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APPROXIMATE JOINT MEASURABILITY OF SPIN ALONG TWO DIRECTIONS

Teiko Heinosaari, Peter Stano, Daniel Reitzner

Published Nov 21, 2012• 4 citations
OpenAlex

Abstract

Abstract. We study the existence of jointly measurable POVM approximations to two non-commuting sharp spin observables. We compare two different ways to specify optimal approximations. 1.

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APPROXIMATE JOINT MEASURABILITY OF SPIN ALONG TWO DIRECTIONS

TEIKO HEINOSAARI, PETER STANO, DANIEL REITZNER

We study the existence of jointly measurable POVM approximations to two non-commuting sharp spin observables. We compare two different ways to specify optimal approximations.

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NOTES ON JOINT MEASURABILITY OF QUANTUM OBSERVABLES

Teiko Heinosaari, Daniel Reitzner, Peter Stano

For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hold. Also some possible candidates which would accompany joint measurability and generalize these apparently useful properties are discussed.

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Approximate joint measurements of qubit observables

Paul Busch, Teiko Heinosaari

Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measurements. The marginal observables found to be optimal approximators are generally not among the coarse-grainings of the observables to be approximated. This yields scope for the improvement of existing joint measurement schemes. Both the quality of the approximations and the intrinsic unsharpness of the approximators are shown to be subject to Heisenberg-type uncertainty relations.

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Joint measurability through Naimark's theorem

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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 $F_1$ 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.

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On coexistence and joint measurability of rank-1 quantum observables

Juha-Pekka Pellonpää

We show that a rank-1 quantum observable (POVM) is jointly measurable with a quantum observable exactly when is a smearing of . If is extreme, rank-1 and discrete, then and are coexistent if and only if they are jointly measurable.

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Structures in the Set of Quantum Observables: Coexistence, Unsharpness, Approximation

Paul Busch, Garrett Birkhoff, John Von Neumann

refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be

•Dec 26, 2014•0 citations
For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hold. Also some possible candidates which would accompany joint measurability and generalize these apparently useful properties are discussed.
Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measurements. The marginal observables found to be optimal approximators are generally not among the coarse-grainings of the observables to be approximated. This yields scope for the improvement of existing joint measurement schemes. Both the quality of the approximations and the intrinsic unsharpness of the approximators are shown to be subject to Heisenberg-type uncertainty relations.
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.
refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be