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arXiv:1404.1477

Joint measurability through Naimark's theorem

Roberto Beneduci

Published Apr 5, 2014• 3 citations
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Abstract

We use Naimark's dilation theorem in order to characterize the joint measurability of two POVMs. Then, we analyze the joint measurability of two commutative POVMs $F_1$ and $F_2$ which are the smearing of two self-adjoint operators $A_1$ and $A_2$ respectively. We prove that the compatibility of $F_1$ and $F_2$ is connected to the existence of two compatible self-adjoint dilations $A_1^+$ and $A_2^+$ of $A_1$ and $A_2$ respectively. As a corollary we prove that each couple of self-adjoint operators can be dilated to a couple of compatible self-adjoint operators. Next, we analyze the joint measurability of the unsharp position and momentum observables and show that it provides a master example of the scheme we propose. Finally, we give a sufficient condition for the compatibility of two effects.

DOI
10.48550/arxiv.1404.1477

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We use Naimark's dilation theorem in order to characterize the joint measurability of two POVMs. Then, we analyze the joint measurability of two commutative POVMs F1F_1F1​ and F2F_2F2​ which are the smearing of two self-adjoint operators A1A_1A1​ and A2A_2A2​ respectively. We prove that the compatibility of F1F_1F1​ and F2F_2F2​ is connected to the existence of two compatible self-adjoint dilations A1+A_1^+A1+​ and A2+A_2^+A2+​ of A1A_1A1​ and A2A_2A2​ respectively. As a corollary we prove that each couple of self-adjoint operators can be dilated to a couple of compatible self-adjoint operators. Next, we analyze the joint measurability of the unsharp position and momentum observables and show that it provides a master example of the scheme we propose. Finally, we give a sufficient condition for the compatibility of two effects.
Abstract After a short review of the main properties of unsharp observables we provide examples of commutative unsharp observables obtained as the projection of sharp observables. In particular, we consider a family of PVMs defined on the tensor product Hilbert space H ⊗ K which include the one introduced by Ozawa in his modification of the von Neumann position measurement model and show that their projections onto H are commutative POVMs. Then, we make some observations on the concept of Naimark dilation and its connections with the integral representation of commutative POVMs. Finally, we introduce a partial order relation in the set of Naimark dilations, show that there is a minimal element and that it is unique. Minimality is a consequence of the partial ordered structure and coincides with the usual definition of minimal Naimark dilation. In passing, we use Naimark’s theorem in order to prove some well known properties of positive operator valued measures.
We obtain a formal characterization of the compatibility or otherwise of a set of positive-operator-valued measures (POVMs) via their Naimark extensions. We show that a set of POVMs is jointly measurable if and only if there exists a single Naimark extension, specified by a fixed ancilla state on the same ancilla Hilbert space, that maps them to a set of commuting projective measurements (PVMs). We use our result to obtain an easily checkable sufficient condition for the compatibility of a pair of dichotomic observables in any dimension. This in turn leads to a characterization of the compatibility regions for some important classes of observables including a pair of unsharp qubit observables. Finally, we also outline as to how our result provides an alternate approach to quantifying the incompatibility of a general set of quantum measurements.
The POVM measurement assemblage consisting of a set of positive operator-value measurements (POVMs) is an important tool for describing and studying the non-locality, quantum steering and quantum correlation of composite quantum systems. In this paper, we first introduce the definition of joint measurability of the POVM measurement assemblage MS\mathcal{M}_SMS​ consisting of a finite number of POVMs and then prove that the joint measurability of MS\mathcal{M}_SMS​ is equivalent to its compatibility. We give a general form of a POVM measurement which is jointly measurable with the one-rank projection measurement. It is proved that the POVM measurement assemblage MS\mathcal{M}_SMS​ is jointly measurable if and only if the corresponding “row stochastic operator matrix" MS\textbf{M}_SMS​ composed of MS\mathcal{M}_SMS​ can be expressed as an “operator convex combination" of {0,IS}\{0,I_S\}{0,IS​}—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage M(n0 ,n1)\mathcal{M}(\textbf{n}_{0~},\textbf{n}_{1})M(n0 ​,n1​) induced by two three-dimensional real unit vectors ni(i=0,1)\textbf{n}_{i}(i=0,1)ni​(i=0,1) is jointly measurable if and only if n0=±n1\textbf{n}_{0}=\pm\textbf{n}_{1}n0​=±n1​. As an application, we obtain a necessary and sufficient condition under which a two-qubit quantum state is unsteerable with M(n0,n1)\mathcal{M}(\textbf{n}_{0},\textbf{n}_{1})M(n0​,n1​).
Abstract Measurement incompatibility is one of the cornerstones of quantum theory. This phenomenon appears in many forms, of which the concept of non-joint measurability has received considerable attention in the recent years. In order to characterise this non-classical phenomenon, various analytical and numerical methods have been developed. The analytical approaches have mostly concentrated on the qubit case, as well as to scenarios involving sets of measurements with symmetries, such as position and momentum or sets of mutually unbiased bases. The numerical methods can, in principle, decide any finite-dimensional and discrete joint measurability problem, but they naturally have practical limitations in terms of computational power. These methods exclusively start from a given set of measurements and ask whether the set possesses incompatibility. Here, we take a complementary approach by asking which measurements are compatible with a given measurement. It turns out, that this question can be answered in full generality through a minimal Naimark dilation of the given measurement: the set of interest is exactly those measurements that have a block-diagonal representation in such dilation. We demonstrate the use of the technique through various qubit examples, leading to an alternative characterisation of all compatible pairs of binary qubit measurements, which retrieves the celebrated Busch criterion. We further apply the technique to special examples of trinary and continuous qubit measurements.
We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice’s pair of non-commuting observables in the presence of Bob’s entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice–Bob state, implementing incompatible POVMs at Bob’s end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix – which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables – could be realized if and only if compatible POVMs are employed.