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Experimental scheme for qubit and qutrit SIC-POVMs using multiport devices

Gelo Noel M. Tabia

Published Jul 25, 2012• 3 citations
OpenAlex

Abstract

It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose are symmetric, informationally complete POVMs. SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather non-trivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been emphasized by the Zeilinger group in Vienna. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.

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Symmetric, informationally complete, positive operator-valued measurements (SIC-POVM) are fundamental in quantum information and foundations of quantum theory[1]. For example, SIC-POVMs are crucial for optimal quantum state tomography, quantum key distribution with optimal trade-off between security and key rate, device-independent certification of optimal randomness from one bit of entanglement, and information-based interpretations and operational reconstructions of quantum theory. Experimentally, photonic experiments of qubit and qutrit tomography, and quantum key distribution[2] have aimed for SIC-POVMs. However, these experiments rely on specific models for parts of the experimental setups and do not allow us to test whether the target measurements are truly SIC-POVMs in a "device independent" (DI) way that is, without relying on any modelling of the devices.

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It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose are symmetric, informationally complete POVMs. SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather non-trivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been emphasized by the Zeilinger group in Vienna. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and by performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather nontrivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been demonstrated by several experimental groups around the world. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
Symmetric informationally complete positive operator-valued measurement (SIC-POVM) is one important class of quantum measurement which is crucial for various quantum information processing tasks. SIC-POVMs have the advantage of providing an unbiased estimator for quantum states with the minimal number of outcomes needed for full tomography. We present an experimental approach on a photonic quantum walk which can be used to implement SIC-POVMs on a single-qbubit. The projection measurements of single-photons' positions correspond to elements of SIC-POVM on the polarization of single-photons.
Generalized quantum measurements play a crucial role in quantum mechanics, and symmetric informationally complete positive operator-valued measurements (SIC POVMs) provide a powerful and flexible framework for extracting information from quantum systems. However, the existence of SIC-POVMs in every finite dimension remains an open question, which has stimulated extensive research into alternative classes of POVMs. Recently, Geng etetet alalal. [Phys. Rev. Lett. 126, 100401 (2021)] proposed a broader class of SIC POVM, called semisymmetric informationally complete POVM (semi-SIC POVM), which extends beyond SIC POVM. In this work, we focus on the four-outcome POVMs and experimentally realize the semi-SIC POVMs using a one-dimensional discrete-time quantum walk. Additionally, employing single photons and linear optics, we perform an experimental self-testing of semi-SIC POVMs in the semi-device-independent manner. Our results pave the way for exploring quantum certification with generalized quantum measurements.
Generalized quantum measurements (also known as positive operator-valued measures or POVMs) are of great importance in quantum information and quantum foundations, but often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric, informationally complete-POVM (or SIC-POVM), is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC-POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.
Reconstructing the state of a quantum system, called quantum state tomography, is an essential task in quantum technologies.Standard methods are inefficient, as they scale exponentially with the number of qubits.Better scaling methods, such as SIC-POVM tomography, are not yet widespread due to their measurement complexity.As such, there is an urgent need to find efficient and compact devices for performing tomography of photonic states.Here we present simulations of photonic devices implementing SIC-POVM tomography in photonic chips.We obtain fidelities F = 0.8452 (for qubits) and F = 0.7609 (for qutrits).Our results show that we can design efficient and compact SIC-POVM tomography modules for integrated-photonics quantum chips.Since integrated photonics is a major quantum technology platform, we expect our results to be instrumental in the future development of compact, costeffective quantum devices.Applications include quantum communications, quantum computing, quantum sensing and quantum imaging.
Symmetric, informationally complete, positive operator-valued measurements (SIC-POVM) are fundamental in quantum information and foundations of quantum theory[1]. For example, SIC-POVMs are crucial for optimal quantum state tomography, quantum key distribution with optimal trade-off between security and key rate, device-independent certification of optimal randomness from one bit of entanglement, and information-based interpretations and operational reconstructions of quantum theory. Experimentally, photonic experiments of qubit and qutrit tomography, and quantum key distribution[2] have aimed for SIC-POVMs. However, these experiments rely on specific models for parts of the experimental setups and do not allow us to test whether the target measurements are truly SIC-POVMs in a "device independent" (DI) way that is, without relying on any modelling of the devices.