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).
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).
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.
This chapter is concerned with the measure of quantum states . This requires one to introduce the subtle notion of quantum measurement , an operation that has no counterpart in the classical domain. To this effect, we first need to develop some new tools, starting with Dirac notation , a formalism that is not only very elegant but is relatively simple to handle. The introduction of Dirac notation makes it possible to become familiar with the inner product for quantum states, as well as different properties for operators and states concerning projection , change of basis , unitary transformations , matrix elements , similarity transformations , eigenvalues and eigenstates , spectral decomposition and diagonal representation , matrix trace and density operator or matrix . The concept of density matrix makes it possible to provide a very first and brief hint of the analog of Shannon's entropy in the quantum world, referred to as von Neumann's entropy , to be further developed in Chapter 21. Once we have all the required tools, we can focus on quantum measurement and analyze three different types referred to as basis-state measurements , projection or von Neumann measurements , and POVM measurements. In particular, POVM measurements are shown to possess a remarkable property of unambiguous quantum state discrimination (UQSD), after which it is possible to derive “absolutely certain” information from unknown system states. The more complex case of quantum measurements in composite systems described by joint or tensor states is then considered.
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 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 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′=Trρ∑jNj†Nj∑jNjρNj†,Nj∈N where N1, N N 2,… is an arbitrary finite subset of operators in a subspace N N of ℬ B ( ℋ H ), the algebra of all operators in ℋ H . We shall assume that the noise is moderate in the sense that the dimension of N N is ‘small’ relative to the dimension of ℬ B ( ℋ H ). We call N N the noise space of the channel K K and any element of K K a noise or error operator. If |ψ〉 ∈ 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′=∑jNjψ2∑jNj⟨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 .
This paper develops an energy-domain formulation of time-independent non-relativistic quantum processes, including scattering, wave packets, transition amplitudes, and tunneling. A fixed-energy spectral component is written as Ψ E ( x,z )= ψ E ( x ) χ z ( E ) , while the complete state Ψ( x,z ) is obtained by summing or integrating over the spectral components. The energy derivative is defined on the spectral phase χ z ( E ) , with the energy-dependent spatial eigenfunction ψ E ( x ) retained as an element of the spectral basis. For a specified reference energy E ref , the spectral phase is χ z ( E )=exp[ −i ( E− E ref ) 2 ( z− z 0 )/ℏ ] . Fixed-energy reflection and transmission amplitudes, tunneling probabilities, probability currents, transition matrix elements, and selection rules retain their standard forms because this factor is a common global phase. For superpositions, however, the z-ordering is generated by ( H ^ x − E ref I ^ ) 2 and is generally not identical to the standard time evolution generated by H ^ x . Continuum states are treated with generalized normalization, channel labels, and the appropriate spectral measure. A right-moving Gaussian free-particle packet is worked out explicitly, yielding analytic narrow-band expressions for its centre and spatial width as functions of z . Transition probabilities are defined as Born-rule probabilities conditioned on a specified interaction event; transition rates, lifetimes, delay times, and tunneling times are not claimed to be reproduced without an operational clock. The paper therefore establishes the preservation of fixed-energy stationary observables and a consistent unitary spectral ordering for time-independent Hamiltonians, while clearly identifying the remaining limitations of the proposed dynamics.
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A classical state specifies the quantitative values of all the state variables relevant to a given system. In some situations, we can know the overall state of the system without being sure about the individual states of the constituent parts. On top of the problems inherent in picking out the crucial details of a system, there is another issue with trying to specify an initial state precisely: there is always some uncertainty connected with measurement. The quantum state of a system contains a series of complex numbers related to the probability that the system will collapse into a new state when a measurement takes place. Each of these new states represents a possible result of the measurement, which might be a path or a specific quantitative value of some physical variable.