The notion of quantum group replaces ordinary matrices with matrices with non-commutative elements. In TGD framework I have proposed that the notion should relate to the inclusions of von Neumann algebras allowing one to describe mathematically the notion of finite measurement resolution. In this article I will consider the notion of quantum matrix inspired by the recent view about quantum TGD relying on the notion of finite measurement resolution. Complex matrix elements are replaced with operators expressible as products of non-negative Hermitian operators and unitary operators analogous to the products of modulus and phase as a representation for complex numbers. The condition that determinant and sub-determinants exist is crucial for the well-defined eigenvalue problem in the generalized sense. Strong/weak permutation symmetry of determinant requires its invariance under permutations of rows and/or columns. Weak permutation symmetry means development of determinant with respect to a fixed row or column and does not pose additional conditions. For weak permutation symmetry the permutation of rows/columns would however has a natural interpretation as braiding for the Hermitian operators defined by the moduli of operator valued matrix elements and here quantum group structure emerges. The commutativity of all sub-determinants is essential for the replacement of eigenvalues with eigenvalue spectra of Hermitian operators and sub-determinants define mutually commuting set of operators. Quantum matrices define a more general structure than quantum group but provide a concrete representation and interpretation for quantum group in terms of finite measurement resolution, in particular when q is a root of unity. One can also understand the fractal structure of inclusion sequences of hyper-finite factors resulting by replacing operators appearing as matrix elements with quantum matrices.
The notion of quantum group replaces ordinary matrices with matrices with non-commutative elements. In TGD framework I have proposed that the notion should relate to the inclusions of von Neumann algebras allowing one to describe mathematically the notion of finite measurement resolution. In this article I will consider the notion of quantum matrix inspired by the recent view about quantum TGD relying on the notion of finite measurement resolution. Complex matrix elements are replaced with operators expressible as products of non-negative Hermitian operators and unitary operators analogous to the products of modulus and phase as a representation for complex numbers. The condition that determinant and sub-determinants exist is crucial for the well-defined eigenvalue problem in the generalized sense. Strong/weak permutation symmetry of determinant requires its invariance under permutations of rows and/or columns. Weak permutation symmetry means development of determinant with respect to a fixed row or column and does not pose additional conditions. For weak permutation symmetry the permutation of rows/columns would however has a natural interpretation as braiding for the Hermitian operators defined by the moduli of operator valued matrix elements and here quantum group structure emerges. The commutativity of all sub-determinants is essential for the replacement of eigenvalues with eigenvalue spectra of Hermitian operators and sub-determinants define mutually commuting set of operators. Quantum matrices define a more general structure than quantum group but provide a concrete representation and interpretation for quantum group in terms of finite measurement resolution, in particular when q is a root of unity. One can also understand the fractal structure of inclusion sequences of hyper-finite factors resulting by replacing operators appearing as matrix elements with quantum matrices.
Stochastic matrices and positive maps in matrix algebras have proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states, respectively. Here we introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with ‘diamond’ sets of stochastic matrices and pseudo-positive maps are dealt with.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.
Abstract A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators summing to identity. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. Discrete transformations in the set of quantum measurements can be described by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares) and study their properties. Analyzing the dynamics they induce we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set.
Negentropic entanglement for which number theoretic entropy characterized by p-adic prime is negative so that entanglement carries information, is in key role in TGD inspired theory of consciousness and quantum biology.
The key feature of negentropic entanglement is that density matrix is proportional to unit matrix so that assumption that state function reduction corresponds to measurement of density matrix does not imply state function reduction to one-dimensional sub-space. This special kind of degenerate density matrix emerges naturally for hierarchy heff=nh interpreted in terms of a hierarchy of dark matter phases. I have already earlier considered explicit realizations of negentropic entanglement assuming that E is invariant under group of unitary or orthogonal transformations (also subgroups of unitary group can be considered - say symplectic group). One can however consider much more general options and this leads to a connection with topological quantum computation (TQC). Entanglement matrix E equal to 1/n1/2 factor times unitary matrix U (as a special case orthogonal matrix O) defines a density matrix given by ρ=UU†/n= Idn/n, which is group invariant. One has NE respected by state function reduction if NMP is assumed. This would give huge number of negentropically entangled states providing a representation for some unitary group or its subgroup (such as symplectic group). In principle any unitary representation of any Lie group would allow representation in terms of NE. In principle any unitary representation of any Lie group would allow a representation in terms of NE. In as generalized number theory vision, a natural condition is that matrix elements of E belong to algebraic extension of p-adic numbers used so that discreted algebraic subgroups of unitary or orthogonal group are selected. This realizes evolutionary hierarchy as a hierarchy of p-adic number fields and their algebraic extensions, and one can imagine that evolution of cognition proceeds by generation of negentropically entangled systems with increasing algebraic dimensions and increasing dimension reflecting itself as an increase of largest prime power dividing n and defining p-adic prime in question. One fascinating implication is ability of TGD Universe to emulate itself like Turing machine: unitary S-matrix codes for scattering amplitudes and therefore for and negentropically entangled subsystem could represent sub-matrix for S-matrix as rules representing the laws of physics in approximation that world corresponds to n-dimension Hilbert space. Also limit n→ ∞ makes sense, especially so in p-adic context where real infinity can correspond to finite number in sense of p-adic norm. Here also dimensions n given as products of powers of infinite primes can be formally considered.
Abstract Quantum permutations arise in many aspects of modern “quantum mathematics”. However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.
We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If α and β are entities, we define what it means for α to be a part of β. This relationship is essentially equivalent to α being a function of β and in this case β can be employed to measure α. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map α^ which takes an entity of a certain type to one of a lower type. For example, if I is an instrument, then I^ is the unique observable measured by I. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels.