Bounding the Minimum Time of a Quantum Measurement

Nathan Shettell1, Federico Centrone2, and Luis Pedro García-Pintos3,4

1Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
3Joint Center for Quantum Information and Computer Science and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
4Theoretical Division (T4), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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Abstract

Measurements take a singular role in quantum theory. While they are often idealized as an instantaneous process, this is in conflict with all other physical processes in nature. In this Letter, we adopt a standpoint where the interaction with an environment is a crucial ingredient for the occurrence of a measurement. Within this framework, we derive lower bounds on the time needed for a measurement to occur. Our bound scales proportionally to the change in entropy of the measured system, and decreases as the number of of possible measurement outcomes or the interaction strength driving the measurement increases. We evaluate our bound in two examples where the environment is modelled by bosonic modes and the measurement apparatus is modelled by spins or bosons.

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[1] N. Bohr et al., The quantum postulate and the recent development of atomic theory, Vol. 3 (Printed in Great Britain by R. & R. Clarke, Limited, 1928).

[2] E. P. Wigner, Review of the quantum-mechanical measurement problem, Science, Computers, and the Information Onslaught , 63 (1984).
https:/​/​doi.org/​10.1016/​B978-0-12-404970-3.50011-2

[3] J. Bub and I. Pitowsky, Two dogmas about quantum mechanics, Many worlds , 433 (2010).

[4] M. Schlosshauer, J. Kofler, and A. Zeilinger, A snapshot of foundational attitudes toward quantum mechanics, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44, 222 (2013).
https:/​/​doi.org/​10.1016/​j.shpsb.2013.04.004

[5] W. Heisenberg, The physical principles of the quantum theory (Courier Corporation, 1949).

[6] H. P. Stapp, The copenhagen interpretation, American journal of physics 40, 1098 (1972).
https:/​/​doi.org/​10.1119/​1.1986768

[7] J. von Neumann, Mathematical foundations of quantum mechanics: New edition (Princeton university press, 2018).

[8] Č. Brukner, On the quantum measurement problem, in Quantum [Un] Speakables II (Springer International Publishing, 2017) pp. 95–117.
https:/​/​doi.org/​10.1007/​978-3-319-38987-5_5

[9] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of modern physics 75, 715 (2003).
https:/​/​doi.org/​10.1103/​RevModPhys.75.715

[10] W. H. Zurek, Quantum darwinism, Nature physics 5, 181 (2009).
https:/​/​doi.org/​10.1038/​nphys1202

[11] M. Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics, Reviews of Modern physics 76, 1267 (2005).
https:/​/​doi.org/​10.1103/​RevModPhys.76.1267

[12] M. A. Schlosshauer, Decoherence: and the quantum-to-classical transition (Springer Science & Business Media, 2007).

[13] H. D. Zeh, On the interpretation of measurement in quantum theory, Foundations of Physics 1, 69 (1970).
https:/​/​doi.org/​10.1007/​BF00708656

[14] E. Joos and H. D. Zeh, The emergence of classical properties through interaction with the environment, Zeitschrift für Physik B Condensed Matter 59, 223 (1985).
https:/​/​doi.org/​10.1007/​BF01725541

[15] M. Schlosshauer, Quantum decoherence, Physics Reports 831, 1 (2019).
https:/​/​doi.org/​10.1016/​j.physrep.2019.10.001

[16] M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J. Raimond, and S. Haroche, Observing the progressive decoherence of the “meter” in a quantum measurement, Physical Review Letters 77, 4887 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.4887

[17] A. N. Jordan and A. N. Korotkov, Uncollapsing the wavefunction by undoing quantum measurements, Contemporary Physics 51, 125 (2010).
https:/​/​doi.org/​10.1080/​00107510903385292

[18] Z. K. Minev, S. O. Mundhada, S. Shankar, P. Reinhold, R. Gutiérrez-Jáuregui, R. J. Schoelkopf, M. Mirrahimi, H. J. Carmichael, and M. H. Devoret, To catch and reverse a quantum jump mid-flight, Nature 570, 200 (2019).
https:/​/​doi.org/​10.1038/​s41586-019-1287-z

[19] M. Carlesso, S. Donadi, L. Ferialdi, M. Paternostro, H. Ulbricht, and A. Bassi, Present status and future challenges of non-interferometric tests of collapse models, Nature Physics 18, 243 (2022).
https:/​/​doi.org/​10.1038/​s41567-021-01489-5

[20] H.-P. Breuer, F. Petruccione, et al., The theory of open quantum systems (Oxford University Press on Demand, 2002).

[21] N. Margolus and L. B. Levitin, The maximum speed of dynamical evolution, Physica D: Nonlinear Phenomena 120, 188 (1998).
https:/​/​doi.org/​10.1016/​S0167-2789(98)00054-2

[22] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Quantum speed limit for physical processes, Physical review letters 110, 050402 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.050402

[23] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Quantum speed limits in open system dynamics, Phys. Rev. Lett. 110, 050403 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.050403

[24] S. Deffner and E. Lutz, Quantum speed limit for non-markovian dynamics, Physical review letters 111, 010402 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.010402

[25] L. P. García-Pintos, S. B. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, Unifying quantum and classical speed limits on observables, Physical Review X 12, 011038 (2022).
https:/​/​doi.org/​10.1103/​PhysRevX.12.011038

[26] P. Strasberg, K. Modi, and M. Skotiniotis, How long does it take to implement a projective measurement?, European Journal of Physics 43, 035404 (2022).
https:/​/​doi.org/​10.1088/​1361-6404/​ac5a7a

[27] W. H. Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?, Physical review D 24, 1516 (1981).
https:/​/​doi.org/​10.1103/​PhysRevD.24.1516

[28] One may be concerned by a `fuzzy' definition of measurement that relies on the state of the system merely becoming close to $\rho ^ \mathcal {QA}_ \mathcal {M}$. More definite, objective notions arise if quantum gravity implies fundamental uncertainties in measurements GambiniLPPullin2019.

[29] V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74, 197 (2002).
https:/​/​doi.org/​10.1103/​RevModPhys.74.197

[30] F. Hiai and D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Communications in mathematical physics 143, 99 (1991).
https:/​/​doi.org/​10.1007/​BF02100287

[31] While alternative bounds on the entropy rate have been derived [55-57], the main advantage of Eq.(7) is that in involves standard deviations instead of operator norms, which typically results in tighter bounds [25].

[32] D. Reeb and M. M. Wolf, Tight bound on relative entropy by entropy difference, IEEE Transactions on Information Theory 61, 1458 (2015).
https:/​/​doi.org/​10.1109/​TIT.2014.2387822

[33] J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, Deep strong coupling regime of the jaynes-cummings model, Physical review letters 105, 263603 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.105.263603

[34] T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas, and H. J. Wörner, Streaking of 43-attosecond soft-x-ray pulses generated by a passively cep-stable mid-infrared driver, Optics express 25, 27506 (2017).
https:/​/​doi.org/​10.1364/​OE.25.027506

[35] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Reviews of Modern Physics 59, 1 (1987).
https:/​/​doi.org/​10.1103/​RevModPhys.59.1

[36] W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Towards quantum superpositions of a mirror, Physical Review Letters 91, 130401 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.130401

[37] L. A. Kanari-Naish, J. Clarke, M. R. Vanner, and E. A. Laird, Can the displacemon device test objective collapse models?, AVS Quantum Science 3, 045603 (2021).
https:/​/​doi.org/​10.1116/​5.0073626

[38] R. Penrose, On gravity's role in quantum state reduction, General relativity and gravitation 28, 581 (1996).
https:/​/​doi.org/​10.1007/​BF02105068

[39] R. Gambini, R. A. Porto, and J. Pullin, Fundamental decoherence from quantum gravity: a pedagogical review, General Relativity and Gravitation 39, 1143 (2007).
https:/​/​doi.org/​10.1007/​s10714-007-0451-1

[40] M. P. Blencowe, Effective field theory approach to gravitationally induced decoherence, Phys. Rev. Lett. 111, 021302 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.021302

[41] D. Walls, M. Collet, and G. Milburn, Analysis of a quantum measurement, Physical Review D 32, 3208 (1985).
https:/​/​doi.org/​10.1103/​PhysRevD.32.3208

[42] M. Brune, S. Haroche, J.-M. Raimond, L. Davidovich, and N. Zagury, Manipulation of photons in a cavity by dispersive atom-field coupling: Quantum-nondemolition measurements and generation of ‘‘schrödinger cat’’states, Physical Review A 45, 5193 (1992).
https:/​/​doi.org/​10.1103/​PhysRevA.45.5193

[43] Alternatively, one could have chosen an alternative $H_ \text {int}$ to avoid the commutativity issue, e.g. $H_ \text {int} = b^\dagger b\sum_k g_k(a_k^\dagger + a_k)$ [41], however said Hamiltonian is representative of coupling the Fock states to the environmental modes, which is unrealistic and thus not typically used.

[44] The scaling of $1/​|\alpha |$ in our bounds seemingly disagrees with the one found in Refs. brune1992manipulation,brune1996observing, where they found a decoherence time that scales as $1/​|\alpha |^2$. The difference is due to the different choice of interaction Hamiltonian brune1992manipulation.

[45] B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Deterministically encoding quantum information using 100-photon schrödinger cat states, Science 342, 607 (2013).
https:/​/​doi.org/​10.1126/​science.1243289

[46] F. Pokorny, C. Zhang, G. Higgins, A. Cabello, M. Kleinmann, and M. Hennrich, Tracking the dynamics of an ideal quantum measurement, Phys. Rev. Lett. 124, 080401 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.080401

[47] M.-J. Hu, Y. Chen, Y. Ma, X. Li, Y. Liu, Y.-S. Zhang, and H. Miao, Scalable simulation of quantum measurement process with quantum computers, arXiv e-prints , arXiv (2022).
https:/​/​doi.org/​10.48550/​ARXIV.2206.14029

[48] J. D. Bekenstein, Universal upper bound on the entropy-to-energy ratio for bounded systems, Phys. Rev. D 23, 287 (1981).
https:/​/​doi.org/​10.1103/​PhysRevD.23.287

[49] S. Deffner and E. Lutz, Generalized clausius inequality for nonequilibrium quantum processes, Physical review letters 105, 170402 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.105.170402

[50] K. Jacobs, Quantum measurement and the first law of thermodynamics: The energy cost of measurement is the work value of the acquired information, Physical Review E 86, 040106 (2012).
https:/​/​doi.org/​10.1103/​PhysRevE.86.040106

[51] M. Navascués and S. Popescu, How energy conservation limits our measurements, Phys. Rev. Lett. 112, 140502 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.140502

[52] S. Deffner, J. P. Paz, and W. H. Zurek, Quantum work and the thermodynamic cost of quantum measurements, Physical Review E 94, 010103 (2016).
https:/​/​doi.org/​10.1103/​PhysRevE.94.010103

[53] Y. Guryanova, N. Friis, and M. Huber, Ideal projective measurements have infinite resource costs, Quantum 4, 222 (2020).
https:/​/​doi.org/​10.22331/​q-2020-01-13-222

[54] R. Gambini, L. P. García-Pintos, and J. Pullin, Single-world consistent interpretation of quantum mechanics from fundamental time and length uncertainties, Phys. Rev. A 100, 012113 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.012113

[55] S. Bravyi, Upper bounds on entangling rates of bipartite hamiltonians, Phys. Rev. A 76, 052319 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.76.052319

[56] S. Deffner, Energetic cost of hamiltonian quantum gates, EPL (Europhysics Letters) 134, 40002 (2021).
https:/​/​doi.org/​10.1209/​0295-5075/​134/​40002

[57] B. Mohan, S. Das, and A. K. Pati, Quantum speed limits for information and coherence, New Journal of Physics 24, 065003 (2022).
https:/​/​doi.org/​10.1088/​1367-2630/​ac753c

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