## Final Report Summary - TOP-ANYON-ENT (Complex Topology, Anyons, and Entanglement)

A great deal of the importance of Topology and Geometry these days is due to their applications to condensed-matter physics and quantum information theory. The project addressed a few related problems of this kind. The first one concerns the behaviour of indistingushable particles on graphs and in particular quantum statistics. The second focuses on quantum correlations between distinguishable particles in a general setting.

A. Particles on Graphs

A graph is a set of points, called vertices, along with edges that connect some of the vertices. An example of a graph that is used by thousands of people on a daily basis is the London Underground. There are also graphs that are too small to be seen with the naked eye, which are fabricated from nano-wires no more than a few millionths of a millimeter in diameter. The objects that inhabit these nano-graphs are particles so small that their motion is governed by the laws of quantum mechanics rather than classical Newtonian physics.

When quantum mechanics is applied to particles moving in our three-dimensional world, the fact that they are indistinguishable vastly restricts their possible collective behaviour to just two types, called fermions and bosons (and which of these types applies is determined by another quantum mechanical property of the particles, called spin). For example, electrons are fermions, and this fact is responsible for a fundamental law called the Pauli Exclusion Principle, which in turn underpins our understanding of the periodic table of elements. The two alternatives taken together - fermions or bosons - are called quantum statistics. In fact quantum statistics is the most elementary topological property of the system of n indistinguishable particles living in a space X which describes what happens to the many-body wave function when particles are exchanged. Abelian quantum statistics is determined by the topological properties of the many particle configuration space, i.e. by abelian representations of the fundamental group which is called the first homology group. In fact abelian quantum statistics can be viewed as a flat connection on the configuration space. The flatness of this connection ensures that there are no classical forces associated with it.

In this project we considered situation when particles live on a graph. We managed to identified the key topological determinants of the quantum statistics in this situation. The structure of the first homology group of the configuration space determines quantum statistics and contains information about one-particle Aharonov-Bohm phases that not contribute to quantum statistics. The key features of a graph that decide about quantum statistics are connectivity and planarity:

For 1-connected graphs, the number of anyon phases depends on the number of particles.

For 2-connected graphs, quantum statistics stabilizes with respect to the number of particles.

For 3-connected graphs (which are also 2-connected), the usual bosonic/fermionic statistics is the only possibility, whereas for planar graphs a single anyon phase is supported.

From the quantum statistics perspective 3-connected graphs mimic two dimensional plane when they are planar and the three dimensional real space when not.

The major mathematical achievement of the project is the development of new methods for calculating the homology groups of graph configuration spaces. They combine tools from algebraic topology and graph theory with a set of combinatorial relations derived from the analysis of certain small canonical graphs. The main advantage of this approach is that it allows to easily predict the results of many complicated key calculations. Using our approach we were able to reproof, in a much simpler way, all the known results concerning homology groups of graph configuration spaces. Moreover, we also obtained new results. The most important one is the compact form formulae for all higher homology groups of the configuration spaces for tree graphs, i.e. graphs that have no cycles. From the perspective of physics the structure of the higher homology groups can have interesting implication on the kinematics of quantum particles on graphs. The study of these properties is the subject of recently submitted ERC Starting Grant.

B. Quantum correlations and quantum computation

The study of correlations is the best way to understand the key differences between quantum and classical physics. It is nowadays well understood that quantum correlations stand behind teleportation, dense coding or the super secure communication. One of the major unsolved problems of the quantum information theory is understanding correlations in systems that consist of more than two particles. In this project we looked at this problem using tools from symplectic geometry. The key concept was the momentum map whose existence is the consequence of the invariance of quantum correlations with respect to unitary operations performed independently on each particle, the so-called local unitary operations. Using geometric aspects of the momentum maps theory developed in the early '80s by Atiyah, Kirwan and Ness we were able to classify critical points of the linear entropy which is an important entanglement measure. Finding these points is the major step forward in understanding the nature of quantum correlations.

We also studied the properties of the multipartite mixed states that belong to symplectic orbits of the local unitary group. For pure states these states are separable or equivalently non-entangled. We discovered that in the mixed case they are so called ClassicalClassical (CC) or ClassicalQuantum (CQ) states i.e. states that form the boundary between classical and quantum correlations. Interestingly these states are also topologically distinguished. The local unitary group orbits through them are the only orbits with non-vanishing Euler-Poincare characteristics.

The last problem that we addressed in this project was the problem of deciding universality of quantum gates. Universal quantum gates play an important role in quantum computing and quantum optics. The ability to effectively manufacture gates operating on many modes, using for example optical networks that couple modes of light, is a natural motivation to consider the universality problems not only for qubits but also for higher dimensional systems, i.e. qudits. For quantum computing with qudits, a universal set of gates consists of all one qudit gates together with an additional two-qudit gate that does not map separable states onto separable states. The set of all one-qudit gates can be, however, generated using a finite number of gates. The set of one-qudit gates S is universal if any gate from SU(d) can be built, with an arbitrary precision, using gates from S. It is known that almost all sets of qudit gates are universal, i.e nonuniversal sets S of the given cardinality are of measure zero and can be characterised by vanishing of a finite number of polynomials in the gates entries and their conjugates. Surprisingly, however, these polynomials are not known and it is hard to find operationally simple criteria that decide one-qudit gates universality. The main obstruction in the considered problem is the lack of classification of finite subgroups of SU(d) for d > 4. We made a major step forward in this project showing that one can still provide some reasonable conditions for universality of one-qudit gates without this knowledge.We formulated an approach that allows to decide universality of S by checking the spectra of the gates and solving some linear equations whose coefficients are polynomial in the entries of the gates and their complex conjugates. Moreover, for non-universal S, our method indicates what type of gates can be added to make S universal. Further problems connected to universal gates and in particular the speed with which gates are approximated are subject of the recently obtained grant by the fellow.

Project website: and choose PROJECTS.

A. Particles on Graphs

A graph is a set of points, called vertices, along with edges that connect some of the vertices. An example of a graph that is used by thousands of people on a daily basis is the London Underground. There are also graphs that are too small to be seen with the naked eye, which are fabricated from nano-wires no more than a few millionths of a millimeter in diameter. The objects that inhabit these nano-graphs are particles so small that their motion is governed by the laws of quantum mechanics rather than classical Newtonian physics.

When quantum mechanics is applied to particles moving in our three-dimensional world, the fact that they are indistinguishable vastly restricts their possible collective behaviour to just two types, called fermions and bosons (and which of these types applies is determined by another quantum mechanical property of the particles, called spin). For example, electrons are fermions, and this fact is responsible for a fundamental law called the Pauli Exclusion Principle, which in turn underpins our understanding of the periodic table of elements. The two alternatives taken together - fermions or bosons - are called quantum statistics. In fact quantum statistics is the most elementary topological property of the system of n indistinguishable particles living in a space X which describes what happens to the many-body wave function when particles are exchanged. Abelian quantum statistics is determined by the topological properties of the many particle configuration space, i.e. by abelian representations of the fundamental group which is called the first homology group. In fact abelian quantum statistics can be viewed as a flat connection on the configuration space. The flatness of this connection ensures that there are no classical forces associated with it.

In this project we considered situation when particles live on a graph. We managed to identified the key topological determinants of the quantum statistics in this situation. The structure of the first homology group of the configuration space determines quantum statistics and contains information about one-particle Aharonov-Bohm phases that not contribute to quantum statistics. The key features of a graph that decide about quantum statistics are connectivity and planarity:

For 1-connected graphs, the number of anyon phases depends on the number of particles.

For 2-connected graphs, quantum statistics stabilizes with respect to the number of particles.

For 3-connected graphs (which are also 2-connected), the usual bosonic/fermionic statistics is the only possibility, whereas for planar graphs a single anyon phase is supported.

From the quantum statistics perspective 3-connected graphs mimic two dimensional plane when they are planar and the three dimensional real space when not.

The major mathematical achievement of the project is the development of new methods for calculating the homology groups of graph configuration spaces. They combine tools from algebraic topology and graph theory with a set of combinatorial relations derived from the analysis of certain small canonical graphs. The main advantage of this approach is that it allows to easily predict the results of many complicated key calculations. Using our approach we were able to reproof, in a much simpler way, all the known results concerning homology groups of graph configuration spaces. Moreover, we also obtained new results. The most important one is the compact form formulae for all higher homology groups of the configuration spaces for tree graphs, i.e. graphs that have no cycles. From the perspective of physics the structure of the higher homology groups can have interesting implication on the kinematics of quantum particles on graphs. The study of these properties is the subject of recently submitted ERC Starting Grant.

B. Quantum correlations and quantum computation

The study of correlations is the best way to understand the key differences between quantum and classical physics. It is nowadays well understood that quantum correlations stand behind teleportation, dense coding or the super secure communication. One of the major unsolved problems of the quantum information theory is understanding correlations in systems that consist of more than two particles. In this project we looked at this problem using tools from symplectic geometry. The key concept was the momentum map whose existence is the consequence of the invariance of quantum correlations with respect to unitary operations performed independently on each particle, the so-called local unitary operations. Using geometric aspects of the momentum maps theory developed in the early '80s by Atiyah, Kirwan and Ness we were able to classify critical points of the linear entropy which is an important entanglement measure. Finding these points is the major step forward in understanding the nature of quantum correlations.

We also studied the properties of the multipartite mixed states that belong to symplectic orbits of the local unitary group. For pure states these states are separable or equivalently non-entangled. We discovered that in the mixed case they are so called ClassicalClassical (CC) or ClassicalQuantum (CQ) states i.e. states that form the boundary between classical and quantum correlations. Interestingly these states are also topologically distinguished. The local unitary group orbits through them are the only orbits with non-vanishing Euler-Poincare characteristics.

The last problem that we addressed in this project was the problem of deciding universality of quantum gates. Universal quantum gates play an important role in quantum computing and quantum optics. The ability to effectively manufacture gates operating on many modes, using for example optical networks that couple modes of light, is a natural motivation to consider the universality problems not only for qubits but also for higher dimensional systems, i.e. qudits. For quantum computing with qudits, a universal set of gates consists of all one qudit gates together with an additional two-qudit gate that does not map separable states onto separable states. The set of all one-qudit gates can be, however, generated using a finite number of gates. The set of one-qudit gates S is universal if any gate from SU(d) can be built, with an arbitrary precision, using gates from S. It is known that almost all sets of qudit gates are universal, i.e nonuniversal sets S of the given cardinality are of measure zero and can be characterised by vanishing of a finite number of polynomials in the gates entries and their conjugates. Surprisingly, however, these polynomials are not known and it is hard to find operationally simple criteria that decide one-qudit gates universality. The main obstruction in the considered problem is the lack of classification of finite subgroups of SU(d) for d > 4. We made a major step forward in this project showing that one can still provide some reasonable conditions for universality of one-qudit gates without this knowledge.We formulated an approach that allows to decide universality of S by checking the spectra of the gates and solving some linear equations whose coefficients are polynomial in the entries of the gates and their complex conjugates. Moreover, for non-universal S, our method indicates what type of gates can be added to make S universal. Further problems connected to universal gates and in particular the speed with which gates are approximated are subject of the recently obtained grant by the fellow.

Project website: and choose PROJECTS.