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InTech eBooks

A Basis for Statistical Theory and Quantum Theory

S. Inge

Published Apr 3, 2013• 1 citation
Publisher

Abstract

Here Tr is the trace operator. The discussion in [1] goes on by arguing that R must be Hermitian (have real eigenvalues) and that ρ ought to be positive with trace 1. An important special case is when ρ is one-dimensional: ρ = |ψ〉〈ψ| for a vector |ψ〉. Then the state is pure, and is equivalently specified by the vector |ψ〉. In general the formula (1) is a consequence of Born’s formula: The probability of observing a pure state |φ〉 when the system is prepared in a pure state |ψ〉 is given by |〈φ|ψ〉|2.

DOI
10.5772/53702

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Here Tr is the trace operator. The discussion in [1] goes on by arguing that R must be Hermitian (have real eigenvalues) and that ρ ought to be positive with trace 1. An important special case is when ρ is one-dimensional: ρ = |ψ〉〈ψ| for a vector |ψ〉. Then the state is pure, and is equivalently specified by the vector |ψ〉. In general the formula (1) is a consequence of Born’s formula: The probability of observing a pure state |φ〉 when the system is prepared in a pure state |ψ〉 is given by |〈φ|ψ〉|2.
The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as tr(A):=∑j=1n⟨βj∣A∣βj⟩tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } tr(A):=j=1∑n​⟨βj​∣A∣βj​⟩ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states ρ=∑j=1nPj∣⟨aj⟨aj∣\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|ρ=j=1∑n​Pj∣⟨aj⟨aj∣ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and pj∈R,pj0,∑j=1npj=1pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1pj∈R,pj​0,j=1∑n​pj​=1
Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state |ψ〉 evolves in time. Using this we can show that the expectation value of an operator evolves as 〈A(t)〉 = 〈ψ(t)| ˆA|ψ(t)〉 (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator ρˆ(t) = |ψ(t)〉〈ψ(t)| (6.2) taken as the outer product of the state vector with itself. From this definition we can write ρˆ = ∑ mn c∗ncm |m〉〈n| (6.3) = ∑ mn ρmn|m〉〈n| (6.4) where the ρmn are the density matrix elements. Expectation values of operator are then given by the trace 〈A(t)〉 = ∑ mn Amnρmn = Tr [ ˆAρ(t)] (6.5) where Tr [ ˆAρ(t)] denotes the trace operation: Tr [ ˆAρ(t)] = ∑ nn′ If ρ is diagonal, then Tr [ ˆAρ(t)] = ∑ Annρnn = ∑ 〈n|A|n〉Pn where Pn = ρnn is the statistical probability of finding the system in state n. These statistical weights must be such that Pn ≤ 1 and∑ Pn = 1 Hence, we conclude that knowing ρ we can compute the statistical average of an operator A.
We look at the problem of communication of states of a finite level quantum system through a channel disturbed by noise. Such a quantum system is described by a finite dimensional complex Hilbert space ℋ H \mathcal{H} H and its states are density operators, in other words, positive hermitian operators of unit trace. We assume that any such state ρ, when transmitted through the channel K K \mathcal{K} K under consideration is received as an output state ρ′ of the form (5.1.1) ρ'= ∑ j N j ρ N j † T r ρ ∑ j N j † N j , N j ∈N ρ′=∑jNjρNj†Trρ∑jNj†Nj,Nj∈N\rho ' = \frac{{\sum\nolimits_j {{N_j}_\rho N_j^\dag } }}{{T{r_\rho }\sum\nolimits_j {N_j^\dag {N_j}} }},{N_j} \in \mathcal{N}ρ′=Trρ​∑j​Nj†​Nj​∑j​Nj​ρ​Nj†​​,Nj​∈N where N1, N N \mathcal{N} N 2,… is an arbitrary finite subset of operators in a subspace N N \mathcal{N} N of ℬ B \mathcal{B} B ( ℋ H \mathcal{H} H ), the algebra of all operators in ℋ H \mathcal{H} H . We shall assume that the noise is moderate in the sense that the dimension of N N \mathcal{N} N is ‘small’ relative to the dimension of ℬ B \mathcal{B} B ( ℋ H \mathcal{H} H ). We call N N \mathcal{N} N the noise space of the channel K K \mathcal{K} K and any element of K K \mathcal{K} K a noise or error operator. If |ψ〉 ∈ ℋ H \mathcal{H} H is a unit vector and ρ = |ψ〉 〈ψ|, the corresponding pure state density operator then the output state ρ′ of (5.1.1) assumes the form (5.1.2) ρ'= ∑ j N j 〈ψ|〈ψ| N j † ∑ j ‖ N j ψ ‖ 2 . ρ′=∑jNj⟨ψ∣⟨ψ∣Nj†∑jNjψ2.\rho ' = \frac{{\sum\nolimits_j {{N_j}\langle \psi |\langle \psi |N_j^\dag } }}{{\sum\nolimits_j {{N_j}\psi {^2}} }}.ρ′=∑j​Nj​ψ2∑j​Nj​⟨ψ∣⟨ψ∣Nj†​​. It is to be noted that this output state need not be pure. If the same input state ρ is transmitted repeatedly the noise operators {N j } in (5.1.1) can differ for different transmissions and the output states can be different. However, the noise operators come from the same noise space N N \mathcal{N} N .
The density matrix ρ is defined and its basic properties are derived in detail: ρ is a Hermitian non-negative operator with unit trace, its eigenvalues are non-negative and lie in the closed interval [0, 1]; if ρ is a projector, it projects onto a one-dimensional sub-space, the trace of ρ 2 is ≤ 1, the equality holding only if ρ is a projector. A necessary and sufficient condition for ρ to be a projector is that all vectors in its definition are identical up to a phase. Pure cases are sets that can be described by a state vector, unlike mixtures . The density matrix of a mixture is not a projection operator, and this may be viewed as its distinguishing feature with respect to pure cases. The mean value of an observable can be reexpressed with the help of the density matrix, and a Schrödinger equation for density matrices holds. Last, but not least, pure cases cannot unitarily evolve into mixtures . The applications of the formalism presented in this chapter deal with orientation of spin-½ particles, polarization of pairs of photons, thermal-equilibrium states. In the second part, quantum entanglement is presented, with emphasis on the existence of state vectors for entangled pairs of particles that belong to the tensor product of Hilbert spaces H 1 ⊗ H 2 , but are not of the simple product form ψ 1 ⊗ ψ 2 . The only correct interpretation of such states has a statistical nature, associated to series of repeated measurements performed by two observers on the constituents of the entangled pair. Hidden-variable theories, with the associated Bell inequalities, are introduced, and a way of using entangled quantum states is described which makes it possible to transfer the polarization state of a photon to another photon. This property provides the foundation of the modern investigations of quantum teleportation.
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.
In quantum mechanics, a physical observable is described by a (linear) operator Q on a Hilbert space H and a (pure) state is described by a unit vector Ψ in H. The expectation value (the average of measured values for a large number of repeated measurements) of the observable Q in the State Ψ is given by 1.1.1 ψ( Q )=( ψQψ ) ]] <![CDATA[ψ(Q)=(ψ,Qψ)\psi \left( Q \right) = \left( {\psi ,Q\psi } \right)ψ(Q)=(ψ,Qψ) More general states called mixtures are described by a positive operator ρ of trace 1, called a density matrix, for which the expectation value is 1.1.2 ψ( Q )=trρQ ]] <![CDATA[ψ(Q)=trρQ\psi \left( Q \right) = tr\rho Qψ(Q)=trρQ and is at most countable sum of states of the form (1.1.1). ((1.1.1) is a special case where ρ is a one-dimensional projection operator.)