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Quantum walk as a generalized measuring device.

Paweł Kurzyński1, Antoni Wójcik2

  • 1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore and Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland.

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Summary
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We demonstrate how a one-dimensional quantum walk can perform generalized measurements, specifically positive operator value measures (POVMs), on a single qubit. This quantum walk approach can generate any rank 1 and rank 2 POVM elements for a qubit.

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Area of Science:

  • Quantum Information Science
  • Quantum Computing
  • Quantum Measurement Theory

Background:

  • Quantum measurements are typically described by projection operators, limiting them to von Neumann measurements.
  • Generalized measurements, described by positive operator value measures (POVMs), offer greater flexibility but are complex to implement.
  • Quantum walks, inspired by classical random walks, are a powerful tool for quantum information processing.

Purpose of the Study:

  • To demonstrate the implementation of generalized quantum measurements (POVMs) using a one-dimensional discrete-time quantum walk.
  • To show that a quantum walk can generate arbitrary rank 1 and rank 2 POVM elements for a single qubit.
  • To extend the concept of von Neumann measurements to POVMs by incorporating internal system evolution.

Main Methods:

  • Utilizing a one-dimensional discrete-time quantum walk model.
  • Engineering the quantum walk parameters to generate specific POVM elements.
  • Mapping particle position measurements in the quantum walk to POVM element measurements on a qubit.

Main Results:

  • A one-dimensional discrete-time quantum walk can implement generalized measurements (POVMs) on a single qubit.
  • Any set of rank 1 and rank 2 POVM elements for a single qubit can be generated by a properly engineered quantum walk.
  • The measurement of a particle at position x=i in the quantum walk corresponds to measuring a POVM element E_i on the qubit.

Conclusions:

  • Quantum walks provide a novel and effective framework for realizing generalized quantum measurements.
  • The proposed quantum walk model naturally extends the von Neumann measurement scheme to POVMs.
  • This work opens new avenues for implementing advanced quantum measurement protocols in quantum information processing.