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Optimal Joint Measurements of Qubit Observables

In quantum mechanics, two incompatible sharp observables and can nevertheless be simultaneously approximated by noisy measurements and at the expense of tolerating some inaccuracy. By quantifying this inaccuracy using a measure of error denoted here by , we want to find the pair that most accurately approximates under the constraint of joint measurability, and in doing so characterize the lower bound of the admissible region of error pairs spanned by all pairs of compatible approximators.
This Demonstration will find optimal approximators and the lower bound of the admissible region for the case of two sharp qubit observables, denoted by Bloch vectors and , with an adjustable degree of incompatibility . By moving the vector corresponding to , the optimizing vector that minimizes the error for while still being compatible with is found. The corresponding error pair is shown on the right-hand side with a red dot. Use the checkboxes to show the vectors of fixed error around and , the regions of joint measurability around and , and to automatically find the nearest pair of optimizing approximators (). Their corresponding error pair is given by a green dot on the right-hand side plot.

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This Demonstration considers a one-qubit system. Assume that the observables in question are dichotomic (two-valued) and unbiased; such an observable is expressed as the set of positive operators (commonly referred to as a POVM) where , with Bloch vector , , and is a vector whose elements are the three Pauli matrices. This is the Bloch vector plotted in this Demonstration. The incompatible observables have normalized Bloch vectors , with associated Hermitian operators , , and the degree of incompatibility between is given by , where denotes the operator norm. The approximating observables and are described in terms of the (generally subnormalized) Bloch vectors and , respectively. It is known that for these observables to be jointly measurable, it is necessary and sufficient that they satisfy . If we fix , for instance, then we define an ellipsoid of joint measurability with respect to :
,
and similarly we can define the ellipsoid for observables jointly measurable with . It can be shown that the optimal approximators lie within the plane spanned by and [1], and so we restrict to this circle within the Bloch sphere, and reduce to ellipses inscribed in this circle. As can be seen, by moving the location of , the shape of the ellipse changes: it will be near a line along as nears the surface of the circle, and it approaches a circle as nears the origin. The measure of error used is given by the metric of Bloch vectors in the qubit case (for a generic physical interpretation in terms of differences of the associated probability distributions, see [1, 2]),
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Geometrically, the Bloch vectors associated with observables of constant error relative to lie in a circle around the endpoint of , and so, while moving around this circle we are able to keep fixed, all the while changing the ellipse of joint measurability, allowing us to find different observables that minimize the error . Assuming that we fix , by minimizing the error under the constraint of joint measurability we find that the minimizing vector must lie on the edge of the ellipse ; that is, , and the circle of radius centered at must be tangent to at . This is demonstrated by choosing the first-order optimization, where the vector with the smallest associated error is found under the constraint of joint measurability. An optimal pair of approximating observables will be such that their associated Bloch vectors will lie on the surface of their respective joint measurability ellipses, and these ellipses will be tangent to the fixed error circles at that point. This is demonstrated by choosing the optimal approximators option, which finds the closest pair of observables that minimize the pair of errors.
As originally proven in [3], the minimum error curve in the plot is given parametrically, with :
,
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It is shown in [4] that this boundary curve of the admissible error region reflects a tradeoff between the two approximation errors and the degrees of the unsharpness of the approximating observables that is governed by the degree of incompatibility of the target observables .
References
[1] P. Busch and T. Heinosaari, "Approximate Joint Measurements of Qubit Observables," Quantum Information and Computation, 8(8), 2008 pp. 797–818.
[2] P. Busch, P. Lahti and R. F. Werner, "Heisenburg Uncertainty for Qubit Measurements," Physical Review A, 89(1), 2014 012129. doi:10.1103/PhysRevA.89.012129.
[3] S. Yu and C. H. Oh, "Optimal Joint Measurement of Two Observables of a Qubit." arxiv.org/abs/1402.3785.
[4] T. Bullock and P. Busch, "Measurement Uncertainty: The Problem of Characterising Optimal Error Bounds." arxiv.org/abs/1512.00104.
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