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Jackson CS, Caves CM. Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1254. [PMID: 37761553 PMCID: PMC10529125 DOI: 10.3390/e25091254] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2023] [Revised: 08/08/2023] [Accepted: 08/11/2023] [Indexed: 09/29/2023]
Abstract
We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument's evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.
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Affiliation(s)
| | - Carlton M. Caves
- Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA
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Atalaya J, Hacohen-Gourgy S, Siddiqi I, Korotkov AN. Correlators Exceeding One in Continuous Measurements of Superconducting Qubits. PHYSICAL REVIEW LETTERS 2019; 122:223603. [PMID: 31283299 DOI: 10.1103/physrevlett.122.223603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2018] [Indexed: 06/09/2023]
Abstract
We consider the effect of phase backaction on the correlator ⟨I(t)I(t+τ)⟩ for the output signal I(t) from continuous measurement of a qubit. We demonstrate that the interplay between informational and phase backactions in the presence of Rabi oscillations can lead to the correlator becoming larger than 1, even though |⟨I⟩|≤1. The correlators can be calculated using the generalized "collapse recipe," which we validate using the quantum Bayesian formalism. The recipe can be further generalized to the case of multitime correlators and arbitrary number of detectors, measuring non-commuting qubit observables. The theory agrees well with experimental results for continuous measurement of a transmon qubit. The experimental correlator exceeds the bound of 1 for a sufficiently large angle between the amplified and informational quadratures, causing the phase backaction. The demonstrated effect can be used to calibrate the quadrature misalignment.
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Affiliation(s)
- Juan Atalaya
- Department of Electrical and Computer Engineering, University of California, Riverside, California 92521, USA
| | - Shay Hacohen-Gourgy
- Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA
- Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA
- Department of Physics, Technion, Haifa 3200003, Israel
| | - Irfan Siddiqi
- Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA
- Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA
| | - Alexander N Korotkov
- Department of Electrical and Computer Engineering, University of California, Riverside, California 92521, USA
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Ochoa MA, Belzig W, Nitzan A. Simultaneous weak measurement of non-commuting observables: a generalized Arthurs-Kelly protocol. Sci Rep 2018; 8:15781. [PMID: 30361691 PMCID: PMC6202392 DOI: 10.1038/s41598-018-33562-0] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2018] [Accepted: 09/19/2018] [Indexed: 11/09/2022] Open
Abstract
In contrast to a projective quantum measurement, in a weak measurement the system is only weakly perturbed while only partial information on the measured observable is obtained. A simultaneous measurement of non-commuting observables cannot be projective, however the strongest possible such measurement can be defined as providing their values at the smallest uncertainty limit. Starting with the Arthurs and Kelly (AK) protocol for such measurement of position and momentum, we derive a systematic extension to a corresponding weak measurement along three steps: First, a plausible form of the weak measurement operator analogous to the Gaussian Kraus operator, often used to model a weak measurement of a single observable, is obtained by projecting a naïve extension (valid for commuting observable) onto the corresponding Gabor space. Second, we show that the so obtained set of measurement operators satisfies the normalization condition for the probability to obtain given values of the position and momentum in the weak measurement operation, namely that this set constitutes a positive operator valued measure (POVM) in the position-momentum space. Finally, we show that the so-obtained measurement operator corresponds to a generalization of the AK measurement protocol in which the initial detector wavefunctions is suitable broadened.
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Affiliation(s)
- Maicol A Ochoa
- Department of Chemistry, University of Pennsylvania, Philadelphia, PA, 19104, USA.
| | - Wolfgang Belzig
- Department of Physics, University of Konstanz, D-78457, Konstanz, Germany.
| | - Abraham Nitzan
- Department of Chemistry, University of Pennsylvania, Philadelphia, PA, 19104, USA. .,School of Chemistry, Tel Aviv University, Tel Aviv, 69978, Israel.
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Hacohen-Gourgy S, Martin LS, Flurin E, Ramasesh VV, Whaley KB, Siddiqi I. Quantum dynamics of simultaneously measured non-commuting observables. Nature 2016; 538:491-494. [DOI: 10.1038/nature19762] [Citation(s) in RCA: 88] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2016] [Accepted: 08/02/2016] [Indexed: 11/09/2022]
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Vool U, Shankar S, Mundhada SO, Ofek N, Narla A, Sliwa K, Zalys-Geller E, Liu Y, Frunzio L, Schoelkopf RJ, Girvin SM, Devoret MH. Continuous Quantum Nondemolition Measurement of the Transverse Component of a Qubit. PHYSICAL REVIEW LETTERS 2016; 117:133601. [PMID: 27715126 DOI: 10.1103/physrevlett.117.133601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2016] [Indexed: 06/06/2023]
Abstract
Quantum jumps of a qubit are usually observed between its energy eigenstates, also known as its longitudinal pseudospin component. Is it possible, instead, to observe quantum jumps between the transverse superpositions of these eigenstates? We answer positively by presenting the first continuous quantum nondemolition measurement of the transverse component of an individual qubit. In a circuit QED system irradiated by two pump tones, we engineer an effective Hamiltonian whose eigenstates are the transverse qubit states, and a dispersive measurement of the corresponding operator. Such transverse component measurements are a useful tool in the driven-dissipative operation engineering toolbox, which is central to quantum simulation and quantum error correction.
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Affiliation(s)
- U Vool
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - S Shankar
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - S O Mundhada
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - N Ofek
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - A Narla
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - K Sliwa
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - E Zalys-Geller
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - Y Liu
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - L Frunzio
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - R J Schoelkopf
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - S M Girvin
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
| | - M H Devoret
- Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
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Ruskov R, Combes J, Mølmer K, Wiseman HM. Qubit purification speed-up for three complementary continuous measurements. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2012; 370:5291-5307. [PMID: 23091209 DOI: 10.1098/rsta.2011.0516] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We consider qubit purification under simultaneous continuous measurement of the three non-commuting qubit operators σ(x), σ(y), σ(z). The purification dynamics is quantified by (i) the average purification rate and (ii) the mean time of reaching a given level of purity, 1-ε. Under ideal measurements (detector efficiency η=1), we show in the first case an asymptotic mean purification speed-up of 4 when compared with a standard (classical) single-detector measurement. However, by the second measure-the mean time of first passage T(ε) of the purity-the corresponding speed-up is only 2. We explain these speed-ups using the isotropy of the qubit evolution that provides an equivalence between the original measurement directions and three simultaneous measurements, one with an axis aligned along the Bloch vector and the other with axes in the two complementary directions. For inefficient detectors, η=1 - δ < 1, the mean time of first passage T(δ,ε)increases because qubit purification competes with an isotropic qubit dephasing. In the asymptotic high-purity limit (ε,δ≪1), we show that the increase possesses a scaling behaviour: ΔT(δ,ε) is a function only of the ratio δ/ε. The increase ΔT(δ,ε) is linear for small arguments, but becomes exponential ~exp(δ,2ε) for δ/ε large.
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Affiliation(s)
- Rusko Ruskov
- Lundbeck Foundation Theoretical Center for Quantum System Research, Department of Physics and Astronomy, Århus University, Denmark
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