Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
Quantum randomness evidently transcends the classical framework of random variables defined on a single comprehensive Kolmogorov probability space. One prominent example is the quantum double-slit experiment owing to Feynman (Feynman 1951 In Second Berkeley Symposium on Mathematical Statistics and Probability (ed. J. Neyman), pp. 533-541 (doi:10.1525/9780520411586-039)). A related non-quantum example, inspired by Boole (Boole 1862 Phil. Trans. R. Soc. Lond. 152, 225-252 (doi:10.1098/rstl.1862.0015)) and Vorob'ev (Vorob'ev 1962 Theory Probab. Appl.7, 147-163 (doi:10.1137/1107014)), has three two-valued random variables X, Y and Z, where the pairs X, Y and X, Z are perfectly correlated, yet Y, Z are perfectly anti-correlated. Such examples can be accommodated using a 'multi-measurable' space with several different sigma-algebras of measurable events. This concept, owing to Vorob'ev (Vorob'ev 1962 Theory Probab. Appl.7, 147-163 (doi:10.1137/1107014)), allows construction of (i) a measurable metaspace whose elements combine a point in the original sample space with a variable 'contextual' Boolean algebra; (ii) a parametric family of 'probability metaspaces', each of which is a Kolmogorov probability space that represents a two-stage stochastic process where a random choice from the original sample space is preceded by the random choice of a contextual Boolean algebra in the multi-measurable space. Subsequent work will explore how quantum experimental results can be described using a quantum measurement tree with one or more preparation nodes, where an experimental configuration is determined that governs the probability distribution of relevant quantum observables. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
The nature of measurements, and their place within quantum theory, has engaged physicists and philosophers for generations [Bra92; Sch03]. Much of that interest centered on variables such as position and momentum of free particles, whose values form a continuum. The present monograph deals with discrete quantum states; the measurements are those required to specify as completely as possible a particular discrete quantum state Ψ or, more generally, a density matrix ρ defined within a finite-dimensional Hilbert space. General remarks General system . At the outset we assume that the possible quantum states are a small number – the N essential states used in formulating the time-dependent Schrödinger equation or specifying the dimensions of the density matrix. To completely characterize a density matrix for such a system we require the N 2 elements. Of these the N diagonal elements are real valued, while the off-diagonal elements of the upper right side are complex conjugates of those on the lower left. Thus with allowance for the requirement of unit trace a total of N 2 – 1 real numbers suffice to completely specify the density matrix. These values must be consistent with the constraints discussed in Sec. 16.6.3. Pure state . If it is known that the system is in a pure state, we require the magnitude and phase of N probability amplitudes. These are constrained by normalization, and so only 2 N –1 real numbers are needed. Out of these 2 N – 1 parameters the overall phase of the statevector is usually not of interest (but see Chap. 20).
Abstract In the literature, there is a version of ‘quantum set theory’ studied under the name ortho-sets (also known as orthogonality spaces). An ortho-set is a set X equipped with a so-called q-distinctness relation ≠ q . In this article, we will introduce and study a version of ‘non-commutative probability measures’ (called q-states) and two versions of ‘non-commutative function algebras’ on ortho-sets. More precisely, given a Hilbert space H , a B ( H ) + -valued q-state on an ortho-set ( X , ≠ q ) is a map α : X → B ( H ) + such that for every ‘maximal pairwise q-distinct subset’ D ⊆ X , the weak ∗ -sum ∑ x ∈ D α ( x ) converges to the identity operator on H . Furthermore, we define an ortho-representation of ( X , ≠ q ) to be a map µ from X to the ortho-lattice of projections on H such that the images of two q-distinct elements of X are orthogonal, and the supremum ⋁ x ∈ E μ ( x ) for every ‘ortho-dense subset’ E ⊆ X is the identity operator on H <
Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
Needless to emphasize, the home of these quantum states and observables are the linear vector spaces or alternatively the Hilbert spaces. This chapter has crucial recipes relating to operations like inverse of matrices, tridiagonalization, QR decomposition, Gram-Schmidt orthogonalization, a wide spectrum of functions and powers of matrices, trace norm, Hilbert-Schmidt norm, singular value decomposition and so on. The recipies in this chapter can be not only used in quantum information theory but it can also be used as standalone recipies for any person working in linear algebra as well.