A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite, and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.     

Geometry of skew information-based quantum coherence

We study the skew information-based coherence of quantum states and derive explicit formulas for Werner states and isotropic states in a set of autotensors of mutually unbiased bases (MUBs). We also give surfaces of skew information-based coherence for Bell-diagonal states and a special class of X states in both computational basis and in MUBs. Moreover, we depict the surfaces of the skew information-based coherence for Bell-diagonal states under various types of local nondissipative quantum channels. The results show similar as well as different features compared with relative entropy of coherence and l₁ norm of coherence.     

Uncertainty relations for quantum coherence with respect to mutually unbiased bases

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.     

Construction of a Family of Maximally Entangled Bases in ℂd ⊗ ℂd′

In this paper, we present a new method for the construction of maximally entangled states in Cd⊗Cd′ when d′≥2d. A systematic way of constructing a set of maximally entangled bases (MEBs) in Cd⊗Cd′ was established. Both cases when d′ is divisible by d and not divisible by d are discussed. We give two examples of maximally entangled bases in C2⊗C4, which are mutually unbiased bases. Finally, we found a new example of an unextendible maximally entangled basis (UMEB) in C2⊗C5.     

Realization of Bipartite Positive-Operator-Value Measurements of Two-Photon Polarization States

In this letter, we propose a scheme of a special quantum optical Fredkin gate assisted by optical manip- ulations and postselection from the coincidence measurements, and then modify it with cross-Kerr nonlinearities to be suitable for the realization of all possible positive operator-valued measurements of bipartite polarization states. This scheme is feasible in the lab with the current experimental technology.     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Mutually unbiased projectors and duality between lines and bases in finite quantum systems

Quantum systems with variables in the ring Z(d)$\mathbb{Z}\left(d\right)$ are considered, and the concepts of weak mutually unbiased bases and mutually unbiased projectors are discussed. The lines through the origin in the Z(d)×Z(d)$\mathbb{Z}\left(d\right)\times \mathbb{Z}\left(d\right)$ phase space, are classified into maximal lines (sets of d$d$ points), and sublines (sets of d_i$d_i$ points where d_i|d$d_i|d$). The sublines are intersections of maximal lines. It is shown that there exists a duality between the properties of lines (resp., sublines), and the properties of weak mutually unbiased bases (resp., mutually unbiased projectors).     

Quantum Isometries of the Finite Noncommutative Geometry of the Standard Model

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M × F, where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.     

Quantum Logic and Non-Commutative Geometry

We propose a general scheme for the “logic” of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non-Commutative Geometry. It involves Baire^*-algebras, the non-commutative version of measurable functions, arising as envelope of the C ^*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of physical states in the standard (W ^*-)algebraic approach to classical mechanics.     

Information Between Quantum Systems via POVMs

The concepts of conditional entropy and information between subsystems of a composite quantum system are generalized to include arbitrary indirect measurements (POVMs). Some properties of those quantities differ from those of their classical counterparts; certain equalities and inequalities of classical information theory may be violated. PACS: 03.67.-a.     

Supersymmetric Quantum Theory and Non-Commutative Geometry

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-) K?hler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail. Received: 1 April 1997 / Accepted: 24 November 1998     

Quantum Measurements,Energy Conservation and Quantum Clocks

A spin chain extending from Alice to Bob with nearest neighbors interactions, initially in its ground state, is considered. Assuming that Bob measures the last spin of the chain, the energy of the spin chain has to increase, at least on average, due to the measurement disturbance. Presumably, the energy is provided by Bob's measurement apparatus. Assuming that, simultaneously to Bob's measurement, Alice measures the first spin, it is shown that either energy is not conserved, – implausible – or the projection postulate doesn't apply, and that there is signalling. An explicit measurement model shows that energy is conserved (as expected), but that the spin chain energy increase is not provided by the measurement apparatus(es), that the projection postulate is not always valid – illustrating the Wigner–Araki–Yanase (WAY) theorem – and that there is signalling, indeed. The signalling is due to the non‐local interaction Hamiltonian. This raises the question of whether a suitable quantum‐information‐inspired model of such non‐local Hamiltonians can be developed.     

Finite State and Finite Stop Quantum Languages

We propose the concept of finite stop quantum automata (ftqa) based on Hilbert space and compare it with the finite state quantum automata (fsqa) proposed by Moore and Crutchfield (Theoretical Computer Science 237(1–2), 2000, 275–306). The languages accepted by fsqa form a proper subset of the languages accepted by ftqa. In addition, the fsqa form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. We introduce complex-valued acceptance degrees and two types of finite stop quantum automata based on them: the invariant ftqa (icftq) and the variant ftqa (vcftq). The languages accepted by icftq form a proper subset of the languages accepted by vcftq. In addition, the icftq form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. In this way, we establish two proper inclusion relations (fsqa) ⊂ (ftqa) and (icftq) ⊂ (vcftq), where the symbol means languages, and two infinite language hierarchies (fsqa) ⊂ (fsqa), (icftq) (icftq).