Erratum: “Symmetry principles in quantum systems theory” [J. Math. Phys. <b>52</b>, 113510 (2011)]

Zeier, Robert; Schulte‐Herbrüggen, Thomas

doi:10.1063/1.4904017

Cited by 10 publications

(51 citation statements)

References 4 publications

(1 reference statement)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…A linear topology (unbranched chain) is sufficient for any quantum algorithm. [40][41][42] • The couplings should be large -so 1 J couplings are preferable.…”

Section: Design Of Moleculementioning

confidence: 99%

Engineering of an all-heteronuclear 5-qubit NMR quantum computer

et al. 2015

View full text Add to dashboard Cite

show abstract

“…A linear topology (unbranched chain) is sufficient for any quantum algorithm. [40][41][42] • The couplings should be large -so 1 J couplings are preferable.…”

Section: Design Of Moleculementioning

confidence: 99%

Engineering of an all-heteronuclear 5-qubit NMR quantum computer

et al. 2015

View full text Add to dashboard Cite

show abstract

“…54 of Ref. 4). Let φ denote a faithful representation of a compact semisimple Lie algebra g such that both the alternating square Alt 2 φ and the symmetric square Sym 2 φ are simple.…”

Section: Alternating Symmetric and Tensor Squaresmentioning

confidence: 99%

“…This classification triggered in Ref. 4 a study of tensor squares φ ⊗ φ which are defined for a representation φ of a Lie algebra g as the representation (φ ⊗ φ)(g) ∶= φ(g) ⊗ ½ dim(φ) + ½ dim(φ) ⊗ φ(g) with g ∈ g. The tensor square φ ⊗ φ of the standard (i.e., defining) representation of the Lie algebra su(ℓ+1) corresponding to the special unitary group has the property that the dimension of its commutant com[φ ⊗ φ] has to grow when restricted to a proper subalgebra h, i.e., dim(com[(φ ⊗ φ) h ]) > dim(com[φ ⊗ φ]) = 2. Here, com[ψ] denotes the commutant of a representation ψ of a Lie algebra g and consists of all complex matrices commuting with all ψ(g) for g ∈ g. This discussion can be summarized as follows.…”

Section: Introductionmentioning

confidence: 99%

“…This led to control-theoretic applications in Ref. 4 where a controlled Schrödinger equation provides certain initial directions (i.e., Lie-algebra generators) in which a quantum system can be steered. In general, these directions do not linearly span but only generate the a priori unknown Lie algebra of all achievable directions, e.g., the threedimensional, infinitesimal rotations around the x-and y-axes generate an additional infinitesimal rotation around the z-axis via the Lie commutator.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On squares of representations of compact Lie algebras

Zeier

Zimborás

2015

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems

show abstract

“…The achievable operations will be characterized by the system Lie algebra, while the reachable sets of states are given by the respective pure state orbits. Dynamic Lie algebras and reachability questions have been intensively studied in the literature for qudit systems [19,[24][25][26]. However, in the case of fermions these questions have to be reconsidered mainly due to the presence of the parity superselection rule.…”

Section: Introductionmentioning

confidence: 99%

A dynamic systems approach to fermions and their relation to spins

Zimborás

Zeier

Keyl

et al. 2014

EPJ Quantum Technol.

Self Cite

View full text Add to dashboard Cite

The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. It just requires knowing drift and control Hamiltonians of an experimental set-up. Then one can easily determine all the states that can be reached from any given initial state. Likewise all the quantum operations that can be simulated with a given set-up can be identified. Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.

show abstract

Erratum: “Symmetry principles in quantum systems theory” [J. Math. Phys. 52, 113510 (2011)]

Cited by 10 publications

References 4 publications

Engineering of an all-heteronuclear 5-qubit NMR quantum computer

Engineering of an all-heteronuclear 5-qubit NMR quantum computer

On squares of representations of compact Lie algebras

A dynamic systems approach to fermions and their relation to spins

Contact Info

Product

Resources

About