In a Generalised Probabilistic Theory (GPT) equipped additionally with some extra geometric structure we define the morphophoric measurements as those for which the measurement map sending states to distributions of the measurement results is a similarity. In the quantum case, morphophoric measurements generalise the notion of a 2-design POVM, thus in particular that of a SIC-POVM. We show that the theory built on this class of measurements retains the chief features of the QBism approach to the basis of quantum mechanics. In particular, we demonstrate how to extend the primal equation ('Urgleichung') of QBism, designed for SIC-POVMs, to the morphophoric case of GPTs. In the latter setting, the equation takes a different, albeit more symmetric, form, but all the quantities that appear in it can be interpreted in probabilistic and operational terms, as in the original 'Urgleichung'.
We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case.
QBism is an interpretation of quantum theory which views quantum mechanics as standard probability theory supplemented with a few extra normative constraints. The fundamental gambit is to represent states and measurements, as well as time evolution, with respect to an informationally complete reference device. From this point of view, the Born rule appears as a coherence condition on probability assignments across several different experiments which manifests as a deformation of the law of total probability (LTP). In this work, we fully characterize those reference devices for which this deformation takes a "simplest possible" (term-wise affine) form. Working in the framework of generalized probability theories (GPTs), we show that, given any reference measurement, a set of post-measurement reference states can always be chosen to give its probability rule this very form. The essential condition is that the corresponding measure-and-prepare channel be depolarizing. We also relate our construction to Szymusiak and Słomczyński's recently introduced notion of morphophoricity and re-examine critically a matrix-norm-based measure of LTP deformation in light of our results. What stands out for the QBist project from this analysis is that it is not only the pure form of the Born rule that must be understood normatively, but the constants within it as well. It is they that carry the details of quantum theory.
We give an introduction to research associated with the generalised probabilistic theories framework, also known as the convex framework. States are real vectors representing lists of probabilities of measurement outcomes. Convex combinations of the vectors represent probabilistic combinations of different state preparations. Transformations are real matrices. Measurement outcomes are represented by functionals of the states, inner products of the state with a real vector, whose values are the probability of the measurement outcome in question. The framework generalises quantum theory. We describe the operational meaning of the framework, and how the concepts can be defined in terms of cones of states and measurement outcome vectors. We describe how the classical and quantum probability theories are represented in the framework. We describe Bell non-locality and the theory with super-quantum non-locality known as box world. We discuss generalised Hamiltonian mechanics in the discrete case and in continuous phase space, including the role of negativity of the phase space density in contextuality and tunnelling.
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in
Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such
n2 vectors in
Cn for any
n .But the existence is only proved when
n≤15 and
n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in
Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1
ρ∈Mn(C) is called a density matrix (or state) if
ρ≥0 and
′b(ρ)=1.For any density matrix
ρ , we can define a state
ρ^ by
ρ^(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t
(≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators
{Pi}i=1k⊂Mn(C) is called a positive operator valued measure (POVM) if
Pi≥0(1≤i≤k) and
∑i=1kP=I.If
P1 is a projection, then
{Pi}i=1k is called a projection valued measure
(PVM) .
QBism understands quantum mechanics to be probability theory supplemented by additional nonclassical coherence conditions. In this dissertation, we develop these nonclassical coherence conditions from first principles, emphasizing the role of a well chosen reference measurement. After treating standard probability on subjective Bayesian lines, we demonstrate an equivalence between the QBist approach and the existing framework of generalized probabilistic theories. We show that the fundamental nonclassical coherence relation may almost always be taken to be a gentle modification of the law of total probability, and give a coherentist account of when an experimental scenario has a classical explanation. Finally, we show that when the reference measurement is chosen to correspond to a complex projective 3-design, the shape of quantum state space is implicit in the probabilities which characterize the reference measurement itself. Thus coherence with this single reference measurement, properly understood, implies coherence with all of finite dimensional quantum mechanics. We then attempt a modest reconstruction of quantum theory along these lines, one practical consequence of which is a method for self-testing complex projective
t-designs for
t≥3 in a theory-agnostic way.
An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.