Péter Vrana

Three-qubit operators, the split Cayley hexagon of order two, and black holes

The set of 63 real generalized Pauli matrices of three-qubits can be factored into two subsets of 35 symmetric and 28 antisymmetric elements. This splitting is shown to be completely embodied in the properties of the Fano plane; the elements of the former set being in a bijective correspondence with the 7 points, 7 lines and 21 flags, whereas those of the latter set having their counterparts in 28 anti-flags of the plane. This representation naturally extends to the one in terms of the split Cayley hexagon of order two. 63 points of the hexagon split into 9 orbits of 7 points (operators) each under the action of an automorphism of order 7. 63 lines of the hexagon carry three points each and represent the triples of operators such that the product of any two gives, up to a sign, the third one. Since this hexagon admits a full embedding in a projective 5-space over GF(2), the 35 symmetric operators are also found to answer to the points of a Klein quadric in such space. The 28 antisymmetric matrices can be associated with the 28 vertices of the Coxeter graph, one of two distinguished subgraphs of the hexagon. The P SL2(7) subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter sub-geometry is found to be intricately related to the E7-symmetric black-hole entropy formula in string theory. It is also conjectured that the full geometry/symmetry of the hexagon should manifest itself in the corresponding black-hole solutions. Finally, an intriguing analogy with the case of Hopf sphere fibrations and a link with coding theory are briefly mentioned.

The Veldkamp space of multiple qubits

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign, and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections. Among the geometric hyperplanes, one can find the set of self-dual operators with respect to the Wootters spin-flip operation well known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases, a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case, the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.

Special entangled quantum systems and the Freudenthal construction

We consider special quantum systems containing both distinguishable and identical constituents. It is shown that for these systems the Freudenthal construction based on cubic Jordan algebras naturally defines entanglement measures invariant under the group of stochastic local operations and classical communication (SLOCC). For this type of multipartite entanglement the SLOCC classes can be explicitly given. These results enable further explicit constructions of multiqubit entanglement measures for distinguishable constituents by embedding them into identical fermionic ones. We also prove that the Plucker relations for the embedding system provide a sufficient and necessary condition for the separability of the embedded one. We argue that this embedding procedure can be regarded as a convenient representation for quantum systems of particles which are really indistinguishable but for some reason they are not in the same state of some inner degree of freedom.

Asymptotic entanglement transformation between W and GHZ states

We investigate entanglement transformations with stochastic local operations and classical communication in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory. Results well-known in that field imply that GHZ states can be transformed into W states at rate 1 for any number of parties. As a generalization, we find that the asymptotic conversion rate from GHZ states to Dicke states is bounded as the number of subsystems increases and the number of excitations is fixed. By generalizing constructions of Coppersmith and Winograd and by using monotones introduced by Strassen, we also compute the conversion rate from W to GHZ states.We investigate entanglement transformations with stochastic local operations and classical communication in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory. Results well-known in that field imply that GHZ states can be transformed into W states at rate 1 for any number of parties. As a generalization, we find that the asymptotic conversion rate from GHZ states to Dicke states is bounded as the number of subsystems increases and the number of excitations is fixed. By generalizing constructions of Coppersmith and Winograd and by using monotones introduced by Strassen, we also compute the conversion rate from W to GHZ states.

The role of topology in quantum tomography

We investigate quantum tomography in scenarios where prior information restricts the state space to a smooth manifold of lower dimensionality. By considering stability we provide a general framework that relates the topology of the manifold to the minimal number of binary measurement settings that is necessary to discriminate any two states on the manifold. We apply these findings to cases where the subset of states under consideration is given by states with bounded rank, fixed spectrum, given unitary symmetry or taken from a unitary orbit. For all these cases we provide both upper and lower bounds on the minimal number of binary measurement settings necessary to discriminate any two states of these subsets.

Local unitary invariants for multipartite quantum systems

A method is presented to obtain local unitary invariants for multipartite quantum systems consisting of fermions or distinguishable particles. The invariants are organized into infinite families, in particular, the generalization to higher dimensional single particle Hilbert spaces is straightforward. Many well-known invariants and their generalizations are also included.

On the algebra of local unitary invariants of pure and mixed quantum states

We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we conjecture that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2m equals the number of conjugacy classes of index m subgroups in a free group on k-1 generators. Similarly, we conjecture that the inverse limit in the case of k-partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two conjectures are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m=2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.

Fault-ignorant quantum search

We investigate the problem of quantum searching on a noisy quantum computer. Taking a fault-ignorant approach, we analyze quantum algorithms that solve the task for various different noise strengths, which are possibly unknown beforehand. We prove lower bounds on the runtime of such algorithms and thereby find that the quadratic speedup is necessarily lost (in our noise models). However, for low but constant noise levels the algorithms we provide (based on Groverʼs algorithm) still outperform the best noiseless classical search algorithm.

Entanglement Distillation from Greenberger–Horne–Zeilinger Shares

We study the problem of converting a product of Greenberger–Horne–Zeilinger (GHZ) states shared by subsets of several parties in an arbitrary way into GHZ states shared by every party. Such a state can be described by a hypergraph on the parties as vertices and with each hyperedge corresponding to a GHZ state shared among the parties incident with it. Our result is that if SLOCC transformations are allowed, then the best asymptotic rate is the minimum of bipartite log-ranks of the initial state, which in turn equals the minimum cut of the hypergraph. This generalizes a result by Strassen on the asymptotic subrank of the matrix multiplication tensor.

Reconstructing quantum states from single-party information

The possible compatibility of density matrices for single-party subsystems is described by linear constraints on their respective spectra. Whenever some of those quantum marginal constraints are saturated, the total quantum state has a specific, simplified structure. We prove that these remarkable global implications of extremal local information are stable; i.e., they hold approximately for spectra close to the boundary of the allowed region. Application of this general result to fermionic quantum systems allows us to characterize natural extensions of the Hartree-Fock ansatz and to quantify their accuracy by resorting to one-particle information, only: The fraction of the correlation energy not recovered by such an ansatz can be estimated from above by a simple geometric quantity in the occupation number picture.