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Many-Body Physics from a Quantum Information Perspective

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Modern Theories of Many-Particle Systems in Condensed Matter Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 843))

Abstract

The quantum information approach to many-body physics has been very successful in giving new insights and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many-body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.

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Notes

  1. 1.

    Due to entanglement swapping [13], one must suitably enlarge the notion of preparation of entangled states. So, an entangled state between two particles can be prepared if and only if either the two particles (call them A and B) themselves come together to interact at a time in the past, or two other particles (call them C and D) do the same, with C having interacted beforehand with A and D with B.

  2. 2.

    A unitary operator on \({\mathcal{H}}_{A}\otimes {\mathcal{H}}_{B}\) is said to be “nonlocal” if it is not of the form \(U_A \otimes U_B,\) where \(U_A\) is a unitary operator acting on \({\mathcal{H}}_{A}\) and \(U_B\) acts on \({\mathcal{H}}_{B}.\)

  3. 3.

    A hyperplane is a linear subspace with dimension smaller by one than the dimension of the space itself.

  4. 4.

    Let \({\mathcal{H}}\) be some Hilbert space. Then the set \({\mathcal{B}}({\mathcal{H}})\) of linear bounded operators acting on \({\mathcal{H}}\) is also a Hilbert space with the Hilbert-Schmidt scalar product \(\langle A|B\rangle=\hbox{tr}(A^{\dagger} B) \,(A,B\in{\mathcal{B}}({\mathcal{H}})).\)

  5. 5.

    By \(\Uplambda_1\circ\Uplambda_2\) we denote the composition of two maps \(\Uplambda_i \,(i=1,2),\) i.e., a map that acts on a given operator X as \(\Uplambda_1\circ\Uplambda_2(X)=\Uplambda_1(\Uplambda_2(X)).\)

  6. 6.

    For a more detailed axiomatic description, and other properties of entanglement measures, the reader is encouraged to consult, e.g., Refs. [8, 50, 51].

  7. 7.

    In the standard basis \(\sigma_{y}\) is given by \(\sigma_{y}=-i|0\rangle\langle1|+i|1\rangle\langle 0|.\)

  8. 8.

    In general the gamma function is defined through

    $$ \Upgamma(z)=\int\limits_{0}^{\infty}t^{z-1}e^{-t}\hbox{d}t \qquad (z\in{\mathbb{C}}). (6.26) $$

    For z being positive integers z=n the gamma function is related to the factorial function via \(\Upgamma(n)=(n-1)!\)

  9. 9.

    The bigamma function is defined as \(\Uppsi(z)=\Upgamma^{\prime}(z)/\Upgamma(z)\) and for natural z = n it takes the form

    $$ \Uppsi(n)=-\gamma+\sum_{k=1}^{n-1}{\frac{1}{n}} (6.34) $$

    with \(\gamma\) being the Euler constant, of which exact value is not necessary for our consideration as it vanishes in Eq. 6.33.

  10. 10.

    For results concerning other kind of systems one can consult Ref. [69].

  11. 11.

    Let us shortly recall that the notation \(f(x)=O(g(x))\) means that there exist a positive constant c and \(x_{0}>0\) such that for any \(x\geq x_{0}\) it holds that \(f(x)\leq cg(x).\)

  12. 12.

    The notation \(f(x)=\Upomega(g(x))\) means that there exist \(c>0\) and \(x_{0}>0\) such that \(f(x)\geq cg(x)\) for all \(x\geq x_{0}.\)

  13. 13.

    Recall that the quantum Rényi entropy is defined as

    $$ S_{\alpha}={\frac{1} {1-\alpha}}\log_{2}\left[\hbox{Tr}\left(\varrho^{\alpha}\right)\right] (6.43) $$

    where \(\alpha\in[0,\infty].\) For \(\alpha=0\) one has \(S_{0}(\varrho)=\log_{2}\hbox{rank}(\varrho)\) and \(S_{\infty}=-\log_{2}\lambda_{\rm max}\) with \(\lambda_{\rm max}\) being the maximal eigenvalue of \(\varrho.\)

  14. 14.

    It should be noticed that one can have much stronger condition for such scaling of entropy. To see this explicitly, say that R is a cubic region \(R=\{1,\ldots,n\}^{D}\) meaning that \(|\partial R|=n^{D-1}\) and \(|R|=n^{D}.\) Then since \(\lim\nolimits_{n\to\infty}[(\log n)/n^{\epsilon}]=0\) for any (even arbitrarily small) \(\epsilon>0,\) one easily checks that \(S(\varrho_{R})/|\partial R|^{1+\epsilon}\to 0\) for \(|\partial R|\to \infty.\)

  15. 15.

    The conditional Shannon entropy is defined as \(H(A|B)=H(A,B)-H(B).\)

  16. 16.

    This is the reason why the Glauber model is also known as the kinetic Ising model (KIM).

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Acknowledgments

We are grateful to Ll. Masanes for helpful discussion. We acknowledge the support of Spanish MEC/MINCIN projects TOQATA (FIS2008-00784) and QOIT (Consolider Ingenio 2010), ESF/MEC project FERMIX (FIS2007-29996-E), EU Integrated Project SCALA, EU STREP project NAMEQUAM, ERC Advanced Grant QUAGATUA, Caixa Manresa, AQUTE, and Alexander von Humboldt Foundation Senior Research Prize.

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  1. ICFO–Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860, Barcelona, Castelldefels, Spain

    R. Augusiak, F. M. Cucchietti & M. Lewenstein

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  1. R. Augusiak
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  2. F. M. Cucchietti
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  3. M. Lewenstein
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Correspondence to R. Augusiak or M. Lewenstein .

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Editors and Affiliations

  1. , Departamento de Física/IFLP, Universidad Nacional de La Plata, La Plata, 1900, Argentina

    Daniel C. Cabra

  2. , Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, Göttingen, 37077, Germany

    Andreas Honecker

  3. , Laboratoire de Physique Théorique, Université Paul Sabatier, Route de Narbonne 118, Toulouse Cedex 4, 31062, France

    Pierre Pujol

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Augusiak, R., Cucchietti, F., Lewenstein, M. (2012). Many-Body Physics from a Quantum Information Perspective. In: Cabra, D., Honecker, A., Pujol, P. (eds) Modern Theories of Many-Particle Systems in Condensed Matter Physics. Lecture Notes in Physics, vol 843. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10449-7_6

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