme. The coherent state $|\alpha \rangle$ is sent once, while at each loop a new coherent state $|2^{(k-1)/2}\rangle \beta$ for $k=\{1,..n\}$, with $n=logM$. The switch ensures a closed loop after the input state $|\alpha \rangle$ passes through. The clicks are collected in the detector $D_0$ for all the $n$ instances of the beam-splitter interaction. In the ideal case, when $|\alpha \rangle =|\beta \rangle$, the detector $D_0$ does not click across any of the $n$ instances. If the two states are different, however, there is a finite probability of a click across the $n$ instances.

Figure 7

Comparison of the completeness for $M=2$ and $M=4$ schemes in presence of experimental noise, as a function of the amplitude of the measured state. Here, we have chosen the experimental imperfection parameters $\eta =0.9,\nu =0.99$ as observed in standard optical setups.

Figure 8

Comparison of the soundness for $M=2$ and $M=4$ schemes in presence of experimental noise parameters $\eta =0.9$, $\nu =0.99$, for a fixed program state amplitude $\beta =1$.

Figure 9

Alice and Bob prepare coherent state fingerprints, $|1_x\rangle$ and $|1_y\rangle$, as a sequence of coherent pulses in $n$ modes, with the $j\text{th}$ mode amplitude encoding the information of $j\text{th}$ component of the vector. This is then interfered sequentially in the balanced beam splitter and the results are analyzed in single-photon threshold detector $D_0$.

Figure 10

Improved quantum fingerprinting protocol where Alice sends a single copy of ${|1\rangle }_x$ to the referee while Bob sends $M-1$ copies of the fingerprint $|1_y\rangle$ to the referee. Here the referee employs the amplifier scheme with $M-1$ beam splitters and $logM$ threshold detectors $\{D_0,D_1,\cdots ,D_{logM-1}\}$.