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arXiv:quant-ph/0608072 · Electronic Notes in Theoretical Computer Science

POVMs and Naimark's Theorem Without Sums

Bob Coecke, Éric Oliver Paquette

Published Jul 1, 2008• 49 citations
arXivHTMLPDF

Abstract

We provide a definition of POVM in terms of abstract tensor structure only. It is justified in two distinct manners. i. At this abstract level we are still able to prove Naimark's theorem, hence establishing a bijective correspondence between abstract POVMs and abstract projective measurements (cf. [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035]) on an extended system, and this proof is moreover purely graphical. ii. Our definition coincides with the usual one for the particular case of the Hilbert space tensor product. We also provide a very useful normal form result for the classical object structure introduced in [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035].

DOI
10.1016/j.entcs.2008.04.015

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We provide a definition of POVM in terms of abstract tensor structure only. It is justified in two distinct manners. i. At this abstract level we are still able to prove Naimark's theorem, hence establishing a bijective correspondence between abstract POVMs and abstract projective measurements (cf. [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035]) on an extended system, and this proof is moreover purely graphical. ii. Our definition coincides with the usual one for the particular case of the Hilbert space tensor product. We also provide a very useful normal form result for the classical object structure introduced in [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035].
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.
Abstract In this chapter we first review projective or von Neumann measurements from the point of view of the projection postulate and the Lüders rule, before turning to more general measurements modelled using positive operator-valued measures (POVMs). After examining their mathematical description, we discuss symmetric informationally complete (SIC) POVMs and Zauner’s conjecture, as well as the representation of POVMs as projective measurements on a larger Hilbert space, a result called Naimark dilation. We then a model for non-ideal projective measurements and analyse the properties of such measurements in relation to the properties unbiasedness, faithfulness, and non-invasiveness of ideal projective measurements. Finally, we consider the problem of distinguishing non-orthogonal quantum states and examine how the use of POVMs can be beneficial for this task.
Abstract Sums play a prominent role in the formalisms of quantum mechanics, whether for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capability. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are †-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras.
In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom.
We revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.
Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces.Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product.The corresponding axioms define classical spaces as objects that allow copying and deleting data.Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data.The sums turn out to be an implicit implementation of this capability.Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus.In category-theoretic terms, classical data types are †-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras. QUANTUM MEASUREMENTS WITHOUT SUMS1 But while computing devices do manipulate strings of 0s and 1s, and high-level modern programming is a matter of providing a convenient interface with that process, the language for quantum information and computation we seek is not a convenient superstructure, but the meaningful infrastructure. 2A closely related knot-theoretical scheme has been put forward by Kauffman in [23].7 See [17] for a discussion why we call this 'formal decohering'.