Quantum Information meets Quantum Matter

“Quantum Information meets Quantum Matter”, it sounds like the beginning of a perfect romance story. It is probably not the kind that makes an Oscar movie, but it does get many physicists excited, physicists including Bei, Duanlu, Xiaogang and me. Actually we find the story so compelling that we decided to write a book about it, and it all started one day in 2011 when Bei popped the question ‘Do you want to write a book about it?’ during one of our conversations.

This idea quickly sparked enthusiasm among the rest of us, who have all been working in this interdisciplinary area and are witness to its rising power. In fact Xiao-Gang has had the same idea of book writing for some time. So now here we are, four years later, posting the first version of the book on arXiv last week.  (arXiv link)

The book is a condensed matter book on the topic of strongly interacting many-body systems, with a special focus on the emergence of topological order. This is an exciting topic, with new developments everyday. We are not trying to cover the whole picture, but rather to present just one perspective – the quantum information perspective – of the story. Quantum information ideas, like entanglement, quantum circuit, quantum codes are becoming ever more popular nowadays in condensed matter study and have lead to many important developments. On the other hand, they are not usually taught in condensed matter courses or covered by condensed matter books. Therefore, we feel that writing a book may help bridge the gap.

We keep the writing in a self-consistent way, requiring minimum background in quantum information and condensed matter. The first part introduces concepts in quantum information that is going to be useful in the later study of condensed matter systems. (It is by no means a well-rounded introduction to quantum information and should not be read in that way.) The second part moves onto explaining one major topic of condensed matter theory, the local Hamiltonians and their ground states, and contains introduction to the most basic concepts in condensed matter theory like locality, gap, universality, etc. The third part then focuses on the emergence of topological order, first presenting a historical and intuitive picture of topological order and then building a more systematic approach based on entanglement and quantum circuit. With this framework established, the fourth part studies some interesting topological phases in 1D and 2D, with the help of the tensor network formalism. Finally part V concludes with the outlook of where this miraculous encounter of quantum information and condensed matter would take us – the unification between information and matter.

We hope that, with such a structure, the book is accessible to both condensed matter students / researchers interested in this quantum information approach and also quantum information people who are interested in condensed matter topics. And of course, the book is also limited by the perspective we are taking. Compared to a standard condensed matter book, we are missing even the most elementary ingredient – the free fermion. Therefore, this book is not to be read as a standard textbook on condensed matter theory. On the other hand, by presenting a new approach, we hope to bring the readers to the frontiers of current research.

The most important thing I want to say here is: this arXiv version is NOT the final version. We posted it so that we can gather feedbacks from our colleagues. Therefore, it is not yet ready for junior students to read in order to learn the subject. On the other hand, if you are a researcher in a related field, please send us criticism, comments, suggestions, or whatever comes to your mind. We will be very grateful for that! (One thing we already learned (thanks Burak!) is that we forgot to put in all the references on conditional mutual information. That will be corrected in a later version, together with everything else.) The final version will be published by Springer as part of their “Quantum Information Science and Technology” series.

I guess it is quite obvious that me writing on the blog of the Institute for Quantum Information and Matter (IQIM) about this book titled “Quantum Information meets Quantum Matter” (QIQM) is not a simple coincidence. The romance story between the two emerged in the past decade or so and has been growing at a rate much beyond expectations. Our book is merely an attempt to record some aspects of the beginning. Let’s see where it will take us.

8 thoughts on “Quantum Information meets Quantum Matter

  1. Here is a purely technical comment. Bookmark the pdf with all the parts, chapters, sections, etc… It makes is much easier to navigate through the book. You can do that automatically in lateX with the hyperref package, I think.

  2. Hi,

    Thanks for making the effort, this book looks really great. I’ve already started reading it.

    However, shouldn’t the sums in (1.14) run from 0,d-1 and not 1,d as is written now? Sorry for being picky, but for students details are important 🙂

    Again, thanks a lot! First impression from the book is really nice.

    Per

  3. Hi, Per, you are absolutely right! And it is very important to get all the details right. Thanks! Will correct the notation.

  4. Quantum Information Meets Quantum Matter is a very good book (as it seems to me), and definitely it is much-needed.

    Perhaps as an additional remark in Section 8.2.1 “Matrix product states: definition and examples”, it might be useful to cite the (vast!) mathematical literature relating to secant varieties of Segre varieties, together with a remark that for inner dimension D=1 the matrix product state cuts-out a Segre variety from the immersing Hilbert space, and for larger D, with the A-matrices restricted to be diagonal, the matrix product states cut-out rank-D secant varieties of Segre varieties.

    Hopefully I have understood the unnatural (to mathematicians) matrix/tensor product state notation correctly.

    Conclusion  It’s good for student readers of Quantum Information Meets Quantum Matter to appreciate, that many of the simplest and most natural questions about these states — both algebraic questions and geometric questions — are puzzling even to experts.

    Some student-friendly math-literature references are appended.

    @misc{cite-key, Title = {Secant varieties of
    {S}egre varieties}, Author = {M. Catalisano and A.
    V. Geramita and A. Gimigliano}, Howpublished =
    {Coimbra {L}ectures: {T}opics in tensors, a summer
    school by {S}hmuel {F}riedland}, Month = {July
    6--8}, Year = {2011}}
    
    @article{cite-key, Author = {M.V. Catalisano and
    A.V. Geramita and A. Gimigliano}, Journal =
    {Linear Algebra and its Applications}, Number =
    {1-3}, Pages = {263 - 285}, Title = {Ranks of
    tensors, secant varieties of Segre varieties and
    fat points}, Volume = {355}, Year = {2002}}
    
    @book{cite-key, Title = {Tensors: Geometry and
    Applications}, Address = {Providence, R.I.},
    Author = {Landsberg, Joseph M.}, Publisher =
    {American Mathematical Society}, Year = {2012}}
    
    @article{cite-key, Author = {Landsberg, J. M. and
    Weyman, J.}, Pages = {685--697}, Title = {On the
    ideals and singularities of secant varieties of
    {S}egre varieties}, Volume = {39}, Year = {2007}}
    • Hi, John,

      Thanks for bringing this up. Indeed, as you said, ‘many of the simplest and most natural questions about these states are puzzling even to experts’ and I am sure there are extensive studies of this subject beyond what we are able to present here. I am afraid it is not quite possible for us cite the vast literature on secant varieties, given the limited scope of this book and its physically oriented approach. Thank you for mentioning them any way.

      Xie

      • Both Nicolas Bourbaki’s books and Donald Knuth’s books (and dozens more math and physics books) flag difficult problems with the dangerous bend symbol. In Latex this is the code

        
        \usepackage{manfnt} % in the preamble
        \dbend % in the body

        If there is ever an edition of Quantum Information Meets Quantum Matter that includes problems sets, then the following natural-to-students problem would deserve to be flagged:

        ————

        A class of dangerous-bend MPS problems  (a)  With reference to Box 8.1 “Matrix Product State” and Box 8.6 “Gauge Degree of Freedom”, count degrees of freedom in the A-matrices to predict whether the MPS for N=4 sites, d=2 physical dimensions, and D=2 inner dimensions, can represent any state in the Hilbert space. Devise a simple numerical method to verify your prediction. Explain any discrepancies.

        (b)  As in (a) above, but with restriction that the A-matrices are diagonal diagonal, predict whether the MPS for N=3 sites, d=3 physical dimensions, and D=4 inner dimensions, can represent any state in the Hilbert space. Devise a simple numerical method to verify your prediction. Explain any discrepancies.

        (c) (double-flagged)  Develop general rules for predicting the dimensionality of matrix product states, viewed as sets cut-out of Hilbert space. Characterize the singularities of the cut-out state-space.

        ————
        Conclusion  The unexpected difficulties associated to practical problems like these at led me first to doubt my own competence, then to doubt my own sanity, and finally led to an undoubted appreciation of the mathematical literature cited above. Needless to say, further references would be welcome.

  5. Pingback: Condensed matter physics, area laws & LQG? | Page 16

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