Non-abelian Quantum Statistics on Graphs Maciążek, Tomasz; Sawicki, Adam
Communications in mathematical physics,
11/2019, Letnik:
371, Številka:
3
Journal Article
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We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework ...for describing quantum statistics of particles constrained to move in a topological space
X
. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of
X
which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.
For a random set
S
⊂
U
(
d
)
of quantum gates we provide bounds on the probability that
S
forms a
δ
-approximate
t
-design. In particular we have found that for
S
drawn from an exact
t
-design the ...probability that it forms a
δ
-approximate
t
-design satisfies the inequality
P
(
δ
≥
x
)
≤
2
D
t
e
−
|
S
|
x
a
r
c
t
a
n
h
(
x
)
(
1
−
x
2
)
|
S
|
/
2
=
O
(
2
D
t
(
e
−
x
2
1
−
x
2
)
|
S
|
)
, where
D
t
is a sum over dimensions of unique irreducible representations appearing in the decomposition of
U
↦
U
⊗
t
⊗
U
¯
⊗
t
. We use our results to show that to obtain a
δ
-approximate
t
-design with probability
P
one needs
O
(
δ
−
2
(
t
log
(
d
)
−
log
(
1
−
P
)
)
)
many random gates. We also analyze how
δ
concentrates around its expected value
E
δ
for random
S
. Our results are valid for both symmetric and non-symmetric sets of gates.
We identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs ...and networks for which the isoscattering properties are preserved for Formula: see text. The theoretical predictions are confirmed experimentally using Formula: see text units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the Formula: see text scattering matrices Formula: see text of the systems to 2n diagonal elements, while the old measures of isoscattering require all Formula: see text entries. The studied problem generalizes a famous question of Mark Kac "Can one hear the shape of a drum?", originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.
The concept of active spaces simplifies the description of interacting quantum many-body systems by restricting to a neighborhood of active orbitals around the Fermi level. The respective ...wavefunction ansatzes which involve all possible electron configurations of active orbitals can be characterized by the saturation of a certain number of Pauli constraints 0 ≤ n i ≤ 1 , identifying the occupied core orbitals (ni = 1) and the inactive virtual orbitals (nj = 0). In Part I, we generalize this crucial concept of active spaces by referring to the generalized Pauli constraints. To be more specific, we explain and illustrate that the saturation of any such constraint on fermionic occupation numbers characterizes a distinctive set of active electron configurations. A converse form of this selection rule establishes the basis for corresponding multiconfigurational wavefunction ansatzes. In Part II, we provide rigorous derivations of those findings. Moroever, we extend our results to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.
We have explained and comprehensively illustrated in Part I (Schilling et al 2019 arXiv:1908.10938) that the generalized Pauli constraints suggest a natural extension of the concept of active spaces. ...In the present Part I (Schilling et al 2019 arXiv:1908.10938)I, we provide rigorous derivations of the theorems involved therein. This will offer in particular deeper insights into the underlying mathematical structure and will explain why the saturation of generalized Pauli constraints implies a specific simplified structure of the corresponding many-fermion quantum state. Moreover, we extend the results of Part I (Schilling et al 2019 arXiv:1908.10938) to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.
For a Haar random set <inline-formula> <tex-math notation="LaTeX">\mathcal {S}\subset U(d) </tex-math></inline-formula> of quantum gates we consider the uniform measure <inline-formula> <tex-math ...notation="LaTeX">\nu_{\mathcal {S}} </tex-math></inline-formula> whose support is given by <inline-formula> <tex-math notation="LaTeX">\mathcal {S} </tex-math></inline-formula>. The measure <inline-formula> <tex-math notation="LaTeX">\nu _{\mathcal {S}} </tex-math></inline-formula> can be regarded as a <inline-formula> <tex-math notation="LaTeX">\delta (\nu _{\mathcal {S}},t) </tex-math></inline-formula>-approximate <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-design, <inline-formula> <tex-math notation="LaTeX">t\in \mathbb {Z}_{+} </tex-math></inline-formula>. We propose a random matrix model that aims to describe the probability distribution of <inline-formula> <tex-math notation="LaTeX">\delta (\nu _{\mathcal {S}},t) </tex-math></inline-formula> for any <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>. Our model is given by a block diagonal matrix whose blocks are independent, given by Gaussian or Ginibre ensembles, and their number, size and type is determined by <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>. We prove that, the operator norm of this matrix, <inline-formula> <tex-math notation="LaTeX">\delta ({t}) </tex-math></inline-formula>, is the random variable to which <inline-formula> <tex-math notation="LaTeX">\sqrt {|\mathcal {S}|}\delta (\nu _{\mathcal {S}},t) </tex-math></inline-formula> converges in distribution when the number of elements in <inline-formula> <tex-math notation="LaTeX">\mathcal {S} </tex-math></inline-formula> grows to infinity. Moreover, we characterize our model giving explicit bounds on the tail probabilities <inline-formula> <tex-math notation="LaTeX">\mathbb {P}(\delta (t)>2+\epsilon) </tex-math></inline-formula>, for any <inline-formula> <tex-math notation="LaTeX">\epsilon >0 </tex-math></inline-formula>. We also show that our model satisfies the so-called spectral gap conjecture, i.e. we prove that with the probability 1 there is <inline-formula> <tex-math notation="LaTeX">t\in \mathbb {Z}_{+} </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">\sup _{k\in \mathbb {Z}_{+}}\delta (k)=\delta (t) </tex-math></inline-formula>. Numerical simulations give convincing evidence that the proposed model is actually almost exact for any cardinality of <inline-formula> <tex-math notation="LaTeX">\mathcal {S} </tex-math></inline-formula>. The heuristic explanation of this phenomenon, that we provide, leads us to conjecture that the tail probabilities <inline-formula> <tex-math notation="LaTeX">\mathbb {P}(\sqrt {\mathcal {S}}\delta (\nu _{\mathcal {S}},t)>2+\epsilon) </tex-math></inline-formula> are bounded from above by the tail probabilities <inline-formula> <tex-math notation="LaTeX">\mathbb {P}(\delta (t)>2+\epsilon) </tex-math></inline-formula> of our random matrix model. In particular our conjecture implies that a Haar random set <inline-formula> <tex-math notation="LaTeX">\mathcal {S}\subset U(d) </tex-math></inline-formula> satisfies the spectral gap conjecture with the probability 1.
One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show ...how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the
N
th tensor power of any irreducible representation of
S
U
(
N
)
contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.
Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states ...which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.
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