[Paper Review] Optimal joint measurement of two observables of a qubit

Sixia Yu, Changhun Oh|arXiv (Cornell University)|Feb 16, 2014

Quantum Mechanics and Applications3 citations

TL;DR

This paper provides the exact tradeoff relation between worst-case errors in jointly measuring two incompatible observables of a qubit, deriving a complete characterization of all achievable error pairs. It constructs explicit, ancilla-free optimal joint measurements that saturate the derived bounds, offering a tight, state-independent uncertainty relation for qubit observables with arbitrary asymmetry in errors.

ABSTRACT

Heisenberg's uncertainty relations for measurement quantify how well we can jointly measure two complementary observables and have attracted much experimental and theoretical attention recently. Here we provide an exact tradeoff between the worst-case errors in measuring jointly two observables of a qubit, i.e., all the allowed and forbidden pairs of errors, especially asymmetric ones, are exactly pinpointed. For each pair of optimal errors we provide an optimal joint measurement that is realizable without introducing any ancilla and entanglement. Possible experimental implementations are discussed and Toronto experiment [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)] can be readily adapted to an optimal joint measurement of two orthogonal observables.

Motivation & Objective

• To determine the complete set of all possible and forbidden pairs of worst-case errors in jointly measuring two incompatible observables of a qubit.
• To construct explicit optimal joint measurements that achieve these minimal errors without requiring ancillary systems or entanglement.
• To provide a tight, state-independent uncertainty relation that generalizes previous bounds, especially for asymmetric error pairs.
• To enable direct experimental adaptation of existing tests (e.g., Toronto experiment) to verify the exact error tradeoff.

Proposed method

• The authors define worst-case errors as the maximum trace distance between probability distributions of sharp and unsharp measurements across all input states.
• They model joint measurements via a positive-operator valued measure (POVM) with four outcomes, whose marginals define unsharp observables for A and B.
• Using geometric optimization, they derive conditions for optimality by requiring the vectors $\vec{a}-\vec{m}$ and $\vec{b}-\vec{n}$ to be tangent to ellipsoids defined by the joint measurability constraint.
• The optimal error tradeoff curve is derived by solving the system of equations from the tangency conditions, leading to a parametric form involving $\varphi$ and the angle $\theta$ between $\vec{a}$ and $\vec{b}$.
• The solution yields explicit expressions for $\mathcal{E}_a$ and $\mathcal{E}_b$ in terms of $\theta$, $\varphi$, and the dot product $C = \vec{m} \cdot \vec{n}$, with $\mu = \nu = 1$ selected for minimality.
• The error tradeoff curve is shown to be the envelope of the family of linear bounds $\epsilon_a \sin\varphi + \epsilon_b \cos\varphi \geq \sqrt{1 + \sin\theta \sin 2\varphi} - 1$.

Experimental results

Research questions

• RQ1 What is the exact set of all possible pairs of worst-case errors for jointly measuring two incompatible qubit observables?
• RQ2 Can optimal joint measurements be constructed without ancilla or entanglement while saturating the error tradeoff bounds?
• RQ3 How does the error tradeoff behave under asymmetric error conditions, and what is the minimal achievable error for one observable when the other is measured with minimal error?
• RQ4 Is the derived tradeoff curve the tightest possible, and does it generalize previous bounds such as the BLW sum bound?
• RQ5 Can existing experimental setups, such as the Toronto experiment, be directly adapted to test the exact error tradeoff?

Key findings

• The exact tradeoff between worst-case errors $\mathcal{E}_a$ and $\mathcal{E}_b$ for measuring two qubit observables $A = \vec{a} \cdot \vec{\sigma}$ and $B = \vec{b} \cdot \vec{\sigma}$ is given by the parametric curve $\mathcal{E}_a = \sqrt{1 - C^2 \cos^2\varphi} - \sin\varphi$, $\mathcal{E}_b = \sqrt{1 - C^2 \sin^2\varphi} - \cos\varphi$, with $C = \vec{m} \cdot \vec{n}$ and $\theta = \angle(\vec{a}, \vec{b})$.
• The optimal error pair is achieved when $\mu = \nu = 1$, ensuring the minimal possible errors for both observables.
• The error tradeoff curve is the envelope of the family of linear bounds $\epsilon_a \sin\varphi + \epsilon_b \cos\varphi \geq \sqrt{1 + \sin\theta \sin 2\varphi} - 1$ for all $\varphi \in [0, \pi/2]$, which generalizes the BLW sum bound at $\varphi = \pi/4$.
• For orthogonal observables ($\theta = \pi/2$), the minimal error for one observable is $\sin\theta = 1$ when the other is measured without error, and the optimal error pair lies on the curve defined by $\mathcal{E}_a^2 + \mathcal{E}_b^2 = 2 - 2\sqrt{1 - \mathcal{E}_a^2\mathcal{E}_b^2}$.
• The optimal joint measurement is realizable via a POVM without ancilla or entanglement, making it experimentally feasible.
• The worst-case errors $\mathcal{E}_a = |x| + |\vec{a} - \vec{m}|$ and $\mathcal{E}_b = |y| + |\vec{b} - \vec{n}|$ are state-independent and fully characterize the intrinsic precision of the measurement apparatus.