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Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

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This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange–Hamilton mechanical systems. A set of basic geometric and quantum mechanics and probability concepts together with methods of computation are developed in general covariant form for curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideas relating the Grigori Perelman’s entropy for geometric flows and associated statistical thermodynamic systems to the quantum von Neumann entropy, classical and quantum relative and conditional entropy, mutual information, etc. We formulate the concept of the entanglement entropy of quantum geometric information flows and study properties and inequalities for quantum, thermodynamic and geometric entropies characterizing such systems.

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Notes

  1. We follow such conventions: the “horizontal” indices, h-indices, run values \( i,j,k,\ldots =1,2,3,4;\) the vertical indices, v-vertical, run values \( a,b,c\ldots =5,6,7,8\); respectively, the v-indices can be identified/ contracted with h-indices 1, 2, 3, 4 for lifts on total (co) bundles, when \(\alpha =(i,a),\beta =(j,b),\gamma =(k,c),\ldots =1,2,3,\ldots 8.\) There are used letters labeled by an abstract left up/low symbol “\(\ ^{\shortmid }\)” (for instance, \(\ ^{\shortmid }u^{\alpha }\) and \(\ ^{\shortmid }g_{\alpha \beta }) \) in order to emphasize that certain geometric/ physical objects are defined on \(T^{*}V.\) Similar formulas can be derived on TV for geometric objects labeled without “\(\ ^{\shortmid }\).” Boldface symbols are used for geometric objects on spaces endowed with nonlinear connection structure [see below formula (3)].

  2. The coefficients of the canonical N-connection are computed following formulas \(\ ^{\shortmid }\widetilde{\mathbf {N}}=\left\{ \ ^{\shortmid }\widetilde{N} _{ij}:=\frac{1}{2}\left[ \{\ \ ^{\shortmid }\widetilde{g}_{ij},\widetilde{H} \}-\frac{\partial ^{2}\widetilde{H}}{\partial p_{k}\partial x^{i}}\ ^{\shortmid }\widetilde{g}_{jk}-\frac{\partial ^{2}\widetilde{H}}{\partial p_{k}\partial x^{j}}\ ^{\shortmid }\widetilde{g}_{ik}\right] \right\} \), where \(\ \ ^{\shortmid }\widetilde{g}_{ij}\) is inverse to \(\ \ ^{\shortmid } \widetilde{g}^{ab}\) (2). The canonical N-adapted (co) frames are

    $$\begin{aligned} \ ^{\shortmid }\widetilde{\mathbf {e}}_{\alpha }=\left( \ ^{\shortmid }\widetilde{ \mathbf {e}}_{i}=\frac{\partial }{\partial x^{i}}-\ ^{\shortmid }\widetilde{N} _{ia}(x,p)\frac{\partial }{\partial p_{a}},\ ^{\shortmid }e^{b}=\frac{ \partial }{\partial p_{b}}\right) ;\ \ ^{\shortmid }\widetilde{\mathbf {e}}^{\alpha }=\left( \ ^{\shortmid }e^{i}=\mathrm{d}x^{i},\ ^{\shortmid }\mathbf {e}_{a}=\mathrm{d}p_{a}+\ ^{\shortmid }\widetilde{N}_{ia}(x,p)\mathrm{d}x^{i}\right) , \end{aligned}$$

    being characterized by corresponding anholonomy relations \(\ [\ ^{\shortmid } \widetilde{\mathbf {e}}_{\alpha },\ ^{\shortmid }\widetilde{\mathbf {e}} _{\beta }]=\ ^{\shortmid }\widetilde{\mathbf {e}}_{\alpha }\ ^{\shortmid } \widetilde{\mathbf {e}}_{\beta }-\ ^{\shortmid }\widetilde{\mathbf {e}}_{\beta }\ ^{\shortmid }\widetilde{\mathbf {e}}_{\alpha }=\ ^{\shortmid }\widetilde{W} _{\alpha \beta }^{\gamma }\ ^{\shortmid }\widetilde{\mathbf {e}}_{\gamma },\) with anholonomy coefficients \(\widetilde{W}_{ia}^{b}=\partial _{a}\widetilde{ N}_{i}^{b},\) \(\widetilde{W}_{ji}^{a}=\widetilde{\Omega }_{ij}^{a},\) and \(\ ^{\shortmid }\widetilde{W}_{ib}^{a}=\partial \ ^{\shortmid }\widetilde{N} _{ib}/\partial p_{a}\) and \(\ ^{\shortmid }\widetilde{W}_{jia}= {\ ^{\shortmid }}\widetilde{\Omega }_{ija}.\) Such a frame is holonomic (integrable) if the respective anholonomy coefficients are zero.

  3. For simplicity, we shall write, for instance, \(\ ^{\shortmid }\widetilde{ \mathbf {N}}(\tau )\) instead of \(\ ^{\shortmid }\widetilde{\mathbf {N}}(\tau ,\ ^{\shortmid }u)\) if that will not result in ambiguities. Relativistic nonholonomic phase spacetimes can be enabled with necessary types double nonholonomic \((2+2)+(2+2)\) and \((3+1)+(3+1)\) splitting [17, 28, 35,36,37]. Local \((3+1)+(3+1)\) coordinates are labeled in the form \(\ ^{\shortmid }u=\{\ ^{\shortmid }u^{\alpha }=\ ^{\shortmid }u^{\alpha _{s}}=(x^{i_{1}},y^{a_{2}};p_{a_{3}},p_{a_{4}})=(x^{\grave{\imath } },u^{4}=y^{4}=t;p_{\grave{a}},p_{8}=E)\}\) for \(i_{1},j_{1},k_{1},\ldots =1,2;\) \( a_{1},b_{1},c_{1},\ldots =3,4;\) \(a_{2},b_{2},c_{2},\ldots =5,6;\) \( a_{3},b_{3},c_{3},\ldots =7,8.\) The indices \(\grave{\imath },\grave{j},\grave{k} ,\ldots =1,2,3,\) respectively, \(\grave{a},\grave{b},\grave{c},\ldots =5,6,7\) can be used for corresponding spacelike hyper surfaces on a base manifold and typical cofiber.

  4. Hereafter, we shall not write such dependencies in explicit form if that will not result in ambiguities.

  5. We use the symbol r for the replica parameter (and not n as in the typical works in information theory) because the symbol n is used in our works for the dimension of base manifolds.

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Acknowledgements

This research develops the former programs partially supported by IDEI, PN-II-ID-PCE-2011-3-0256, CERN 2012-2014, DAAD-2015 and QGR 2016-2017 and contains certain results for new grant proposals. The UAIC affiliation for S. V. refers to a Project IDEI hosted by that University during 2012–2015, when the bulk of geometric ideas and methods of this and partner works were elaborated (to put a relevant co-/affiliation for further related results was the condition of that grant). Performing rigorous mathematical proofs and respective manuscripts request many years of technical work and further collaborations. S. V. is grateful to D. Singleton, S. Rajpoot and P. Stavrinos for collaboration and supporting his research on geometric methods in physics.

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Authors and Affiliations

  1. Physics Department, California State University at Fresno, Fresno, CA, 93740, USA

    Sergiu I. Vacaru

  2. Department of Theoretical Physics and Computer Modelling, Institute of Applied-Physics and Computer Sciences, Yuriy Fedkovych Chernivtsi National University, 101 Storozhynetska Street, Chernivtsi, 58029, Ukraine

    Sergiu I. Vacaru

  3. Project IDEI - 2011, University “Al. I. Cuza”, Iaşi, Romania

    Sergiu I. Vacaru

  4. SRTV - Studioul TVR Iaşi, 28 Alexandru Lapuşneanu Street, 700057, Iaşi, Romania

    Laurenţiu Bubuianu

  5. University Apollonia, 2 Muzicii Street, 700399, Iaşi, Romania

    Laurenţiu Bubuianu

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  1. Sergiu I. Vacaru
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The UAIC affiliation is for the director of a hosted Project IDEI2012-2-15. Address for post correspondence in 2019–2020 as a visitor senior researcher at YF CNU Ukraine is: 37 Haharina Yu. street, ap. 3, Chernivtsi, Ukraine, 58008.

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Vacaru, S.I., Bubuianu, L. Classical and quantum geometric information flows and entanglement of relativistic mechanical systems. Quantum Inf Process 18, 376 (2019). https://doi.org/10.1007/s11128-019-2487-z

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