Abstract
We introduce superdensity operators as a tool for analyzing quantum information in spacetime. Superdensity operators encode spacetime correlation functions in an operator framework, and support a natural generalization of Hilbert space techniques and Dirac’s transformation theory as traditionally applied to standard density operators. Superdensity operators can be measured experimentally, but accessing their full content requires novel procedures. We demonstrate these statements on several examples. The superdensity formalism suggests useful definitions of spacetime entropies and spacetime quantum channels. For example, we show that the von Neumann entropy of a super-density operator is related to a quantum generalization of the Kolmogorov-Sinai entropy, and compute this for a many-body system. We also suggest experimental protocols for measuring spacetime entropies.
Article PDF
Similar content being viewed by others
References
V.I. Arnold, Mathematical methods of classical mechanics, Springer, Germany (1989).
R. Haag, Local quantum physics: fields, particles, algebras, Springer, Germany (2012).
R.B. Griffiths, Consistent histories and the interpretation of quantum mechanics, J. Statist. Phys. 36 (1984) 219 [INSPIRE].
R.B. Griffiths, Consistent interpretation of quantum mechanics using quantum trajectories, Phys. Rev. Lett. 70 (1993) 2201 [INSPIRE].
R.B. Griffiths, Consistent quantum theory, Cambridge University Press, Cambridge U.K. (2002).
H.F. Dowker and J.J. Halliwell, The quantum mechanics of history: the decoherence functional in quantum mechanics, Phys. Rev. D 46 (1992) 1580 [INSPIRE].
R. Omnès, Interpretation of quantum mechanics, Phys. Lett. A 125 (1987) 169.
R. Omnès, The interpretation of quantum mechanics, Princeton University Press, Princeton U.S.A. (1994).
M. Gell-Mann and J.B. Hartle. Quantum mechanics in the light of quantum cosmology, in Complexity, entropy and the physics of information, W. Zurek ed., Addison-Wesley, U.S.A. (1990).
M. Gell-Mann and J.B. Hartle, Alternative decohering histories in quantum mechanics in the proceedings of the 25th International Conference on High Energy Physics (ICHEP90), August 2–8, Singapore (1990).
J.B. Hartle, The quantum mechanics of cosmology, in Quantum cosmology and baby universes, S. Coleman et al. eds., World Scientific, Singapore (1991).
C.J. Isham, Quantum logic and the histories approach to quantum theory, J. Math. Phys. 35 (1994) 2157 [gr-qc/9308006] [INSPIRE].
C.J. Isham and N. Linden, Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory, J. Math. Phys. 35 (1994) 5452 [gr-qc/9405029] [INSPIRE].
C.J. Isham and N. Linden, Continuous histories and the history group in generalized quantum theory, J. Math. Phys. 36 (1995) 5392 [gr-qc/9503063] [INSPIRE].
C.J. Isham, Topos theory and consistent histories: the internal logic of the set of all consistent sets, Int. J. Theor. Phys. 36 (1997) 785 [gr-qc/9607069] [INSPIRE].
J. Cotler and F. Wilczek, Entangled histories, Phys. Scripta T 168 (2016) 014004 [arXiv:1502.02480] [INSPIRE].
J. Cotler and F. Wilczek, Bell tests for histories, arXiv:1503.06458.
J. Cotler et al., Experimental test of entangled histories, Ann. Phys. 387 (2017) 334 [arXiv:1601.02943] [INSPIRE].
J. Cotler and F. Wilczek, Emporal observables and entangled histories, arXiv:1702.05838.
Y. Aharonov, P.G. Bergmann and J.L. Lebowitz. Time symmetry in the quantum process of measurement, Phys. Rev. B 134 (1964) 1410.
Y. Aharonov and L. Vaidman, Complete description of a quantum system at a given time, J. Phys. A 24 (1991) 2315.
Y. Aharonov et al., Multiple-time states and multiple-time measurements in quantum mechanics, Phys. Rev. A 79 (2009) 052110 [arXiv:0712.0320] [INSPIRE].
J.F. Fitzsimons, J.A. Jones and V. Vedral, Quantum correlations which imply causation, Sci. Rept. 5 (2015) 18281.
G. Lindblad, Nonmarkovian quantum stochastic processes and their entropy, Commun. Math. Phys. 65 (1979) 281 [INSPIRE].
R. Alicki and M. Fannes, Defining quantum dynamical entropy, Lett. Math. Phys. 32 (1994) 75.
R. Alicki and M. Fannes, Quantum dynamical systems, Oxford University Press, Oxford U.K. (2001).
R. Alicki et al., An algebraic approach to the Kolmogorov-Sinai entropy, Rev. Math. Phys. 8 (1996) 167.
R. Alicki, Information-theoretical meaning of quantum-dynamical entropy, Phys. Rev. A 66 (2002) 052302 [quant-ph/0201012].
R. Alicki et al., Quantum dynamical entropy and decoherence rate, J. Phys. A 37 (2004) 5157 [quant-ph/0309194].
M.B. Fannes, B. Haegeman and D. Vanpeteghem, Robustness of dynamical entropy, J. Phys. A 38 (2005) 2103 [math-ph/0411055].
L. Hardy, The operator tensor formulation of quantum theory, Phil. Trans. Roy. Soc. A 370 (2012) 3385 [arXiv:1201.4390] [INSPIRE].
L. Hardy, Operational general relativity: possibilistic, probabilistic and quantum, arXiv:1608.06940 [INSPIRE].
O. Oreshkov, F. Costa and Č. Brukner, Quantum correlations with no causal order, Nature Commun. 3 (2012) 1092 [arXiv:1105.4464] [INSPIRE].
O. Oreshkov and N.J. Cerf. Operational formulation of time reversal in quantum theory, Nature Phys. 11 (2015) 853 [arXiv:1507.07745] [INSPIRE].
F. Costa and S. Shrapnel, Quantum causal modelling, New J. Phys. 18 (2016) 063032 [arXiv:1512.07106] [INSPIRE].
G. Chiribella, G.M. D’Ariano and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80 (2009) 022339 [arXiv:0904.4483] [INSPIRE].
A. Bisio et al., Quantum networks: general theory and applications, Acta Phys. Slovaca 61 (2011) 273 [arXiv:1601.04864].
F.A. Pollock et al., Complete framework for efficient characterisation of non-Markovian processes, arXiv:1512.00589.
S. Milz, F.A. Pollock and K. Modi, Reconstructing open quantum system dynamics with limited control, arXiv:1610.02152.
S. Milz, F.A. Pollock and K. Modi, An introduction to operational quantum dynamics, Open Syst. Inform. Dyn. 24 (2017) 1740016 [arXiv:1708.00769].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
D. Stanford, Many-body chaos at weak coupling, JHEP 10 (2016) 009 [arXiv:1512.07687] [INSPIRE].
G.M. D’Ariano, M.G.A. Paris and M.F. Sacchi, Quantum tomography, Adv. Imag. Electron. Phys. 128 (2003) 206.
A. Bisio et al., Optimal quantum tomography, IEEE J. Sel. Top. Quantum Electron. 15 (2009) 1646 [arXiv:1702.08751] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2010).
J. Haah et al., Sample-optimal tomography of quantum states, IEEE Trans. Inf. Theor. 63 (2017) 5628 [arXiv:1508.01797] [INSPIRE].
R. O’Donnell and J. Wright, Efficient quantum tomography, in the proceedings of the 48th annual ACM symposium on Theory of Computing (STOC 2016), June 19–21, MIT, Cambridge, U.S.A. (2016).
M.M. Wolf, F. Verstraete, M.B. Hastings and J.I. Cirac, Area laws in quantum systems: mutual information and correlations, Phys. Rev. Lett. 100 (2008) 070502 [arXiv:0704.3906] [INSPIRE].
Y.B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surv. 32 (1977) 55.
J.P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57 (1985) 617 [INSPIRE].
L.S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys. 108 (2002) 733.
I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42 (2009) 504003.
M.M. Wolf, Quantum channels & operations: guided tour, lecture notes available online (2012).
T.F. Havel, Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups, J. Math. Phys. 44 (2003) 534 [quant-ph/0201127].
A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 (2015).
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
T. Monz et al., 14-qubit entanglement: creation and coherence, Phys. Rev. Lett. 106 (2011) 130506 [arXiv:1009.6126].
R. Islam et al., Emergence and frustration of magnetism with variable-range interactions in a quantum simulator, Science 340 (2013) 583.
N. Ofek et al., Extending the lifetime of a quantum bit with error correction in superconducting circuits, Nature 536 (2016) 441.
C. Song et al., 10-qubit entanglement and parallel logic operations with a superconducting circuit, Phys. Rev. Lett. 119 (2017) 180511 [arXiv:1703.10302].
M. Gärttner et al., Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet, Nature Phys. 13 (2017) 781 [arXiv:1608.08938] [INSPIRE].
S. Kuhr, Quantum-gas microscopes: a new tool for cold-atom quantum simulators, National Sci. Rev. 3 (2016) 170.
S. Trotzky et al., A dynamical quantum simulator, Nature Phys. 8 (2012) 123.
A. Mazurenko et al., A cold-atom Fermi-Hubbard antiferromagnet, Nature 545 (2017) 462.
H. Bernien et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature 551 (2017) 579 [arXiv:1707.04344].
D.A. Abanin and E. Demler, Measuring entanglement entropy of a generic many-body system with a quantum switch, Phys. Rev. Lett. 109 (2012) 020504 [arXiv:1204.2819] [INSPIRE].
A.J. Daley et al., Measuring entanglement growth in quench dynamics of bosons in an optical lattice, Phys. Rev. Lett. 109 (2012) 020505 [arXiv:1205.1521] [INSPIRE].
R. Islam et al., Measuring entanglement entropy through the interference of quantum many-body twins, arXiv:1509.01160.
P. Hauke et al., Measuring multipartite entanglement via dynamic susceptibilities, Nature Phys. 12 (2016) 778 [arXiv:1509.01739].
A.M. Kaufman et al., Quantum thermalization through entanglement in an isolated many-body system, Science 353 (2016) 794.
H. Pichler et al., Measurement protocol for the entanglement spectrum of cold atoms, Phys. Rev. X 6 (2016) 041033 [arXiv:1605.08624] [INSPIRE].
J. Gray et al., Measuring entanglement negativity, arXiv:1709.04923.
J. Cotler et al., Quantum causal structure, to appear.
M. Wen-Long Ma et al., Statistics and phase transitions in sequential quantum measurements of a two-level system, Phys. Rev. A 98 (2018) 012117 [arXiv:1711.02482].
J. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
M. Van Raamsdonk, Building up space-time with quantum entanglement, Int. J. Mod. Phys. D 19 (2010) 2429 [Gen. Rel. Grav. 42 (2010) 2323] [arXiv:1005.3035] [INSPIRE].
B. Czech et al., The gravity dual of a density matrix, Class. Qaunt. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement “thermodynamics”, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
T. Faulkner et al., Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement, arXiv:1405.2933 [INSPIRE].
D.L. Jafferis et al., Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1711.03119
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Cotler, J., Jian, CM., Qi, XL. et al. Superdensity operators for spacetime quantum mechanics. J. High Energ. Phys. 2018, 93 (2018). https://doi.org/10.1007/JHEP09(2018)093
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2018)093