Open quantum systems are harder to track than open classical systems

Prahlad Warszawski1 and Howard M. Wiseman2

1Centre of Excellence in Engineered Quantum Systems (Australian Research Council), School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia
2Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia

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

Abstract

For a Markovian (in the strongest sense) open quantum system it is possible, by continuously monitoring the environment, to perfectly track the system; that is, to know the stochastically evolving pure state of the system without altering the master equation. In general, even for a system with a finite Hilbert space dimension $D$, the pure state trajectory will explore an infinite number of points in Hilbert space, meaning that the dimension $K$ of the classical memory required for the tracking is infinite. However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed that tracking of a qubit ($D=2$) is always possible with a bit ($K=2$), and gave a heuristic argument implying that a finite $K$ should be sufficient for any $D$, although beyond $D=2$ it would be necessary to have $K>D$. Our paper is concerned with rigorously investigating the relationship between $D$ and $K_{\rm min}$, the smallest feasible $K$. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with $D>2$, $K_{\rm min}>D$, by a computational proof (via Hilbert Nullstellensatz certificates of infeasibility). That is, beyond $D=2$, $D$-dimensional open quantum systems are provably harder to track than $D$-dimensional open classical systems. We stress that this result allows complete freedom in choice of monitoring scheme, including adaptive monitoring which is, in general, necessary to implement a physically realizable ensemble (as it is known) of just $K$ pure states. Moreover, we develop, and better justify, a new heuristic to guide our expectation of $K_{\rm min}$ as a function of $D$, taking into account the number $L$ of Lindblad operators as well as symmetries in the problem. The use of invariant subspace and Wigner symmetries (that we recently introduced elsewhere, [New J. Phys. https://doi.org/10.1088/1367-2630/ab14b2]) makes it tractable to conduct a numerical search, using the method of polynomial homotopy continuation, to find finite physically realizable ensembles in $D=3$. The results of this search support our heuristic. We thus have confidence in the most interesting feature of our heuristic: in the absence of symmetries, $K_{\rm min} \sim D^2$, implying a quadratic gap between the classical and quantum tracking problems. Explicit adaptive monitoring schemes that realize the discovered finite ensembles are obtained numerically, thus facilitating future experimental investigations.

► BibTeX data

► References

[1] H. M. Wiseman and G. J. Milburn. Quantum measurement and control. Cambridge University Press, 2010. 10.1017/​CBO9780511813948.
https:/​/​doi.org/​10.1017/​CBO9780511813948

[2] H. M. Wiseman and J. A. Vaccaro. Inequivalence of pure state ensembles for open quantum systems: the preferred ensembles are those that are physically realizable. Phys. Rev. Lett., 87 (24): 240402, 2001. 10.1103/​physrevlett.87.240402.
https:/​/​doi.org/​10.1103/​physrevlett.87.240402

[3] R. I. Karasik and H. M. Wiseman. How many bits does it take to track an open quantum system? Phys. Rev. Lett., 106 (2): 020406, 2011a. 10.1103/​physrevlett.106.020406.
https:/​/​doi.org/​10.1103/​physrevlett.106.020406

[4] R. I. Karasik and H. M. Wiseman. Tracking an open quantum system using a finite state machine: stability analysis. Phys. Rev. A, 84 (5): 052120, 2011b. 10.1103/​physreva.84.052120.
https:/​/​doi.org/​10.1103/​physreva.84.052120

[5] S. Daryanoosh, H. M. Wiseman, and T. Brandes. Stochastic feedback control of quantum transport to realize a dynamical ensemble of two nonorthogonal pure states. Phys. Rev. B, 93 (8): 085127, 2016. 10.1103/​physrevb.93.085127.
https:/​/​doi.org/​10.1103/​physrevb.93.085127

[6] P. Warszawski and H. M. Wiseman. Symmetries and physically realizable ensembles for open quantum systems. New J. Phys. 10.1088/​1367-2630/​ab14b2.
https:/​/​doi.org/​10.1088/​1367-2630/​ab14b2

[7] A. Gill. Introduction to the theory of finite-state machines. McGraw-Hill, 1962.

[8] N. Bohr. On the constitution of atoms and molecules. Phil. Mag., 26 (151): 1-25, 1913. 10.1080/​14786441308634955.
https:/​/​doi.org/​10.1080/​14786441308634955

[9] E. Schrödinger. Discussion of probability relations between separated systems. Math. Proc. Camb. Philos. Soc., 31 (4): 555-563, 1935. 10.1017/​s0305004100013554.
https:/​/​doi.org/​10.1017/​s0305004100013554

[10] H. Carmichael. An open systems approach to quantum optics: lectures presented at the Université Libre de Bruxelles, October 28 to November 4, 1991, volume 18. Springer Science & Business Media, 2009. 10.1007/​978-3-540-47620-7.
https:/​/​doi.org/​10.1007/​978-3-540-47620-7

[11] H. M. Wiseman. Quantum theory of continuous feedback. Phys. Rev. A, 49: 2133-2150, Mar 1994. 10.1103/​PhysRevA.49.2133.
https:/​/​doi.org/​10.1103/​PhysRevA.49.2133

[12] It is important to note that the implied number of detectors (equal to $M$) can be greater than the number of Lindblad operators, $L$, describing the ME. This is most easily understood in a quantum optics context: linear interferometers, described by ${S}$, take the field outputs of the system as inputs while $\vec{\beta}$ represents the addition of weak local oscillators (WLOs) to the interferometer outputs prior to photodetection. The merging and splitting of system output fields can also refer to frequency conversion, so that the `linear interferometer' is to be understood in the most general sense. The LOs are required to be weak (that is, $|{\beta}_l|^2$ not much larger than $Tr[c^\dagger _l c_l \rho_{\rm ss}]$) so that a non-negligible amount of information is gathered concerning the system upon each detection, meaning that the quantum trajectory is jump-like rather than diffusive; consequently, the PRE can consist of a finite number of states. We stress that most jump-like unravellings will lead, like diffusive unravellings, to PREs of infinite size. Those that lead to finite PREs, as we are interested in, are exceptional.

[13] J. Dalibard, Y. Castin, and K. Mølmer. Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett., 68: 580-583, Feb 1992. 10.1103/​PhysRevLett.68.580.
https:/​/​doi.org/​10.1103/​PhysRevLett.68.580

[14] C. W. Gardiner, A. S. Parkins, and P. Zoller. Wave-function quantum stochastic differential equations and quantum-jump simulation methods. Phys. Rev. A, 46: 4363-4381, Oct 1992. 10.1103/​PhysRevA.46.4363.
https:/​/​doi.org/​10.1103/​PhysRevA.46.4363

[15] A Barchielli. Stochastic differential equations anda posteriori states in quantum mechanics. Int. J. Theor. Phys., 32 (12): 2221-2233, 1993. 10.1007/​bf00672994.
https:/​/​doi.org/​10.1007/​bf00672994

[16] N. Gisin and I. C. Percival. The quantum-state diffusion model applied to open systems. J. Phys. A: Math. Gen., 25 (21): 5677, 1992. 10.1088/​0305-4470/​25/​21/​023.
https:/​/​doi.org/​10.1088/​0305-4470/​25/​21/​023

[17] H. M. Wiseman and G. J. Milburn. Interpretation of quantum jump and diffusion processes illustrated on the bloch sphere. Phys. Rev. A, 47: 1652-1666, Mar 1993. 10.1103/​PhysRevA.47.1652.
https:/​/​doi.org/​10.1103/​PhysRevA.47.1652

[18] E. Schrödinger. Probability relations between separated systems. Math. Proc. Camb. Phil. Soc., 32: 446, 1936. 10.1017/​s0305004100019137.
https:/​/​doi.org/​10.1017/​s0305004100019137

[19] L. P. Hughston, R. Jozsa, and W. K. Wootters. A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A, 183 (1): 14-18, 1993. 10.1016/​0375-9601(93)90880-9.
https:/​/​doi.org/​10.1016/​0375-9601(93)90880-9

[20] K. A. Kirkpatrick. The Schrödinger-HJW theorem. Found. Phys. Lett., 19 (1): 95, 2006. 10.1007/​s10702-006-1852-1.
https:/​/​doi.org/​10.1007/​s10702-006-1852-1

[21] In the case that the rank of $\rho_{{\rm ss}}$ is less than $D$, then we can project the Lindblad ME onto a smaller dimensional subspace and redefine that as D. Also note that uniqueness of $\rho_{{\rm ss}}$ is assumed in our work for convenience. If the assumption were not made then our results would become dependent upon the particular steady-state under consideration. For example, different steady-states could be of differing rank, and also have a differing effective number of decoherence channels (non-zero $Tr[\hat{c}^\dagger _l\hat{c}_l \rho_{{\rm ss}}]$). These complications could be included, but would distract from the underlying physics that we wish to highlight.

[22] N. Courtois, A. Klimov, J. Patarin, and A. Shamir. Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In Advances in Cryptology—EUROCRYPT 2000, pages 392-407. Springer, 2000. 10.1007/​3-540-45539-6_27.
https:/​/​doi.org/​10.1007/​3-540-45539-6_27

[23] W. Bosma, J. Cannon, and C. Playoust. The MAGMA algebra system i: The user language. J. Symb. Comput., 24 (3): 235-265, 1997. 10.1006/​jsco.1996.0125.
https:/​/​doi.org/​10.1006/​jsco.1996.0125

[24] J. Verschelde. Algorithm 795: Phcpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw., 25 (2): 251-276, 1999. 10.1145/​317275.317286.
https:/​/​doi.org/​10.1145/​317275.317286

[25] D. A. Cox, J. Little, and D. O'Shea. Using algebraic geometry, volume 185. Springer Science & Business Media, 2006. 10.1007/​b138611.
https:/​/​doi.org/​10.1007/​b138611

[26] T. Li. Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta numerica, 6: 399-436, 1997. 10.1017/​s0962492900002749.
https:/​/​doi.org/​10.1017/​s0962492900002749

[27] E. P. Wigner. Gruppentheorie. Vieweg, 1931 pp 251-254. doi.org/​10.1007/​978-3-663-02555-9.
https:/​/​doi.org/​doi.org/​10.1007/​978-3-663-02555-9

[28] E. P. Wigner. Group Theory. Academic Press Inc., 1959 pp 233-236. 10.1016/​b978-0-12-750550-3.x5001-0.
https:/​/​doi.org/​10.1016/​b978-0-12-750550-3.x5001-0

[29] B. Schumacher and M. Westmoreland. Quantum processes systems, and information. Cambridge University Press, 2010. 10.1017/​CBO9780511814006.
https:/​/​doi.org/​10.1017/​CBO9780511814006

[30] L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and real computation: A manifesto. Int. J. Bifurc. Chaos, 6 (01): 3-26, 1996. 10.1142/​s0218127496001818.
https:/​/​doi.org/​10.1142/​s0218127496001818

[31] J. Faugere. A new efficient algorithm for computing Gröbner bases (F4). J. Pure Appl. Algebr., 139 (1): 61-88, 1999. 10.1016/​s0022-4049(99)00005-5.
https:/​/​doi.org/​10.1016/​s0022-4049(99)00005-5

[32] B. Buchberger. A theoretical basis for the reduction of polynomials to canonical forms. ACM SIGSAM Bulletin, 10 (3): 19-29, 1976. 10.1145/​1088216.1088219.
https:/​/​doi.org/​10.1145/​1088216.1088219

[33] Private communication with Dr. Allan Steel.

[34] D. Cox, J. Little, and D. O'Shea. Ideals, varieties, and algorithms, volume 3. Springer, 1992. 10.1007/​978-1-4757-2181-2.
https:/​/​doi.org/​10.1007/​978-1-4757-2181-2

[35] Note that this `tracking', a numerical technique of polynomial homotopy continuation, is unrelated to the efficient `tracking' of open quantum systems, which is the physics problem under investigation.

[36] C. Hill, K. Lee, A. Leykin, T. Duff, A. Jensen, and J. Sommars. Solving polynomial systems via homotopy continuation and monodromy. IMA J. Numer. Anal., 04 2018. 10.1093/​imanum/​dry017.
https:/​/​doi.org/​10.1093/​imanum/​dry017

[37] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of N-level systems. J. Math. Phys., 17 (5): 821-825, 1976. 10.1063/​1.522979.
https:/​/​doi.org/​10.1063/​1.522979

[38] To prove that $L=1$ PREs are not possible for $K=4,5$ for the example MEs chosen, we rely upon ruling out various graphs using the considerations of Sec. 3.1. For the graphs that are not ruled out in this way, we then obtain computational Nullstellensätze showing that no PRE can exist. In contrast to the case where a Nullstellensatz is obtained for a fully connected graph (for the $K=3$), for these $L=1$, $K=4,5$ examples there are many isomorphically different graphs that can be drawn when the number of non-zero rates is less than the maximal number. That is, ruling out the graph with the optimal (largest) number of transitions is not sufficient to prove that no PRE can exist. Instead, we must consider all the isomorphically different graphs with fewer rates also and then either rule them out via Sec. 3.1 or by Nullstellensatz. This difficulty does not arise if a Nullstellensatz can be performed on the fully connected graph as all graphs with fewer transitions can be obtained from this graph just by setting some of the transition rates to zero. Consequently, one must first identify all the relevant graphs (there are about 50) and then obtain Nullstellensätze for them in order to have an overall proof of $K\leq 5$ PRE non-existence for a single example ME. Some graph Nullstellensätze take longer than others, with a total runtime for each ME of around 2 days.

[39] T. Y. Li, T. Sauer, and J. A. Yorke. The cheater’s homotopy: an efficient procedure for solving systems of polynomial equations. SIAM J. Numer. Anal., 26 (5): 1241-1251, 1989. 10.1137/​0726069.
https:/​/​doi.org/​10.1137/​0726069

[40] There is no guarantee that a numerical search for solutions to a polynomial system will actually find all existing solutions. Algorithms have various defined tolerances that may not be tight enough to faithfully explore the search space for a given ME. This is in contrast to the case where a Hilbert Nullstellensatz is found, which is a guarantee that no solutions can exist. However, in the case of square polynomial systems, it has always been the case that there will exist non-PRE solutions to the constraints (that is, complex solutions, whereas PREs require real valued solutions as the real and imaginary equations have been separated out). These complex solutions will mean that a Nullstellensatz will not exist, even in the absence of PREs, and cannot be utilized as an exact proof.

[41] L. Hardy. Quantum theory from five reasonable axioms. arXiv:quant-ph/​0101012, 2001.
arXiv:quant-ph/0101012

[42] A. Monras, A. Beige, and K. Wiesner. Hidden quantum Markov models and non-adaptive read-out of many-body states. Appl. Math. and Comp. Sciences, 3: 93, 2011.

[43] M. Gu, K. Wiesner, E. Rieper, and V. Vedral. Quantum mechanics can reduce the complexity of classical models. Nat. Commun., 3, Mar 2012. 10.1038/​ncomms1761.
https:/​/​doi.org/​10.1038/​ncomms1761

[44] D. Mehta. Numerical polynomial homotopy continuation method and string vacua. Adv. High Energy Phys., 2011, 2011a. 10.1155/​2011/​263937.
https:/​/​doi.org/​10.1155/​2011/​263937

[45] D. Mehta. Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation method. Phys. Rev. E, 84 (2): 025702, 2011b. 10.1103/​physreve.84.025702.
https:/​/​doi.org/​10.1103/​physreve.84.025702

[46] D. Mehta, H. D. Nguyen, and K. Turitsyn. Numerical polynomial homotopy continuation method to locate all the power flow solutions. IET Gener. Transm. Distrib., 10 (12): 2972-2980, 2016. 10.1049/​iet-gtd.2015.1546.
https:/​/​doi.org/​10.1049/​iet-gtd.2015.1546

[47] A. J. Sommese and C. W. Wampler II. The Numerical solution of systems of polynomials arising in engineering and science. World Scientific, 2005. 10.1142/​5763.
https:/​/​doi.org/​10.1142/​5763

[48] E. L. Allgower and K. Georg. Numerical continuation methods: an introduction, volume 13. Springer Science & Business Media, 2012. 10.1137/​1.9780898719154.
https:/​/​doi.org/​10.1137/​1.9780898719154

[49] A. Leykin, J. Verschelde, and Y. Zhuang. Parallel homotopy algorithms to solve polynomial systems. In ICMS, pages 225-234. Springer, 2006. 10.1007/​11832225_22.
https:/​/​doi.org/​10.1007/​11832225_22

[50] G. Malajovich and K. Meer. Computing minimal multi-homogeneous Bézout numbers is hard. Theory Comput. Syst., 40 (4): 553-570, 2007. 10.1007/​s00224-006-1322-y.
https:/​/​doi.org/​10.1007/​s00224-006-1322-y

[51] T. Duff, C. Hill, A. Jensen, K. Lee, A. Leykin, and J. Sommars. MonodromySolver: a Macaulay2 package for solving polynomial systems via homotopy continuation and monodromy. Available at http:/​/​people.math.gatech.edu/​∼aleykin3/​MonodromySolver.
http:/​/​people.math.gatech.edu/​

Cited by

[1] Thomas J. Elliott, Chengran Yang, Felix C. Binder, Andrew J. P. Garner, Jayne Thompson, and Mile Gu, "Extreme dimensional compression with quantum modelling", arXiv:1909.02817.

The above citations are from SAO/NASA ADS (last updated 2019-10-20 14:20:58). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2019-10-20 14:20:56).