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Quantum dynamics of simultaneously measured non-commuting observables


In quantum mechanics, measurements cause wavefunction collapse that yields precise outcomes, whereas for non-commuting observables such as position and momentum Heisenberg’s uncertainty principle limits the intrinsic precision of a state. Although theoretical work1 has demonstrated that it should be possible to perform simultaneous non-commuting measurements and has revealed the limits on measurement outcomes, only recently2,3 has the dynamics of the quantum state been discussed. To realize this unexplored regime, we simultaneously apply two continuous quantum non-demolition probes of non-commuting observables to a superconducting qubit. We implement multiple readout channels by coupling the qubit to multiple modes of a cavity. To control the measurement observables, we implement a ‘single quadrature’ measurement by driving the qubit and applying cavity sidebands with a relative phase that sets the observable. Here, we use this approach to show that the uncertainty principle governs the dynamics of the wavefunction by enforcing a lower bound on the measurement-induced disturbance. Consequently, as we transition from measuring identical to measuring non-commuting observables, the dynamics make a smooth transition from standard wavefunction collapse to localized persistent diffusion and then to isotropic persistent diffusion. Although the evolution of the state differs markedly from that of a conventional measurement, information about both non-commuting observables is extracted by keeping track of the time ordering of the measurement record, enabling quantum state tomography without alternating measurements. Our work creates novel capabilities for quantum control, including rapid state purification4, adaptive measurement5,6, measurement-based state steering and continuous quantum error correction7. As physical systems often interact continuously with their environment via non-commuting degrees of freedom, our work offers a way8,9 to study how notions of contemporary quantum foundations10,11,12,13,14 arise in such settings.

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Figure 1: Multimode single quadrature measurement (SQM).
Figure 2: Validation of simultaneous non-commuting measurements.
Figure 3: Probability distribution of the density matrix.
Figure 4: Magnitude of the quantum back-action.


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We thank J. Dressel, A. Jordan, A. Korotkov, J. Combes, M. Sarovar, U. Vool and J. Atalaya for discussions, and MIT Lincoln laboratories for fabrication of the travelling wave parametric amplifier. L.S.M. and V.V.R. acknowledge support from the National Science Foundation (graduate fellowship grant 1106400). L.S.M. additionally acknowledges support from a Berkeley Fellowship for Graduate Study. This work was supported by the Air Force Office of Scientific Research (grant FA9550-12-1-0378) and the Army Research Office (grant W911NF-15-1-0496).

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Authors and Affiliations



S.H.-G. and L.S.M. wrote the manuscript, conducted the experiment and analysed the data. S.H.-G., L.S.M., E.F. and I.S. conceived of the experiment. S.H.-G. and L.S.M. constructed the experimental set-up with assistance from V.V.R. and E.F. S.H.-G. fabricated the qubit and LJPAs. L.S.M. performed theoretical analysis with assistance from S.H.-G. All authors contributed to discussions and preparation of the manuscript. All work was carried out under the supervision of K.B.W. and I.S.

Corresponding author

Correspondence to Shay Hacohen-Gourgy.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 SQM versus tomographic validation.

Tomographic validation of a set of trajectories initialized in the state y = −1 and tracked for 3 μs, with the axes being perpendicular. Horizontal axis (‘SQM’) indicates Bloch sphere coordinate as predicted by trajectory reconstruction. Vertical axis (‘tomography’) represents the coordinate reconstructed from post-selected tomography data. Error bars (s.e.m.) are derived from the Poisson statistics of the qubit measurement. Data are staggered vertically to allow for clearer visualization by adding one of seven arbitrary offsets. These offsets are chosen according to the number of measurements contributing to each data point.

Extended Data Figure 2 SQM calibration.

Calibration of the SQM phase at the target Rabi frequency using the integrated SQM signal versus the measurement axis (SQM phase) and the Rabi amplitude index. Colour bar is scaled to the outcomes in which the qubit is aligned or anti-aligned to the measurement axis. Rabi amplitude index sweep is performed by sweeping the modulation amplitude to drive approximately 40-MHz Rabi oscillations. The red dashed line indicates the index at which the Rabi frequency was determined to be 40 MHz.

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Hacohen-Gourgy, S., Martin, L., Flurin, E. et al. Quantum dynamics of simultaneously measured non-commuting observables. Nature 538, 491–494 (2016).

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