Quantum dynamics of simultaneously measured non-commuting observables

Abstract

In quantum mechanics, measurements cause wavefunction collapse that yields precise outcomes, whereas for non-commuting observables such as position and momentum Heisenbergs uncertainty principle limits the intrinsic precision of a state.Although theoretical work 1 has demonstrated that it should be possible to perform simultaneous non-commuting measurements and has revealed the limits on measurement outcomes, only recently 2,3 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 purification 4 , adaptive measurement 5,6 , measurement-based state steering and continuous quantum error correction 7 . As physical systems often interact continuously with their environment via non-commuting degrees of freedom, our work offers a way 8,9 to * shayhh@berkeley.edushayhh@berkeley.edu study how notions of contemporary quantum foundations 10-14 arise in such settings.In this work, we implement two high quantum efficiency readouts of the angular momenta about two different axes of an artificial spin-1/2 system and observe in real-time the resulting dynamics. Importantly, our measurements are designed to be individually quantum non-demolition, which ensures that the back-action arises only from the competition between incompatible observables.Our experiment utilizes a single superconducting transmon qubit coupled dispersively to a multimode waveguide cavity 15 . This results in a qubit-state dependent shift of the cavity mode frequency. By applying a microwave tone to a single cavity mode, one can infer the qubit state from the phase of the reflected signal 16 ; this readout scheme has been used extensively with superconducting qubits for quantum information processing, and also to perform weak measurements 17 and track quantum trajectories of a single qubit 18 . In our configuration, each cavity mode constitutes a measure...

Simultaneous continuous measurement of noncommuting observables: Quantum state correlations

“…Such experimental and theoretical efforts have paved the way for interesting applications of CWQMs such as rapid state purification [18], quantum feedback [9,13,16,19,20], and preparation of entangled states [21][22][23][24]. With CWQMs, it is also possible to simultaneously measure non-commuting observables [25][26][27], and the first experimental demonstration of such measurement on a superconducting qubit was realized only last year [28].…”

“…Simultaneous continuous measurement of two qubit observables, σ z and σ ϕ ≡ σ z cos ϕ + σ x sin ϕ, was implemented by weakly coupling a superconducting transmon qubit to two intracavity modes [28]. The measurement was effectively a stroboscopic measurement of a fast rotating qubit.…”

Past observable dynamics of a continuously monitored qubit

“…Note that our model here interleaves the measurements of σ x and σ z for simplicity; in the continuum limit, dt → 0, the measurements become effectively simultaneous [42,64]. As in the last section, the smoothed estimates obtained in this way fit the measurement output better than the expectation values x = Tr ρ rz,p, rx,p σ x and z = Tr ρ rz,p, rx,p σ z obtained from the most informationally complete causal state of the qubit.…”

“…[41,42,64,65]). We assume characteristic collapse timescales τ z and τ x for Gaussian Kraus operators M rz and N rx measuring σ z and σ x , respectively, with τ z < τ x so that the agent Z causes the majority of the measurement backaction.…”

“…To keep this manuscript self-contained, we briefly review the essential details of how a temporal sequence of independent Gaussian measurements models continuous-in-time measurements from a quantum information perspective. This model is a slight idealization of those that describe recent experimental work on quantum state trajectories well [16,18,[25][26][27]32], but deliberately neglects relevant experimental details-such as measurement inefficiency, environmental decoherence, energy relaxation, phase-backaction, and non-Markovian effects from finite detector bandwidth [34,35]-in order to isolate the essential effect of the measurement backaction. (For an alternative recent derivation of a simple continuous measurement model that includes some of these nonidealities in the context of feedback, see also Ref.…”