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A density operator that is a rank-1 projection is known as a pure quantum state, and all quantum states that are not pure are designated mixed. Pure states are also known as wavefunctions . Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system (i.e., P ( x ) = 1 {\displaystyle P(x)=1 ...
In quantum mechanics, given a projection-valued measure of a measurable space to the space of continuous endomorphisms upon a Hilbert space , the projective space P ( H ) {\displaystyle \mathbf {P} (H)} of the Hilbert space H {\displaystyle H} is interpreted as the set of possible ( normalizable ) states φ {\displaystyle \varphi } of a quantum ...
In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum ...
In the context of quantum information theory, the operators {V i} are called the Kraus operators (after Karl Kraus) of Φ. Notice, given a completely positive Φ, its Kraus operators need not be unique. For example, any "square root" factorization of the Choi matrix C Φ = B ∗ B gives a set of Kraus operators. Let
Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below). An operator in n-dimensional space can be written:
When spinors are used to describe the quantum states, the three spin operators (S x, S y, S z,) can be described by 2 × 2 matrices called the Pauli matrices whose eigenvalues are ± ħ / 2 . For example, the spin projection operator S z affects a measurement of the spin in the z direction.
By means of a projection operator, the dynamics is split into a slow, collective part (relevant part) and a rapidly fluctuating irrelevant part. The goal is to develop dynamical equations for the collective part. The Nakajima-Zwanzig (NZ) generalized master equation is a formally exact approach for simulating quantum dynamics in condensed phases.
Q|SI> is a platform embedded in .Net language supporting quantum programming in a quantum extension of while-language. [ 47 ] [ 56 ] This platform includes a compiler of the quantum while-language [ 57 ] and a chain of tools for the simulation of quantum computation, optimisation of quantum circuits, termination analysis of quantum programs ...