arXiv subjects

Marek Mozrzymas

Publications and source records attributed to Marek Mozrzymas.

27 records · Page 2Linked to original sources

arXiv2014

Group representation approach to 1 - N universal quantum cloning machines

In this work, we revisit the problem of finding an admissible region of fidelities obtained after an application of an arbitrary $1 \rightarrow N$ universal quantum cloner which has been recently solved in [A. Kay et al., Quant. Inf. Comput 13, 880 (2013)] from the side of cloning machines. Using group-theory formalism, we show that the allowed region for fidelities can be alternatively expressed in terms of overlaps of pure states with recently found irreducible representations of the commutant $U \otimes U \otimes \ldots \otimes U \otimes U^*$, which gives the characterization of the allowed region where states being cloned are figure of merit. Additionally, it is sufficient to take pure states with real coefficients only, which makes calculations simpler. To obtain the allowed region, we make a convex hull of possible ranges of fidelities related to a given irrep. Subsequently, two cases: $1 \rightarrow 2$ and $1 \rightarrow 3$ cloners, are studied for different dimensions of states as illustrative examples.

quant-ph↗

arXiv2014

Structure and properties of the algebra of partially transposed permutation operators

We consider the structure of algebra of operators, acting in $n-$fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two types of irreducible representations of the algebra. The first one is strictly connected with the representations of the group $S(n-1)$ induced by irreducible representations of the group $S(n-2)$. The second type is structurally connected with irreducible representations of the group $S(n-1)$.

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arXiv2013

Commutant structuture of Ux...xUxU* transformations

In this paper we have found irreducible representations (irreps) of the algebra of partially transposed permutation operators on last subsystem. We give here direct method of irreps construction for an arbitrary number of subsystems n and local dimension d. Our method is inspired by representation theory of symmetric group S(n), theory of Brauer Algebras and Walled Brauer Algebras.

quant-ph↗

arXiv2013

Local random quantum circuits are approximate polynomial-designs - numerical results

We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest neighbour two-qubit gates form an approximate unitary poly(n)-design [F.G.S.L. Brandao et al., arXiv:1208.0692]. Using a group theory formalism, spectral gaps that give a ratio of convergence to a given t-design are evaluated for a different number of qubits n (up to 20) and degrees t (t=2,3,4 and 5), improving previously known results for n=2 in the case of t=2 and 3. Their values lead to a conclusion that the previously used lower bound that bounds spectral gaps values may give very little information about the real situation and in most cases, only tells that a gap is closed. We compare our results to the another lower bounding technique, again showing that its results may not be tight.

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arXiv2012

Convergence to equilibrium under a random Hamiltonian

We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.

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arXiv2011

Distillation of entanglement by projection on permutationally invariant subspaces

We consider distillation of entanglement from two qubit states which are mixtures of three mutually orthogonal states: two pure entangled states and one pure product state. We distill entanglement from such states by projecting n copies of the state on permutationally invariant subspace and then applying one-way hashing protocol. We find analytical expressions for the rate of the protocol. We also generalize this method to higher dimensional systems. To get analytical expression for two qubit case, we faced a mathematical problem of diagonalizing a family of matrices enjoying some symmetries w.r.t. to symmetric group. We have solved this problem in two ways: (i) directly, by use of Schur-Weyl decomposition and Young symmetrizers (ii) showing that the problem is equivalent to a problem of diagonalizing adjacency matrices in a particular instance of a so called algebraic association scheme.

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arXiv2004

Tensor operators and Wigner-Eckart theorem for the quantum superalgebra U_{q}[osp(1\mid 2)]

Tensor operators in graded representations of Z_{2}-graded Hopf algebras are defined and their elementary properties are derived. Wigner-Eckart theorem for irreducible tensor operators for U_{q}[osp(1\mid 2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra U_{q}[osp(1\mid 2)] are considered. The reduced matrix elements for the irreducible tensor operators are calculated. A construction of some elements of the center of U_{q}[osp(1\mid 2)] is given.

math-ph↗

arXiv2004

Wigner-Eckart theorem for tensor operators of Hopf algebras

We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra U$_{t}$(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of U$_{t}$(su(2)) is given.

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arXiv1995

Quantum Deformations of Conformal Algebras Introducing Fundamental Mass Parameters

We consider new class of classical r-matrices for D=3 and D=4 conformal Lie algebras. These r-matrices do satisfy the classical Yang-Baxter equation and as two-tensors belong to the tensor product of Borel subalgebra. In such a way we generalize the lowest order of known nonstandard quantum deformation of sl(2) to the Lie algebras sp(4) = so(5) and sl(4) = so(6). As an exercise we interpret nonstandard deformation of sl(2) as describing quantum D=1 conformal algebra with fundamental mass parameter. Further we describe the D=3 and D=4 conformal bialgebras with deformation parameters equal to the inverse of fundamental masses. It appears that for D=4 the deformation of the Poincare algebra sector coincides with "null plane" quantum Poincare algebra.

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