Pieter Naaijkens' homepagePostdoctoral researcher in mathematical physics. My reserach interests are in using operator algebraic techniques to study topological phases.
https://pieter.naaijkens.nl/
Tue, 21 Feb 2017 11:42:34 -0800Tue, 21 Feb 2017 11:42:34 -0800Jekyll v3.1.6Jones index, data hiding and total quantum dimension<p>The paper <em><a href="http://arxiv.org/abs/1608.02618">Jones index, data hiding and total quantum dimension</a></em>, together with Leander Fiedler and Tobias Osborne is available on the arXiv. We started this project already a few years ago, so I am happy that after changing the scope a few times and finding better ways to explain things, the paper is finaly ready. Below is a high level summary of the paper, explaining why we think it is interesting.</p>
<p>In the paper we study topologically ordered states in the thermodynamic limit. This has
significant advantages over the more prevalent finite dimensional approach,
and opens up the possibility of using deep results only available in infinite
dimensions, as we demonstrate. We illustrate this by a data hiding scheme and
connect these concepts to more familiar quantities such as topological
entanglement entropy. This way we are able to give a new interpretation of
the quantum dimension. We expect our approach to open up new possibilities in
the study of topological phases.</p>
<p>One of the key points of our work is that it forms a bridge between different
fields, from quantum information and topologically ordered condensed matter
systems in physics, to operator algebra and index theory in pure mathematics.
These are quite distinct fields, so a major goal of our paper is to
communicate our results in a way that is accessible to researchers working in
any of these fields. To achieve this, we have included brief discussions of
known results and techniques, along with illustrating how they are related to
our new results. We first illustrate the main intuition behind our results in a finite dimensional
setting, and explain which complications arise there, and how they can be
resolved by going to the thermodynamic limit.</p>
<p>The starting point of our paper is the observation made by one of
us earlier that the total quantum dimension, an invariant related to the
anyonic excitations of the system, can be obtained by considering the Jones
index (or rather, its generalisation) of an inclusion of operator algebras.
What is new in the present paper is that we reinterpret the quantum dimension
(again, in the thermodynamic limit setting) in terms of a secret sharing
scheme. This interpretation can then be made precise using properties of the
index. In particular, it is possible to exactly quantify the amount of
information that can he hidden in this way, and to identify a quantum channel.
This uses deep results on the Jones index, that at the same time can be given
a natural physical interpretation in our context.</p>
<p>To summarise, our work sheds new light on the total quantum dimension. Using
completely different methods than considered before in the literature, we in
the end obtain similar interpretations. We expect that these new methods will
be useful in the further study of topologically ordered systems, for one since
it allows us to tap into a vast wealth of mathematical results. At the same
time, we hope it will be a useful and accessible introduction to the
relatively unknown (but in our opinion extremely useful) operator algebraic
methods.</p>
Mon, 26 Sep 2016 00:00:00 -0700
https://pieter.naaijkens.nl/news/research/2016/09/26/quantum-dimension.html
https://pieter.naaijkens.nl/news/research/2016/09/26/quantum-dimension.htmlnewsresearchGround states of abelian quantum double models<p>A new preprint (with Matthew Cha and Bruno Nachtergaele) is available on the arXiv: <em><a href="http://arxiv.org/abs/1608.04449">The complete set of infinite volume ground states for Kitaev’s abelian quantum double models</a></em>.
In this paper we consider abelian quantum double models, realised as a local quantum spin system by Kitaev’s description for any finite abelian group <script type="math/tex">G</script>.<sup id="fnref:kitaev"><a href="#fn:kitaev" class="footnote">1</a></sup>
This includes the well-known toric code model, which corresponds to a choice of <script type="math/tex">G = \mathbb{Z}_2</script> for the abelian group.</p>
<p>We consider the model on an infinite <script type="math/tex">\mathbb{Z}^2</script> lattice, with edges between nearest neigbours. Each edge contains a quantum spin system of dimension <script type="math/tex">\vert G\vert</script>. Dynamics are implemented by commuting local projectors, acting on stars (the four edged coming out of a vertex), and plaquettes (the four edges making up a square in the lattice). These projectors detect the absence of a charge at a star or plaquette. There is a unique frustration free ground state (in the thermodyamic limit).<sup id="fnref:afh"><a href="#fn:afh" class="footnote">2</a></sup> However, classifying all ground states is much harder: to find all ground states, one has to find all states that satisfy</p>
<script type="math/tex; mode=display">\omega(A^* \delta(A)) \geq 0.</script>
<p>Here <script type="math/tex">\delta(A) = \lim_{\Lambda \to \infty} [H_\Lambda, A]</script>, with <script type="math/tex">A</script> a local observable, is the generator obtained from the local Hamiltonians. There are many results on the existence and structural properties of the set of ground states of quantum spin models, but surprisingly few results find the complete ground state space of concrete models. In fact, as far as we are aware, our result is the first example in two dimensions.</p>
<p>It turns out that the set of ground states is related to the charge structure of the model. It is possible, using local operators, to create pairs of excitations (or charges) from any state. These excitations are labelled by <script type="math/tex">(\chi,c)</script>, where <script type="math/tex">\chi</script> is a character of <script type="math/tex">G</script> and <script type="math/tex">c \in G</script>. To find the total charge, one just has to take the product of all the charges. The local operators create a pair of inverse charges at each end, so they do not change the total charge of the system. One can, however, send of one of the charges to infinity, leaving a state with a <em>single</em> charge. These states lead to inequivalent representations or superselection sectors.<sup id="fnref:fn"><a href="#fn:fn" class="footnote">3</a></sup></p>
<p>It turns out that these states are ground states, and in fact are essentially all possible ground states. Our main result can be summarised as follows:</p>
<p><strong>Theorem</strong> <em>Let <script type="math/tex">G</script> be a finite abelian group. Then each ground state can be written as a convex combination of at most <script type="math/tex">\vert G \vert^2</script> disjoint states. The pure states in each of these sectors are equivalent to a charged stated, whose equivalence classes are labelled by <script type="math/tex">(\chi, c)</script> for <script type="math/tex">\chi</script> a character of <script type="math/tex">G</script>, and <script type="math/tex">c \in G</script>.</em></p>
<p>There are two important ideas behind our proof: (i) we can define “local” charge projections, that project onto all states with a specific total charge in a region <script type="math/tex">\Lambda</script>, and (ii) a specific choice of boundary conditions, so that states with a single excitation in the bulk are ground states of the local Hamiltonian with these boundary conditions.</p>
<p>Then, by taking any ground state, we can make a sequence of finite volume ground states (with the appropriate boundary conditions), by projecting onto the different charged sectors in a box of size <script type="math/tex">L \times L</script>. We show that this limit converges in the weak<script type="math/tex">*</script>-topology to a ground state, and that if we take the right convex combination of these states, we get our original state back. Then, to conclude the proof, we can consider a pure state in one of the sectors. By using some topological properties of the model, in particular that there is some freedom in choosing paths necessary to define the local operators that create a pair of charges, it is possible to show that the expectation values of local operators that are sufficiently far away, are indistinguishable from the expectation value in the charged states that can be constructed explicitly. It follows that the states must be equivalent, in the sense that their GNS representations are unitarily equivalent.</p>
<h4 id="footnotes">Footnotes</h4>
<div class="footnotes">
<ol>
<li id="fn:kitaev">
<p>A. Kitaev, <em>Fault-tolerant quantum computation by anyons</em>, <a href="http://dx.doi.org/10.1016/S0003-4916(02)00018-0">Ann. Physics</a> <strong>303</strong>, 2–30, 2003. [<a href="http://arxiv.org/abs/quant-ph/9707021">arXiv</a>] <a href="#fnref:kitaev" class="reversefootnote">↩</a></p>
</li>
<li id="fn:afh">
<p>R. Alicki, M. Fannes, M. Horodecki, <em>A statistical mechanics view on Kitaev’s proposal for quantum memories</em>, <a href="http://dx.doi.org/10.1088/1751-8113/40/24/012">J. Phys. A.: Math. Theor.</a> <strong>40</strong>, 6451–6467, 2007. <a href="http://arxiv.org/abs/quant-ph/0702102">[arXiv]</a> <a href="#fnref:afh" class="reversefootnote">↩</a></p>
</li>
<li id="fn:fn">
<p>L. Fiedler, P. Naaijkens, <em>Haag duality for Kitaev’s quantum double model for abelian groups</em>, <a href="http://dx.doi.org/10.1142/S0129055X1550021X">Rev. Math. Phys.</a> <strong>27</strong>, 1550021, 2015. <a href="http://arxiv.org/abs/1406.1084">[arXiv]</a> <a href="#fnref:fn" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Sun, 21 Aug 2016 00:00:00 -0700
https://pieter.naaijkens.nl/news/research/2016/08/21/ground-states-qdouble.html
https://pieter.naaijkens.nl/news/research/2016/08/21/ground-states-qdouble.htmlnewsresearchHaag duality in Kitaev's quantum double models<p>Our paper on Haag duality in quantum double models has been published in <a href="http://dx.doi.org/10.1142/S0129055X1550021X">Reviews in Mathematical Physics</a>. This is the first paper of my co-author <a href="http://www.leander-fiedler.de">Leander Fiedler</a>, a PhD student in Hannover with whom I have worked closely together while I was a postdoc there, so this is very exciting. Our paper generalises results first obtained for the toric code (where <script type="math/tex">G = \mathbb{Z}_2</script>)<!--_--> to arbitrary abelian groups <script type="math/tex">G</script>. In particular, we prove that Haag duality holds in the translational invariant ground state and uncover the superselection structure of the model.</p>
<p>So what exactly is Haag duality? We consider Kitaev’s quantum double model in two dimensions in the thermodynamic limit. For simplicity, we assume that we have a square lattice, with at each edge between neighbouring vertices a <script type="math/tex">\vert G\vert</script>-dimensional Hilbert space. This leads in a natural way to a <script type="math/tex">C^*</script>-algebra<!--*--> <script type="math/tex">\mathfrak{A}</script> of observables, called the quasi-local algebra.<sup id="fnref:quasilocal"><a href="#fn:quasilocal" class="footnote">1</a></sup> This is very general. The physical input comes from specifying dynamics for the system in terms of local Hamiltonians, in our case those first introduced by Kitaev.<sup id="fnref:kitaev"><a href="#fn:kitaev" class="footnote">2</a></sup> The precise form is not important here: the main point is that this allows us to select ground states for these dynamics. In the present case, there are many ground states, but for our purposes there is a preferred one: there is a unique translational invariant ground state (which is pure).<sup id="fnref:gs"><a href="#fn:gs" class="footnote">3</a></sup></p>
<p>Once we have selected a state <script type="math/tex">\omega_0</script> on <script type="math/tex">\mathfrak{A}</script> we can obtain a representation <script type="math/tex">\pi_0</script> on a Hilbert space <script type="math/tex">\mathcal{H}</script>, such that the state <script type="math/tex">\omega_0</script> is implemented by a vector <script type="math/tex">\Omega \in \mathcal{H}</script>. This can be done via the <a href="https://en.wikipedia.org/wiki/Gelfand–Naimark–Segal_construction">GNS construction</a>. Now consider a “cone region” of the system: take a point in the lattice and two semi-infinite lines emanating from them. We say that, roughly speaking, the sites within the two lines form a cone <script type="math/tex">\Lambda</script>.<sup id="fnref:cone"><a href="#fn:cone" class="footnote">4</a></sup> Associated to the cone is the <script type="math/tex">C^*</script>-<!--*-->algebra <script type="math/tex">\mathfrak{A}(\Lambda)</script> of quasi-local observables localised in that cone. The algebra of all quasi-local observables <em>outside</em> the cone is denoted by <script type="math/tex">\mathfrak{A}(\Lambda)</script>. We can then define two new algebras:</p>
<script type="math/tex; mode=display">\mathcal{R}_\Lambda := \pi_0(\mathfrak{A}(\Lambda))'',
\quad
\mathcal{R}_{\Lambda^c} := \pi_0(\mathfrak{A}(\Lambda^c))'',</script>
<p>where the prime means taking the commutant in <script type="math/tex">\mathfrak{B}(\mathcal{H})</script>. By taking the double commutant we obtain <a href="https://en.wikipedia.org/wiki/Von_Neumann_algebra">von Neumann algebras</a>. It should be noted that these algebras strongly depend on the choice of representation <script type="math/tex">\pi_0</script>. By locality, the property that observables localised in disjoint regions commute, it is easy to see that <script type="math/tex">\mathcal{R}_\Lambda \subset \mathcal{R}_{\Lambda^c}'</script>. <em>Haag duality</em> then is the claim that the reverse inclusion also holds.</p>
<p>The proof relies on a good understanding of how the Hilbert space can be build by acting with local observables on the ground state vector <script type="math/tex">\Omega</script>. It turns out that we can label a convenient orthonormal basis by looking at “charge configurations” of anyons. These configurations can be obtained by acting with local operators associated to paths on the lattice on the ground state vector. A good understanding of the commutation properties of these operators is essential. In this case, these relations have a nice algebraic structure stemming from the representation theory of Hopf algebras. It allows us to look at the subspace of all charge configurations in the cone <script type="math/tex">\Lambda</script> (and trivial charge outside), and see how <script type="math/tex">\mathcal{R}_\Lambda</script> and <script type="math/tex">\mathcal{R}_{\Lambda^c}'</script> act when restricted to this subspace. By proving a density result, it is possible to show that they are each others commutants. This result then extends to the whole Hilbert space.</p>
<p>Haag duality is mainly used as a technical tool in the study of the superselection sectors of the model. For a short introduction of the main ideas behind this construction, which originates from algebraic quantum field theory, see <a href="http://dx.doi.org/10.1007/978-3-319-21353-8_9">here</a> (or the <a href="http://arxiv.org/abs/1508.07170">arXiv version</a>). In essence, it allows us to replace representations of <script type="math/tex">\mathfrak{A}</script>, which satisfy a physically reasonable selection criterion,<sup id="fnref:superselect"><a href="#fn:superselect" class="footnote">5</a></sup> by endomorphisms of <script type="math/tex">\mathfrak{A}</script>. These endomorphisms have a much richer structure than the representations. For example, two endomorphisms can be composed. Physically speaking this is the same as “fusion”: bringing two charges together and see how they combine.<sup id="fnref:fusionrule"><a href="#fn:fusionrule" class="footnote">6</a></sup> By studying these endomorphisms we can uncover all the relevant physical properties of the anyonic charges in the model. As expected, they are described by the representation theory of the quantum double of the group <script type="math/tex">G</script>.</p>
<h4 id="footnotes">Footnotes</h4>
<div class="footnotes">
<ol>
<li id="fn:quasilocal">
<p>This is the algebra of all observables that can be approximated arbitrarily well (in norm) by observables that act only on a finite number of sites. It is the inductive limit of local algebras <script type="math/tex">\mathfrak{A}(\Lambda) = \bigotimes_{x \in \Lambda} M_{\vert G\vert}(\mathbb{C})</script>. <a href="#fnref:quasilocal" class="reversefootnote">↩</a></p>
</li>
<li id="fn:kitaev">
<p>Kitaev, A.: Fault-tolerant quantum computation by anyons. <em>Ann. Physics.</em> <strong>303</strong>, 2–30 (2003). <a href="#fnref:kitaev" class="reversefootnote">↩</a></p>
</li>
<li id="fn:gs">
<p>The current version on the arXiv is a bit imprecise on this point. This is the ground state that we use there, but the claim that this is the only ground state is not correct. See the published version for the correct statement. <a href="#fnref:gs" class="reversefootnote">↩</a></p>
</li>
<li id="fn:cone">
<p>The precise shape is not that important, as long as it is an infinite region without any holes and big enough that any finite set can be translated into it. <a href="#fnref:cone" class="reversefootnote">↩</a></p>
</li>
<li id="fn:superselect">
<p>In particular, we look at representations that look like the ground state representation <em>outside</em> of arbitrary cones <script type="math/tex">\Lambda</script>. Alternatively, this means that the charges we are interested in, which correspond to equivalence classes of such representations, can be localised <em>inside</em> a cone, and can also be moved around to be localised inside any other cone. <a href="#fnref:superselect" class="reversefootnote">↩</a></p>
</li>
<li id="fn:fusionrule">
<p>This is analogous to the representation theory of compact groups, where the tensor product of two irreducible representations can be decomposed into a direct sum of such representations. <a href="#fnref:fusionrule" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>
Mon, 21 Dec 2015 00:00:00 -0800
https://pieter.naaijkens.nl/news/research/2015/12/21/Haag-duality.html
https://pieter.naaijkens.nl/news/research/2015/12/21/Haag-duality.htmlnewsresearchNew website uploaded<p>The website has been updated and redesigned, to reflect my move to UC Davis.</p>
Sun, 11 Oct 2015 11:00:00 -0700
https://pieter.naaijkens.nl/news/2015/10/11/new-website.html
https://pieter.naaijkens.nl/news/2015/10/11/new-website.htmlnewsMarie Sklodowska-Curie project started<p>I have started working on the `Operator-algebraic Approach to Topological Phases’ (OATP) project funded by the European Union. This project will run for three years. See <a href="/mariecurie/">here</a> for more details.</p>
Tue, 01 Sep 2015 02:00:00 -0700
https://pieter.naaijkens.nl/news/2015/09/01/Marie-Curie-started.html
https://pieter.naaijkens.nl/news/2015/09/01/Marie-Curie-started.htmlnews