Improved error-tradeoff and error-disturbance relations in terms of measurement error components

Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report various error-disturbance tradeoff relations by decomposing the measurement errors and disturbance into two different components, namely, operator bias and fuzziness. Our uncertainty relations reveal the tradeoffs between these two components of errors, and imply various conditionally valid error-tradeoff relations for the unbiased and projective measurements. We also design a quantum circuit to measure the two components of the error and disturbance.

Figure 1

Forbidden regions (shaded) of errors $({\epsilon }_{\mathcal{A}},{\epsilon }_{\mathcal{B}})$ implied by various error-tradeoff relations in the case of ${\sigma }_A={\sigma }_B=c_{AB}=1$. The region below the thin solid curve is forbidden by Ozawa's relation Eq. (11), while the region below the thick solid curve is forbidden by Branciard's tight (for pure states) error-tradeoff relation [13], which in this case reads ${\epsilon }_{\mathcal{A}}^2+{\epsilon }_{\mathcal{B}}^2\ge 1$. The conditionally valid error-tradeoff relations, implied by our relation Eq. (10), forbid more errors $({\epsilon }_{\mathcal{A}},{\epsilon }_{\mathcal{B}})$ for the cases of two unbiased measurements (dashed blue curve), unbiased for $A$ and projective for $B$ (dash-dotted red line), and unbiased for $A$ and unconditional for $B$ (dotted green curve).

Figure 2

Error-disturbance uncertainty relations in the case of the experiment in Ref. [15]. (a) The left hand sides (LHS) of Ozawa's relation (dashed green curve) and that of our relation (10) (dash-dotted blue curve) as well as the product of the error and disturbance (dotted red curve) are plotted. The red shading indicates the region where these relations are violated. (b) The components of the error and disturbance are given. Here, the observables of interest are $A=X$ and $B=Y$, and the state of the spin is the eigenstate of $Z$ with eigenvalue $+1$, where $X,Y$, and $Z$ are the Pauli matrices. The measurement performed for $A$ is the projective measurement of $X_{\phi }:=cos\phi X+sin\phi Y$.

Figure 3

Quantum circuit to measure (a) the fuzziness ${\overline{\epsilon }}_{\mathcal{A}}$ and ${\overline{\epsilon }}_{\mathcal{B}}$, (b) the total error ${\epsilon }_{\mathcal{A}}$, (c) the total disturbance ${\eta }_{\mathcal{B}}$, and (d) $|{\langle [\overline{\mathcal{A}},\overline{\mathcal{B}}]\rangle }_{\rho }|$. The dashed boxes stand for a generalized measurement used to approximate that measurement of $A$, which is without loss of generality implemented by the indirect measurement model.