Distinguishability times and asymmetry monotone-based quantum speed limits in the Bloch ball

T.J. Volkoff1 and K.B. Whaley2

1Department of Physics, Konkuk University, Seoul 05029, Korea
2Berkeley Quantum Information and Computation Center and Department of Chemistry, UC Berkeley, Berkeley, California 94720, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

For both unitary and open qubit dynamics, we compare asymmetry monotone-based bounds on the minimal time required for an initial qubit state to evolve to a final qubit state from which it is probabilistically distinguishable with fixed minimal error probability (i.e., the minimal error distinguishability time). For the case of unitary dynamics generated by a time-independent Hamiltonian, we derive a necessary and sufficient condition on two asymmetry monotones that guarantees that an arbitrary state of a two-level quantum system or a separable state of $N$ two-level quantum systems will unitarily evolve to another state from which it can be distinguished with a fixed minimal error probability $\delta \in [0,1/2]$. This condition is used to order the set of qubit states based on their distinguishability time, and to derive an optimal release time for driven two-level systems such as those that occur, e.g., in the Landau-Zener problem. For the case of non-unitary dynamics, we compare three lower bounds to the distinguishability time, including a new type of lower bound which is formulated in terms of the asymmetry of the uniformly time-twirled initial system-plus-environment state with respect to the generator $H_{SE}$ of the Stinespring isometry corresponding to the dynamics, specifically, in terms of $\Vert [H_{SE},\rho_{\text{av}}(\tau)]\Vert_{1}$, where $\rho_{\text{av}}(\tau):={1\over \tau}\int_{0}^{\tau}dt\, e^{-iH_{SE}t}\rho \otimes \vert 0\rangle_{E}\langle 0\vert_{E} e^{iH_{SE}t}$.

The ability to reliably distinguish the states of a quantum system at different points in time is central to the performance of all high-precision quantum technologies, from atomic clocks to quantum sensors. However, the minimal time required to reliably distinguish such states, i.e., the "distinguishability time", depends sensitively on the quantum states involved, on their dynamics, on the method that is used to discriminate those states, and on what level of reliability is considered acceptable. In this work, we develop a framework for calculating distinguishability times that takes all these aspects of the problem into account, providing exact results or bounding the distinguishability time from below using quantities called "asymmetry monotones". Such lower bounds are referred to as quantum speed limits. Our framework leads to a completely new method for deriving tight quantum speed limits that approximate the true distinguishability time We apply our results to qubits, the basic building blocks of quantum clocks and quantum processors. Because of the simple description of qubit quantum states as points in a closed three-dimensional ball, we can directly compare our new quantum speed limits to established bounds for complex dynamical models that arise from coupling qubits to a realistic noisy environment.

► BibTeX data

► References

[1] Lloyd, S. Ultimate physical limits to computation. Nature 406, 1047 (2000). URL https:/​/​doi.org/​10.1038/​35023282.
https:/​/​doi.org/​10.1038/​35023282

[2] Kitaev, A., Shen, A. & Vyalyi, M. Classical and Quantum Computation (American Mathematical Society, 1999).

[3] Bergou, J., Herzog, U. & Hillery, M. Discrimination of quantum states. Lect. Notes Phys. 649, 417 (2004). URL https:/​/​doi.org/​10.1080/​09500340903477756.
https:/​/​doi.org/​10.1080/​09500340903477756

[4] Samsonov, B. Minimum error discrimination problem for pure qubit states. Phys. Rev. A 80, 052305 (2009). URL https:/​/​doi.org/​10.1103/​PhysRevA.80.052305.
https:/​/​doi.org/​10.1103/​PhysRevA.80.052305

[5] Bae, J. & Kwek, L.-C. Quantum state discrimination and its applications. J. Phys. A: Math. Theor. 48, 083001 (2015). URL https:/​/​doi.org/​10.1088/​1751-8113/​48/​8/​083001.
https:/​/​doi.org/​10.1088/​1751-8113/​48/​8/​083001

[6] Nair, R., Yen, B., Guha, S., Shapiro, J. & Pirandola, S. Symmetric $m$-ary phase discrimination using quantum-optical probe states. Phys. Rev. A 86, 022306 (2012). URL https:/​/​doi.org/​10.1103/​PhysRevA.86.022306.
https:/​/​doi.org/​10.1103/​PhysRevA.86.022306

[7] Tan, K. C., Volkoff, T., Kwon, H. & Jeong, H. Quantifying the coherence between coherent states. Phys. Rev. Lett. 119, 190405 (2017). URL https:/​/​doi.org/​10.1103/​PhysRevLett.119.190405.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.190405

[8] Holevo, A. Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).

[9] Volkoff, T. & Whaley, K. Macroscopicity of quantum superpositions on a one-parameter unitary path in Hilbert space. Phys. Rev. A 90, 062122 (2014). URL https:/​/​doi.org/​10.1103/​PhysRevA.90.062122.
https:/​/​doi.org/​10.1103/​PhysRevA.90.062122

[10] Giovannetti, V., Lloyd, S. & Maccone, L. Quantum limits to dynamical evolution. Phys. Rev. A 67, 052109 (2003). URL https:/​/​doi.org/​10.1103/​PhysRevA.67.052109.
https:/​/​doi.org/​10.1103/​PhysRevA.67.052109

[11] Pires, D., Cianciaruso, M., Céleri, L., Adesso, G. & Soares-Pinto, D. Generalized geometric quantum speed limits. Phys. Rev. X 6, 021031 (2016). URL https:/​/​doi.org/​10.1103/​PhysRevX.6.021031.
https:/​/​doi.org/​10.1103/​PhysRevX.6.021031

[12] Pires, D., Céleri, L. & Soares-Pinto, D. Geometric lower bound for a quantum coherence measure. Phys. Rev. A 91, 042330 (2015). URL https:/​/​doi.org/​10.1103/​PhysRevA.91.042330.
https:/​/​doi.org/​10.1103/​PhysRevA.91.042330

[13] Deffner, S. Geometric quantum speed limits: a case for Wigner phase space. New J. Phys. 19, 103018 (2017). URL https:/​/​doi.org/​10.1088/​1367-2630/​aa83dc.
https:/​/​doi.org/​10.1088/​1367-2630/​aa83dc

[14] Morozova, E. & Chentsov, N. Markov invariant geometry on manifolds of states. J. Math. Sci. 56, 2648 (1991). URL https:/​/​doi.org/​10.1007/​BF01095975.
https:/​/​doi.org/​10.1007/​BF01095975

[15] Marvian, I., Spekkens, R. & Zanardi, P. Quantum speed limits, coherence, and asymmetry. Phys. Rev. A 93, 052331 (2016). URL https:/​/​doi.org/​10.1103/​PhysRevA.93.052331.
https:/​/​doi.org/​10.1103/​PhysRevA.93.052331

[16] Helstrom, C. Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

[17] Hayashi, A., Horibe, M. & Hashimoto, T. Quantum pure-state identification. Phys. Rev. A 72, 052306 (2005). URL https:/​/​doi.org/​10.1103/​PhysRevA.72.052306.
https:/​/​doi.org/​10.1103/​PhysRevA.72.052306

[18] Fanizza, M., Mari, A. & Giovanetti, V. Optimal universal learning machines for quantum state discrimination. arXiv 1805.03477v1 (2018).

[19] del Campo, A., Egusquiza, I., Plenio, M. & Huelga, S. Quantum Speed Limits in Open System Dynamics. Phys. Rev. Lett. 110, 050403 (2013). URL https:/​/​doi.org/​10.1103/​PhysRevLett.110.050403.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.050403

[20] Deffner, S. & Lutz, E. Quantum Speed Limit for Non-Markovian Dynamics. Phys. Rev. Lett. 111, 010402 (2013). URL https:/​/​doi.org/​10.1103/​PhysRevLett.111.010402.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.010402

[21] Taddei, M., Escher, B., Davidovich, L. & de Matos Filho, R. Quantum Speed Limit for Physical Processes. Phys. Rev. Lett. 110, 050402 (2013). URL https:/​/​doi.org/​10.1103/​PhysRevLett.110.050402.
https:/​/​doi.org/​10.1103/​PhysRevLett.110.050402

[22] Dodonov, V. & Dodonov, A. Energy-time and frequency-time uncertainty relations: exact inequalities. Phys. Scr. 90, 074049 (2015). URL https:/​/​doi.org/​10.1088/​0031-8949/​90/​7/​074049.
https:/​/​doi.org/​10.1088/​0031-8949/​90/​7/​074049

[23] Deffner, S. & Campbell, S. Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control. J. Phys. A: Math. Theor. 50, 453001 (2017). URL https:/​/​doi.org/​10.1088/​1751-8121/​aa86c6.
https:/​/​doi.org/​10.1088/​1751-8121/​aa86c6

[24] Commins, E. D. Quantum mechanics: an experimentalist's approach (Cambridge University Press, 2014).

[25] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: nonrelativistic theory (Pergamon Press, 1977).

[26] Mandelstam, L. & Tamm, I. The uncertainty relation between energy and time in nonrelativistic quantum mechanics. J. Phys. USSR 9, 249 (1945). URL https:/​/​doi.org/​10.1007/​978-3-642-74626-0_8.
https:/​/​doi.org/​10.1007/​978-3-642-74626-0_8

[27] Anandan, J. & Aharonov, Y. Geometry of Quantum Evolution. Phys. Rev. Lett. 65, 1697 (1990). URL https:/​/​doi.org/​10.1103/​PhysRevLett.65.1697.
https:/​/​doi.org/​10.1103/​PhysRevLett.65.1697

[28] Margolus, N. & Levitin, L. The maximum speed of dynamical evolution. Physica D 120, 188 (1996). URL https:/​/​doi.org/​10.1016/​S0167-2789(98)00054-2.
https:/​/​doi.org/​10.1016/​S0167-2789(98)00054-2

[29] Horesh, N. & Mann, A. Intelligent states for the Aharonov-Anandan parameter-based uncertainty relation. J. Phys. A.: Math. Gen. 31, L609 (1998). URL https:/​/​doi.org/​10.1088/​0305-4470/​31/​36/​003.
https:/​/​doi.org/​10.1088/​0305-4470/​31/​36/​003

[30] Fuchs, C. & van de Graaf, J. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Th. 45, 1216 (1999). URL https:/​/​doi.org/​10.1109/​18.761271.
https:/​/​doi.org/​10.1109/​18.761271

[31] Jiang, Z. Quantum Fisher information for states in exponential form. Phys. Rev. A 89, 032128 (2014). URL https:/​/​doi.org/​10.1103/​PhysRevA.89.032128.
https:/​/​doi.org/​10.1103/​PhysRevA.89.032128

[32] Pang, S. & Brun, T. Quantum metrology of a general Hamiltonian parameter. Phys. Rev. A 90, 022117 (2014). URL https:/​/​doi.org/​10.1103/​PhysRevA.90.022117.
https:/​/​doi.org/​10.1103/​PhysRevA.90.022117

[33] Jing, X.-X., Liu, J., Xiong, H.-N. & Wang, X. Maximal quantum Fisher information for general su(2) parametrization processes. Phys. Rev. A 92, 012302 (2015). URL https:/​/​doi.org/​10.1103/​PhysRevA.92.012312.
https:/​/​doi.org/​10.1103/​PhysRevA.92.012312

[34] Liu, J., Jing, X.-X. & Wang, X. Quantum metrology with unitary parametrization processes. Sci. Rep. 5, 8565 (2015). URL https:/​/​doi.org/​10.1038/​srep08565.
https:/​/​doi.org/​10.1038/​srep08565

[35] Braunstein, S. & Caves, C. Statistical Distance and the Geometry of Quantum States. Phys. Rev. Lett. 72, 3439 (1994). URL https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439.
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[36] Zhang, J., Vala, J., Sastry, S. & Whaley, K. B. Geometric theory of nonlocal two-qubit operations. Phys. Rev. A 67, 042313 (2003). URL https:/​/​doi.org/​10.1103/​PhysRevA.67.042313.
https:/​/​doi.org/​10.1103/​PhysRevA.67.042313

[37] Marvian, I. & Spekkens, R. Extending Noether's theorem by quantifying the asymmetry of quantum states. Nat. Comm. 5, 3821 (2014). URL https:/​/​doi.org/​10.1038/​ncomms4821.
https:/​/​doi.org/​10.1038/​ncomms4821

[38] Mondal, D., Datta, C. & Sazim, S. Quantum coherence sets the quantum speed limit for mixed states. Phys. Lett. A 380, 689 (2016). URL https:/​/​doi.org/​10.1016/​j.physleta.2015.12.015.
https:/​/​doi.org/​10.1016/​j.physleta.2015.12.015

[39] Carlini, A., Hosoya, A., Koike, T. & Okudaira, Y. Time-Optimal Quantum Evolution. Phys. Rev. Lett. 96, 060503 (2006). URL https:/​/​doi.org/​10.1103/​PhysRevLett.96.060503.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.060503

[40] Mostafazadeh, A. Hamiltonians generating optimal-speed evolutions. Phys. Rev. A 79, 014101 (2009). URL https:/​/​doi.org/​10.1103/​PhysRevA.79.014101.
https:/​/​doi.org/​10.1103/​PhysRevA.79.014101

[41] Stinespring, W. Positive functions on C*-Algebras. Proc. Am. Math. Soc. 6, 211 (1955). URL https:/​/​doi.org/​10.2307/​2032342.
https:/​/​doi.org/​10.2307/​2032342

[42] Davies, E. Quantum Theory of Open Systems (Academic Press , New York, 1976).

[43] Mirkin, N., Toscano, F. & Wisniacki, D. Quantum-speed-limit bounds in an open quantum evolution. Phys. Rev. A 94, 052125 (2016). URL https:/​/​doi.org/​10.1103/​PhysRevA.94.052125.
https:/​/​doi.org/​10.1103/​PhysRevA.94.052125

[44] Laine, E.-M., Piilo, J. & Breuer, H.-P. Measure for the non-Markovianity of quantum processes. Phys. Rev. A 81, 062115 (2010). URL https:/​/​doi.org/​10.1103/​PhysRevA.81.062115.
https:/​/​doi.org/​10.1103/​PhysRevA.81.062115

[45] Cianciaruso, M., Maniscalco, S. & Adesso, G. Role of non-markovianity and backflow of information in the speed of quantum evolution. Phys. Rev. A 96, 012105 (2017). URL https:/​/​doi.org/​10.1103/​PhysRevA.96.012105.
https:/​/​doi.org/​10.1103/​PhysRevA.96.012105

[46] Holevo, A. On quasiequivalence of locally normal states. Theor. Math. Phys. 13, 184 (1972). URL https:/​/​doi.org/​10.1007/​BF01035528.
https:/​/​doi.org/​10.1007/​BF01035528

Cited by

[1] D. Z. Rossatto, D. P. Pires, F. M. de Paula, and O. P. de Sá Neto, "Quantum coherence and speed limit in the mean-field Dicke model of superradiance", Physical Review A 102 5, 053716 (2020).

[2] Jie Sun and Songfeng Lu, "On the time–energy uncertainty relation of Kieu applied to quantum evolutions of Grover’s search", International Journal of Quantum Information 18 05, 2050021 (2020).

[3] A. A. Dìaz V., V. Martikyan, S. J. Glaser, and D. Sugny, "Purity speed limit of open quantum systems from magic subspaces", Physical Review A 102 3, 033104 (2020).

[4] Francesco Campaioli, Felix A. Pollock, and Kavan Modi, "Tight, robust, and feasible quantum speed limits for open dynamics", Quantum 3, 168 (2019).

[5] T J Volkoff and Michael J Martin, "Saturating the one-axis twisting quantum Cramér-Rao bound with a total spin readout", Journal of Physics Communications 8 1, 015004 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-29 05:48:00). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2024-03-29 05:48:00).