Abstract
We consider a device which can be programmed 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 in a non-destructive fashion. 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 generalised 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.
Highlights
A typical experiment with measurement of a quantum state involves preparing a circuit, which is often classically controlled, to perform a vast range of operations on the state
They showed that this probability increases with the number of copies of the state in the program register. They proposed an implementation of a state comparison test, where all output states are measured, with generic quantum states encoded in single photons. They showed that the same implementation provides a programmable projective measurement scheme involving balanced beam splitters and photon number-resolving detectors, in which the states in the program registers approximate the direction of the measurement, which is performed on the state in the input register
Motivated by the coherent state mapping of quantum protocols, we extend the results of Ref. [14] and introduce a programmable device which uses coherent states of light to perform a given projective measurement onto program coherent states, with commercially available passive linear optics components such as balanced beam splitters and singlephoton threshold detectors
Summary
A typical experiment with measurement of a quantum state involves preparing a circuit, which is often classically controlled, to perform a vast range of operations on the state. To succeed with an arbitrarily small desired error probability , this technique needs to perform independent tests on at-least logarithm of inverse- number of copies of both quantum states Generalizing this scenario, Chabaud et al [14] introduced a version of the swap test when one is provided with just a single copy of one of the states in some input register and multiple copies of the other state in the program registers. They proposed an implementation of a state comparison test, where all output states are measured, with generic quantum states encoded in single photons They showed that the same implementation provides a programmable projective measurement scheme involving balanced beam splitters and photon number-resolving detectors, in which the states in the program registers approximate the direction of the measurement, which is performed on the state in the input register. VII by giving a concrete improvement of the quantum fingerprinting protocol to solve the Euclidean distance problem [6]
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