Weak Measurement of a Superconducting Qubit Reconciles Incompatible Operators

Traditional uncertainty relations dictate a minimal amount of noise in incompatible projective quantum measurements. However, not all measurements are projective. Weak measurements are minimally invasive methods for obtaining partial state information without projection. Recently, weak measurements were shown to obey an uncertainty relation cast in terms of entropies. We experimentally test this entropic uncertainty relation with strong and weak measurements of a superconducting transmon qubit. A weak measurement, we find, can reconcile two strong measurements’ incompatibility, via backaction on the state. Mathematically, a weak value—a preselected and postselected expectation value—lowers the uncertainty bound. Hence we provide experimental support for the physical interpretation of the weak value as a determinant of a weak measurement’s ability to reconcile incompatible operations.

Figure 1

Our experimental setup involves a superconducting transmon qubit coupled dispersively to a microwave cavity. The cavity’s state is sketched in phase space, defined by quadratures $I$ and $Q$. Coherent states probe the cavity, acquiring a phase shift (red and blue circles) dependent on the qubit’s state. The transmitted-probe quadrature that contains qubit-state information is demodulated and digitized into discrete measurement outcomes $j$.

Figure 2

Characterization of entropic uncertainties: (a) We subject a state $\rho$ to one of three measurements. The measurements’ entropies are defined as the detectors’ von Neumann entropies. (b) Entropies measured for the state $\rho =|0⟩⟨0|$. Bands indicate statistical error from finite sampling (approximately 10 000 repetitions per angle). $H(\mathcal{I}{)}_{\rho }$ and $H(F{)}_{\rho }$ characterize projective measurements. $H(AF{)}_{\rho }-H(A{)}_{\rho }$ quantifies the change, caused by the weak measurement, in the second measurement’s entropy, when ${\theta }_A=\pi /4$. $H(\mathcal{I}{)}_{\rho }+H(F{)}_{\rho }$ maximizes when ${\theta }_F=\pi /2$, such that $F=X$, while $\mathcal{I}=Z$. The second measurement’s entropy change, $H(AF{)}_{\rho }-H(A{)}_{\rho }$, maximizes at ${\theta }_F=0.53\pi$.

Figure 3

Measurements of the entropic uncertainty relation: (a) The entropy $H(AF{)}_{\rho }$. (b) Detail of $H(AF{)}_{\rho }$ versus ${\theta }_A$ (markers), compared to theory (dashed line), at ${\theta }_F=\pi /2$. Bands indicate statistical error that results from finite sampling (approximately 140 000 repetitions per angle). (c) Bloch-plane sketch indicating the $A$ measurement’s backaction (dashed arrow) on the initial state. (d) The bound of Eq. (7). The dashed line indicates the bound’s theoretical maximum.