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In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, Naimark's theorem says that if {} = is a POVM acting on a Hilbert space of dimension , then there exists a PVM {} = acting on a Hilbert space ′ of dimension ′ and an isometry: ′ such that for all ,
The Hilbertian tensor product of H 1 and H 2, sometimes denoted by H 1 ^ H 2, is the Hilbert space obtained by completing H 1 ⊗ H 2 for the metric associated to this inner product. [87] An example is provided by the Hilbert space L 2 ([0, 1]).
In quantum field theory, it is expected that the Hilbert space is also the space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined. However, it is often hard to define a concrete Borel measure on the classical configuration space, since the integral theory on infinite ...
Tensor products of Hilbert spaces arise often in quantum mechanics. If some particle is described by the Hilbert space H 1 , {\displaystyle H_{1},} and another particle is described by H 2 , {\displaystyle H_{2},} then the system consisting of both particles is described by the tensor product of H 1 {\displaystyle H_{1}} and H 2 ...
The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures.
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent ...
In quantum mechanics a state space is a separable complex Hilbert space.The dimension of this Hilbert space depends on the system we choose to describe. [1] [2] The different states that could come out of any particular measurement form an orthonormal basis, so any state vector in the state space can be written as a linear combination of these basis vectors.
The Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space () = = = (()) (())Here , the complex scalars, consists of the states corresponding to no particles, the states of one particle, () the states of two identical particles etc.