Abstract
We develop a mathematical formalism that allows to study decoherence with a great level generality, so as to make it appear as a geometrical phenomenon between reservoirs of dimensions. It enables us to give quantitative estimates of the level of decoherence induced by a purely random environment on a system according to their respectives sizes, and to exhibit some links with entanglement entropy.
Similar content being viewed by others
Notes
It is actually very important that the decoherence process (in particular a measurement) is not instantaneous. Otherwise, it would be impossible to explain why an unstable nucleus continuously measured by a Geiger counter is not frozen due to the quantum Zeno effect. See the wonderful model of [6, §8.3 and §8.4] that quantifies the effect of continuous measurement on the decay rate.
We use the quotation marks because, on \({\mathbb {S}}^{n}\) equipped with its complex scalar product, this set doesn’t look like a cap as it does in the real case. QM is nothing but a geometrical way of calculating probabilities (in which the total probability formula is not true, so that it looks like all possible histories interfere), but the geometry in use is quite different from the intuitive one given by the familiar real scalar product. It is noteworthy to remark that the universe, through its quantum statistics, obeys very precisely the geometry of the complex scalar product, and more generally the geometry induced by its canonical extension on tensor products of Hilbert spaces.
References
Zeh, H.D.: On the interpretation of measurement in quantum theory. Found. Phys. 1, 69–76 (1970). https://doi.org/10.1007/BF00708656
Zurek, W.H.: Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? Phys. Rev. D 24, 1516 (1981). https://doi.org/10.1103/PhysRevD.24.1516
Zurek, W.H.: Decoherence, Einselection, and the quantum origins of the classical. Rev. Modern Phys. 75, 715 (2003). https://doi.org/10.1103/RevModPhys.75.715
Joos, E.: Decoherence through interaction with the environment. In: Decoherence and the appearance of a classical world in quantum theory, pp. 35–136. Springer (1996). https://doi.org/10.1007/978-3-662-03263-3
Di Biagio, A., Rovelli, C.: Stable facts, relative facts. Found. Phys 51, 1–13 (2021). https://doi.org/10.1007/s10701-021-00429-w
Fonda, L., Ghirardi, G., Rimini, A.: Decay theory of unstable quantum systems. Rep. Progr. Phys. 41, 587 (1978). https://doi.org/10.1088/0034-4885/41/4/003
Saloff-Coste, L.: Precise estimates on the rate at which certain diffusions tend to equilibrium. Math. Zeitschrift 217, 641–677 (1994). https://doi.org/10.1007/BF02571965
Spengler, C., Huber, M., Hiesmayr, B.C.: Composite parameterization and Haar measure for all unitary and special unitary groups. J. Math. Phys. 53, 013501 (2012). https://doi.org/10.1063/1.3672064
Zurek, W.H.: Environment-induced superselection rules. Phys. Rev. D 26, 1862 (1982). https://doi.org/10.1103/PhysRevD.26.1862
Moran, P.: The closest pair of n random points on the surface of a sphere. Biometrika 66, 158–162 (1979). https://doi.org/10.1093/BIOMET/66.1.158
Rankin, R.A.: The closest packing of spherical caps in n dimensions. Glasgow Math. J. 2, 139–144 (1955)
Zhang, K.: Spherical cap packing asymptotics and rank-extreme detection. IEEE Trans. Inf. Theor. 63, 4572–4584 (2017)
Poulin, D., Qarry, A., Somma, R., Verstraete, F.: Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space. Phys. Rev. Lett. 106, 170501 (2011). https://doi.org/10.1103/PhysRevLett.106.170501
Orús, R.: A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Ann. Phys. 349, 117–158 (2014). https://doi.org/10.1016/j.aop.2014.06.013
Schlosshauer, M.: Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Modern Phys. 76, 1267 (2005). https://doi.org/10.1103/RevModPhys.76.1267
Le Hur, K.: Entanglement entropy, decoherence, and quantum phase transitions of a dissipative two-level system. Ann. Phys. 323, 2208–2240 (2008). https://doi.org/10.1016/j.aop.2007.12.003
Merkli, M., Berman, G., Sayre, R., Wang, X., Nesterov, A.I.: Production of entanglement entropy by decoherence. Open Sys. Inf. Dyna. 25, 1850001 (2018). https://doi.org/10.1142/S1230161218500014
Wyner, A.D.: Random packings and coverings of the unit n-sphere. Bell Syst. Techn. J. 46, 2111–2118 (1967)
Acknowledgements
I would like to gratefully thank my PhD supervisor Dimitri Petritis for the great freedom he grants me in my research, while being nevertheless always present to guide me. I also thank my friends Dmitry Chernyak and Matthieu Dolbeault for illuminating discussions.
Funding
no funding was received for conducting this study.
Author information
Authors and Affiliations
Contributions
Antoine Soulas wrote the whole manuscript.
Corresponding author
Ethics declarations
Competing interest
the author has no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Annex: Decoherence Estimated by the Entanglement Entropy with the Environment
Annex: Decoherence Estimated by the Entanglement Entropy with the Environment
We establish here the formula (3): we first derive the inequality (1), and then look for a relation between \(\eta\) and the linear entropy or the entanglement entropy. Inserting the second into the first directly yields (3).
1.1 Relation Between \(\eta\) and the Level of Classicality
Let’s keep the notations of §2, where we defined \(\rho _{\mathcal {S}}^{(q)}(t) = \sum _{i \ne j} c_i \overline{c_j} {\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle } {|{i}\rangle }{\langle {j}|}\). We have \(|\hspace{-1.111pt}|\hspace{-1.111pt}| \rho _{\mathcal {S}}^{(q)}(t)|\hspace{-1.111pt}|\hspace{-1.111pt}| \leqslant \eta (t)\) because for all vectors \({|{\Psi }\rangle } = \sum _k \alpha _k {|{k}\rangle } \in \mathcal {H}_{\mathcal {S}}\) of norm 1,
Now, if F is a subspace of \(\mathcal {H}_{\mathcal {S}}\) (i.e. a probabilistic event), let \((\varphi _k)_k\) be an orthonormal basis of F. We have:
In a nutshell: \({\mathbb {P}}_{\text {quantum}} = {\mathbb {P}}_{\text {classical}} + \mathcal {O}(\eta )\).
1.2 Relation Between \(\eta\) and the Linear Entropy
We define the linear entropy (or purity defect) of a state \(\rho\) to be \(S_{\text {lin}}(\rho ) = 1 - {\text {tr}}(\rho ^2)\). Since \(\mathcal {S}\) is initially in a pure state, the quantity \(\frac{S_{\text {lin}}(\rho _{\mathcal {S}}(t))}{ S_{\text {lin}}(\rho _{\mathcal {S}}^{(d)})}\) goes from 0 at \(t=0\) to almost 1 when \(t \rightarrow +\infty\). It measures the ratio of purity that has already been lost compared to its final ideal value. Recall that \(\rho _{\mathcal {S}}(t) = \sum _{i=1}^d |c_i |^2 {|{i}\rangle }{\langle {i}|} + \sum _{1 \leqslant i \ne j \leqslant d} c_i \overline{c_j} {\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle } {|{i}\rangle }{\langle {j}|}\), so that:
since the last fraction always equals 1 because \(1 = (\sum _i |c_i |^2)(\sum _i |c_i |^2) = \sum _i |c_i |^4 + \sum _{i \ne j} |c_i |^2 |c_j |^2\). Note that, for any given time t, this inequality is actually an equality for the initial state \({|{\Psi _{\mathcal {S}}(0)}\rangle } = c_{i_0} {|{i_0}\rangle } + c_{j_0} {|{j_0}\rangle }\) where \(i_0\) and \(j_0\) denote two indices such that \(\eta (t) = |{\langle {\mathcal {E}_{i_0}(t) \vert \mathcal {E}_{j_0}(t)}\rangle } |\). Thus:
1.3 Relation Between \(\eta\) and the Entanglement Entropy
The entanglement entropy is always much harder to manipulate. We were not able to prove in the general case a similar result when the linear entropy \(S_{\text {lin}}\) is replaced by the entanglement entropy S, but numerical simulations tend to indicate that the same formula is still (almost) true and that there exists a deep link between the quantity \(1 - \eta ^2(t)\) and the ratio \(\frac{S(\rho _{\mathcal {S}}(t))}{S(\rho _{\mathcal {S}}^{(d)})}\). Here are some considerations to get convinced.
In dimension \(d=2\), if one denotes \(f(t) = {\langle {\mathcal {E}_2(t) \vert \mathcal {E}_1(t)}\rangle }\), one can write \(\rho _{\mathcal {S}}(t) = \begin{pmatrix} |c_1 |^2 &{} c_1 \overline{c_2} f(t) \\ \overline{c_1} c_2 {\overline{f}}(t) &{} |c_1 |^2 \end{pmatrix}\), whose eigenvalues are \(\text {ev}_{\pm } = \frac{1}{2}(1 \pm \sqrt{( |c_1 |^2 - |c_2 |^2)^2 + 4 f^2(t) |c_1 |^2 |c_2 |^2 })\). At large times, \(f \ll 1\) and after some calculations we get at leading order:
The ratio preceding \(\eta ^2\) is a one-parameter real function in \(|c_1 |^2\) (since \(|c_2 |^2 = 1 - |c_1 |^2\)) defined on [0, 1]; it turns out that is takes only values in \([-1, -0.7]\) and tends to \(-1\) only when \(|c_1 |^2\) tends to 0 or 1. Therefore, in dimension 2, we still have (at least at leading order):
In higher dimension, if we suppose that one of the \({\langle {\mathcal {E}_j(t) \vert \mathcal {E}_i(t)}\rangle }\) decreases much slower than the others (assume without loss of generality that it is \({\langle {\mathcal {E}_2(t) \vert \mathcal {E}_1(t)}\rangle }\), still denoted f(t)), then after some time \(\rho _{\mathcal {S}}(t)\) is not very different from:
Using the previous inequality in dimension 2:
and, once again, this bound is attained for an appropriate choice of the \((c_i)_{1\leqslant i \leqslant d}\).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Soulas, A. Decoherence as a High-Dimensional Geometrical Phenomenon. Found Phys 54, 11 (2024). https://doi.org/10.1007/s10701-023-00740-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10701-023-00740-8