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arXiv:2310.12302 · Canadian Journal of Physics

Conditions for the existence of positive operator valued measures

Maximilian Schumacher, G. Alber

Published Jun 4, 2024• 4 citations
arXivHTMLPDF

Abstract

Sufficient and necessary conditions are presented for the existence of ( N, M)-positive operator valued measures (( N, M)-POVMs) valid for arbitrary-dimensional quantum systems. Firstly, a sufficient condition for the existence of ( N, M)-POVMs is presented. It yields a simple relation determining an upper bound on the continuous parameter of an arbitrary ( N, M)-POVM, below which all its POVM elements are guaranteed to be positive semidefinite. Secondly, for arbitrary optimal ( N, M)-POVMs conditions on their existence are derived, which exhibit a close connection to the existence of sets of ( M − 1) N isospectral, traceless, orthonormal, and hermitian operators with particular common spectra. The specific form of the two possible types of spectra depends on whether M is smaller than the dimension of the quantum system under consideration or not. For the special case of M = 2 and dimensions, which are powers of two, these sets of operators always exist and can be expressed in terms of the Clifford algebra of tensor products of Pauli operators.

DOI
10.1139/cjp-2023-0281

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Sufficient and necessary conditions are presented for the existence of ( N, M)-positive operator valued measures (( N, M)-POVMs) valid for arbitrary-dimensional quantum systems. Firstly, a sufficient condition for the existence of ( N, M)-POVMs is presented. It yields a simple relation determining an upper bound on the continuous parameter of an arbitrary ( N, M)-POVM, below which all its POVM elements are guaranteed to be positive semidefinite. Secondly, for arbitrary optimal ( N, M)-POVMs conditions on their existence are derived, which exhibit a close connection to the existence of sets of ( M − 1) N isospectral, traceless, orthonormal, and hermitian operators with particular common spectra. The specific form of the two possible types of spectra depends on whether M is smaller than the dimension of the quantum system under consideration or not. For the special case of M = 2 and dimensions, which are powers of two, these sets of operators always exist and can be expressed in terms of the Clifford algebra of tensor products of Pauli operators.
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in Cn\mathbb{C}^{n}Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such n2n^{2}n2 vectors in Cn\mathbb{C}^{n}Cn for any nnn .But the existence is only proved when n≤15n\leq 15n≤15 and n=19n=19n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in Cn\mathbb{C}^{n}Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1 ρ∈Mn(C)\rho\in M_{n}(\mathbb{C})ρ∈Mn​(C) is called a density matrix (or state) if ρ≥0\rho\geq 0ρ≥0 and ′b(ρ)=1.\prime b(\rho)=1.′b(ρ)=1.For any density matrix ρ\rhoρ , we can define a state ρ^\hat{\rho}ρ^​ by ρ^(X)=h(ρX)\hat{\rho}(X)=h(\rho X)ρ^​(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t (≻(\succ(≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators {Pi}i=1k⊂Mn(C)\{P_{i}\}_{i=1}^{k}\subset M_{n}(\mathbb{C}){Pi​}i=1k​⊂Mn​(C) is called a positive operator valued measure (POVM) if Pi≥0(1≤i≤k)P_{i}\geq 0(1\leq i\leq k)Pi​≥0(1≤i≤k) and ∑i=1kP=I.\sum_{i=1}^{k}P=I.∑i=1k​P=I.If P1P_{1}P1​ is a projection, then {Pi}i=1k\{P_{i}\}_{i=1}^{k}{Pi​}i=1k​ is called a projection valued measure (PVM)(PVM)(PVM) .
Determining the conditions under which positive operator-valued measures (POVMs), the most general class of quantum measurements, outperform projective measurements remains a challenging and largely unresolved problem. Of particular interest are projectively simulable POVMs, which can be realized through probabilistic mixtures of projective measurements, and therefore offer no advantage over projective schemes. Characterizing the boundary between simulable and non-simulable POVMs is, however, a difficult task, and existing tools either fail to scale efficiently, provide limited experimental feasibility or work only for specific POVMs. Here, we introduce and demonstrate a general method to certify non-simulability of a POVM by introducing a complete hierarchy of semidefinite programs. It provides upper bounds on the non-simulability measure of critical visibility of arbitrary POVMs which are tight in many cases and outperform previously known criteria. We experimentally certify the non-simulability of two- and three-dimensional POVMs using a trapped-ion qudit quantum processor by constructing non-simulability witnesses and introduce a modification of our framework that makes them robust against state preparation errors. Finally, we extend our results to the setting where an additional ancilla system is available.
We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.
Various forms of optimality for quantum observables described as normalized positive-operator-valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables that are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient conditions are imposed on informationally complete POVMs within the set of pure states. We also discuss joint and sequential measurements of optimal quantum observables.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.