arXiv subjects

Ingemar Bengtsson

Publications and source records attributed to Ingemar Bengtsson.

At least 37 records · Page 2Linked to original sources

arXiv2017

Self-testing properties of Gisin's elegant Bell inequality

An experiment in which the Clauser-Horne-Shimony-Holt inequality is maximally violated is self-testing (i.e., it certifies in a device-independent way both the state and the measurements). We prove that an experiment maximally violating Gisin's elegant Bell inequality is not similarly self-testing. The reason can be traced back to the problem of distinguishing an operator from its complex conjugate. We provide a complete and explicit characterization of all scenarios in which the elegant Bell inequality is maximally violated. This enables us to see exactly how the problem plays out.

quant-ph↗

arXiv2017

Dimension towers of SICs. I. Aligned SICs and embedded tight frames

Algebraic number theory relates SIC-POVMs in dimension $d>3$ to those in dimension $d(d-2)$. We define a SIC in dimension $d(d-2)$ to be aligned to a SIC in dimension $d$ if and only if the squares of the overlap phases in dimension $d$ appear as a subset of the overlap phases in dimension $d(d-2)$ in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension $d$ there exists an aligned SIC in dimension $d(d-2)$. In all our examples the aligned SIC has lower dimensional equiangular tight frames embedded in it. If $d$ is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If $d-2$ is an odd prime number we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.

quant-ph↗

arXiv2017

On discrete structures in finite Hilbert spaces

We present a brief review of discrete structures in a finite Hilbert space, relevant for the theory of quantum information. Unitary operator bases, mutually unbiased bases, Clifford group and stabilizer states, discrete Wigner function, symmetric informationally complete measurements, projective and unitary t--designs are discussed. Some recent results in the field are covered and several important open questions are formulated. We advocate a geometric approach to the subject and emphasize numerous links to various mathematical problems.

quant-ph↗

arXiv2017

Cliffordtori and unbiased vectors

The existence problem for mutually unbiased bases is an unsolved problem in quantum information theory. A related question is whether every pair of bases admits vectors that are unbiased to both. Mathematically this translates to the question whether two Lagrangian Clifford tori intersect, and a body of results exists concerning it. These (deep!) results are however rather weak when viewed from the point of view of the first problem. We make a detailed study of how the intersections behave in the simplest non-trivial case, that of complex projective 2-space (the qutrit), for which the set of Clifford tori can be usefully parametrized by the unistochastic subset of Birkhoff's polytope. An interesting picture emerges. A foray into higher dimensions is included.

quant-ph↗

arXiv2017

The Number Behind the Simplest SIC-POVM

The simple concept of a SIC poses a very deep problem in algebraic number theory, as soon as the dimension of Hilbert space exceeds three. A detailed description of the simplest possible example is given.

quant-ph↗

arXiv2016

A brief introduction to multipartite entanglement

A concise introduction to quantum entanglement in multipartite systems is presented. We review entanglement of pure quantum states of three--partite systems analyzing the classes of GHZ and W states and discussing the monogamy relations. Special attention is paid to equivalence with respect to local unitaries and stochastic local operations, invariants along these orbits, momentum map and spectra of partial traces. We discuss absolutely maximally entangled states and their relation to quantum error correction codes. An important case of a large number of parties is also analysed and entanglement in spin systems is briefly reviewed.

quant-ph↗

arXiv2016

A Toy Penrose Inequality and its Proof

We formulate and prove a toy version of the Penrose inequality. The formulation mimics the original Penrose inequality in which the scenario is the following: A shell of null dust collapses in Minkowski space and a marginally trapped surface forms on it. Through a series of arguments relying on established assumptions, an inequality relating the area of this surface to the total energy of the shell is formulated. Then a further reformulation turns the inequality into a statement relating the area and the outer null expansion of a class of surfaces in Minkowski space itself. The inequality has been proven to hold true in many special cases, but there is no proof in general. In the toy version here presented, an analogous inequality in (2+1)-dimensional anti-de Sitter space turns out to hold true.

gr-qc↗

arXiv2016

Geometric Phases for Mixed States of the Kitaev Chain

The Berry phase has found applications in building topological order parameters for certain condensed matter systems. The question whether some geometric phase for mixed states can serve the same purpose has been raised, and proposals are on the table. We analyze the intricate behaviour of Uhlmann's geometric phase in the Kitaev chain at finite temperature, and then argue that it captures quite different physics from that intended. We also analyze the behaviour of a geometric phase introduced in the context of interferometry. For the Kitaev chain, this phase closely mirrors that of the Berry phase, and we argue that it merits further investigation.

quant-ph↗

arXiv2015

Thermodynamic Metrics and Black Hole Physics

We give a brief survey of thermodynamic metrics, in particular the Hessian of the entropy function, and how they apply to black hole thermodynamics. We then provide a detailed discussion of the Gibbs surface of Kerr black holes. In particular we analyze its global properties, and extend it to take the entropy of the inner horizon into account. A brief discussion of Kerr-Newman black holes is included.

gr-qc↗

arXiv2015

Black holes: Their large interiors

Christodoulou and Rovelli have remarked on the large interiors possessed by static black holes. We amplify their remarks, and extend them to the spinning case.

gr-qc↗

arXiv2014

States that are far from being stabilizer states

Stabilizer states are eigenvectors of maximal commuting sets of operators in a finite Heisenberg group. States that are far from being stabilizer states include magic states in quantum computation, MUB-balanced states, and SIC vectors. In prime dimensions the latter two fall under the umbrella of Minimum Uncertainty States (MUS) in the sense of Wootters and Sussman. We study the correlation between two ways in which the notion of "far from being a stabilizer state" can be quantified, and give detailed results for low dimensions. In dimension 7 we identify the MUB-balanced states as being antipodal to the SIC vectors within the set of MUS, in a sense that we make definite. In dimension 4 we show that the states that come closest to being MUS with respect to all the six stabilizer MUBs are the fiducial vectors for Alltop MUBs.

quant-ph↗

arXiv2014

Order 3 Symmetry in the Clifford Hierarchy

We investigate the action of the first three levels of the Clifford hierarchy on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop vectors splits into three Clifford orbits. These vectors form configurations with so-called Zauner subspaces, eigenspaces of order 3 elements of the Clifford group highly relevant to the SIC problem. We identify Alltop vectors as the magic states that appear in the context of fault-tolerant universal quantum computing, wherein the appearance of distinct Clifford orbits implies a surprising inequivalence between some magic states.

quant-ph↗

arXiv2014

Galois Unitaries, Mutually Unbiased Bases, and MUB-balanced states

A Galois unitary is a generalization of the notion of anti-unitary operators. They act only on those vectors in Hilbert space whose entries belong to some chosen number field. For Mutually Unbiased Bases the relevant number field is a cyclotomic field. By including Galois unitaries we are able to remove a mismatch between the finite projective group acting on the bases on the one hand, and the set of those permutations of the bases that can be implemented as transformations in Hilbert space on the other hand. In particular we show that there exist transformations that cycle through all the bases in every dimension which is an odd power of an odd prime. (For even primes unitary MUB-cyclers exist.) These transformations have eigenvectors, which are MUB-balanced states (i.e. rotationally symmetric states in the original terminology of Wootters and Sussman) if and only if d = 3 modulo 4. We conjecture that this construction yields all such states in odd prime power dimension.

quant-ph↗

arXiv2014

Classics Illustrated: Limits of Spacetimes

We carefully study the $e \rightarrow m$ and $e \rightarrow 0$ limits of the Reissner-Nordström spacetime using Geroch's definition of limits of spacetimes. This is implemented by embedding the one-parameter family of spacetimes in anti-de Sitter space, and as a result we obtain metrically correct Penrose diagrams. For $e \rightarrow m$ two distinct limits are studied.

gr-qc↗

arXiv2014

Four remarks on spin coherent states

We discuss how to recognize the constellations seen in the Majorana representation of quantum states. Then we give explicit formulas for the metric and symplectic form on SU(2) orbits containing general number states. Their metric and symplectic areas differ unless the states are coherent. Finally we discuss some patterns that arise from the Lieb-Solovej map, and for dimensions up to nine we find the location of those states that maximize the Wehrl-Lieb entropy.

quant-ph↗

arXiv2013

Linear Dependencies in Weyl-Heisenberg Orbits

Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl-Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.

math-ph↗

arXiv2013

Trapped surfaces in Oppenheimer-Snyder black holes

The Oppenheimer-Snyder solution models a homogeneous round dust cloud collapsing to a black hole. Inside its event horizon there is a region through which trapped surfaces pass. We try to determine exactly where the boundary of this region meets the centre of the cloud. We present explicit examples of the relevant trapped (topological) spheres; they extend into the exterior vacuum region, and are carefully matched at the junction between the cloud and the vacuum.

gr-qc↗

arXiv2012

Families of complex Hadamard matrices

What is the dimension of a smooth family of complex Hadamard matrices including the Fourier matrix? We address this problem with a power series expansion. Studying all dimensions up to 100 we find that the first order result is misleading unless the dimension is 6, or a power of a prime. In general the answer depends critically on the prime number decomposition of the dimension. Our results suggest that a general theory is possible. We discuss the case of dimension 12 in detail, and argue that the solution consists of two 13-dimensional families intersecting in a previously known 9-dimensional family. A precise conjecture for all dimensions equal to a prime times another prime squared is formulated.

math-ph↗