Positive Operator-valued Measure Elements Research Articles

How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of SL(2,C)

The generalized Q-function of a spin system can be considered the outcome probability distribution of a state subjected to a measurement represented by the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As fundamental as the SCS POVM is to the 2-sphere phase-space representation of spin systems, it has only recently been reported that the SCS POVM can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of curved-phase-space correspondences for quantum systems. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators of any spin j are shown to represent elements of the Lie group SL(2,C)≅Spin(3,C), a complex version of the usual unitary operators that represent elements of SU(2)≅Spin(3,R). Consequently, the associated POVM elements represent points in the symmetric space SU(2)∖SL(2,C), which can be recognized as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path integral, (Fokker-Planck) diffusion equation, and stochastic differential equations, are applied to show that the continuous isotropic POVM quickly limits to the SCS POVM, placing spherical phase space at the boundary of the fundamental Lie group SL(2,C) in an operationally meaningful way. Two basic mathematical tools are used to analyze the evolving Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan, decomposition associated with the symmetric pair SU(2) ⊂ SL(2,C). Informed by these tools, the three schochastic techniques are applied directly to the Kraus operators in a representation-independent – and therefore geometric – way (independent of any spectral information about the spin components).The Kraus-operator-centric, geometric treatment applies not just to SU(2) ⊂ SL(2,C), but also to any compact semisimple Lie group and its complexification. The POVM associated with the continuous isotropic measurement of Lie-group generators thus corresponds to a type-IV globally Riemannian symmetric space and limits to the POVM of generalized coherent states. This generalization is the focus of a sequel to this article.

Generalised uncertainty relations from finite-accuracy measurements

In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial regionσg> 0. This length scale is the standard deviation of the envelope function,g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a regionσ̃g>0in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by settingσg≃ℏG/c3, the Planck length, andσ̃g≃ℏΛ/3, where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.

Quantum control attack: Towards joint estimation of protocol and hardware loopholes

In this paper we present the quantum control attack on quantum key distribution systems. The cornerstone of the attack is that Eve can use unitary (polar) decomposition of her positive-operator valued measure elements, which allows her to realize the feed-forward operation (quantum control), change the states in the channel after her measurement and impose them to Bob. Below we consider the general eavesdropping strategy and the conditions those should be satisfied to provide the attack successfully. Moreover we consider several types of the attack, each of them is based on a different type of discrimination. We also provide the example on two non-orthogonal states and discuss different strategies in this case.

Naimark extension for the single-photon canonical phase measurement

We address the implementation of the positive operator-valued measure (POVM) describing the optimal M-outcomes discrimination of the polarization state of a single photon. Initially, the POVM elements are extended to projective operators by Naimark theorem, then the resulting projective measure is implemented by a Knill-Laflamme-Milburn scheme involving an optical network and photon counters. We find the analytical expression of the Naimark extension and the detection scheme that realise it for an arbitrary number of outcomes M = 2^N.

Quantum Calibration of Photon-Number-Resolving Detectors Based on Multi-Pixel Photon Counters

In this paper, we reconstructed the positive operator-valued measure (POVM) of a photon-number-resolving detector (PNRD) based on a multi-pixel photon counter (MPPC) by means of quantum detector tomography (QDT) at 791 nm and 523 nm, respectively. MPPC is a kind of spatial-multiplexing PNRD with a silicon avalanche photodiode (Si-APD) array as the photon receiver. Experimentally, the quantum characteristics of MPPC were calibrated at 2 MHz at two different wavelengths. The POVM elements were given by QDT. The fidelity of the reconstructed POVM elements is higher than 99.96%, which testifies that the QDT is reliable to calibrate MPPC at different wavelengths. With QDT and associated Wigner functions, the quantum properties of MPPC can be calibrated more directly and accurately in contrast with those conventional methods of modeling detectors.

Photodetector figures of merit in terms of POVMs

A photodetector may be characterized by various figures of merit such as response time, bandwidth, dark count rate, efficiency, wavelength resolution, and photon-number resolution. On the other hand, quantum theory says that any measurement device is fully described by its positive-operator-valued measure (POVM) which generalizes the textbook notion of the eigenstates of the appropriate hermitian operator (the ‘observable’) as measurement outcomes. Here we show how to define a multitude of photodetector figures of merit in terms of a given POVM. We distinguish classical and quantum figures of merit and issue a conjecture regarding trade-off relations between them. We discuss the relationship between POVM elements and photodetector clicks, and how models of photodetectors may be tested by measuring either POVM elements or figures of merit. Finally, the POVM is advertised as a platform-independent way of comparing different types of photodetectors, since any such POVM refers to the Hilbert space of the incoming light, and not to any Hilbert space internal to the detector.

Quantum detector tomography of a single-photon frequency upconversion detection system.

We experimentally presented a full quantum detector tomography of a synchronously pumped infrared single-photon frequency upconversion detector. A maximum detection efficiency of 37.6% was achieved at the telecom wavelength of 1558 nm with a background noise about 1.0 × 10-3 counts/pulse. The corresponding internal quantum conversion efficiency reached as high as 84.4%. The detector was then systematically characterized at different pump powers to investigate the quantum decoherence behavior. Here the reconstructed positive operator valued measure elements were equivalently illustrated with the Wigner function formalism, where the quantum feature of the detector is manifested by the presence of negative values of the Wigner function. In our experiment, pronounced negativities were attained due to the high detection efficiency and low background noise, explicitly showing the quantum feature of the detector. Such quantum detector could be useful in optical quantum state engineering, quantum information processing and communication.

Experimental realization of generalized qubit measurements based on quantum walks

We report an experimental implementation of a single-qubit generalized measurement scenario, the positive-operator valued measure (POVM), based on a quantum walk model. The qubit is encoded in a single-photon polarization. The photon performs a quantum walk on an array of optical elements, where the polarization-dependent translation is performed via birefringent beam displacers and a change of the polarization is implemented with the help of wave plates. We implement: (i) trine POVM, i.e., the POVM elements uniformly distributed on an equatorial plane of the Bloch sphere; (ii) symmetric-informationally-complete (SIC) POVM; and (iii) unambiguous discrimination of two nonorthogonal qubit states.

Tomography of spatial mode detectors.

Transformation and detection of photons in higher-order spatial modes usually requires complicated holographic techniques. Detectors based on spatial holograms suffer from non-idealities and should be carefully calibrated. We report a novel method for analyzing the quality of projective measurements in spatial mode basis inspired by quantum detector tomography. It allows us to calibrate the detector response using only gaussian beams. We experimentally investigate the inherent inaccuracy of the existing methods of mode transformation and provide a full statistical reconstruction of the POVM (positive operator valued measure) elements for holographic spatial mode detectors.

Quantum characterization of superconducting photon counters

We address the quantum characterization of photon counters based on transition-edge sensors (TESs) and present the first experimental tomography of the positive operator-valued measure (POVM) of a TES. We provide the reliable tomographic reconstruction of the POVM elements up to 11 detected photons and M = 100 incoming photons, demonstrating that it is a linear detector.

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

AbstractThis paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self‐delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program‐size complexity H (s) of a given finite binary string s. In the standard way, H (s) is defined as the length of the shortest input string for U to output s. In the other way, the so‐called universal probability m is introduced first, and then H (s) is defined as –log2 m (s) without reference to the concept of program‐size.Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator‐valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour‐El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi‐POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi‐POVM. The validity of this definition is discussed.In what follows, we introduce an operator version $ \hat H $(s) of H (s) in a Hilbert space of infinite dimension using a universal semi‐POVM, and study its properties. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Pure-state informationally complete and “really” complete measurements

I construct a positive-operator-valued measure (POVM) which has $2d$ rank-1 elements and which is informationally complete for generic pure states in $d$ dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (e-print quant-ph∕0404137). I show that if a rank-1 POVM is required to be informationally complete for all pure states in $d$ dimensions, it must have at least $3d\ensuremath{-}2$ elements. I also show that, in a POVM which is informationally complete for all pure states in $d$ dimensions, for any vector there must be at least $2d\ensuremath{-}1$ POVM elements which do not annihilate that vector.

Minimal Informationally Complete Measurements for Pure States

We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PSI-complete) POVM. We show that a measurement with 2D-1 outcomes cannot be PSI-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PSI-complete POVM has 2D outcomes. We also consider PSI-complete POVMs that have only rank-one POVM elements and construct an example with 3D-2 outcomes, which is a generalization of the tetrahedral measurement for a qubit. The question of the minimal number of elements in a rank-one PSI-complete POVM is left open.