# Quantum bit temporal correlations

**The analogous discussion of the above using QM has profound differences involving how we model reality and time. The QM version of the above classical experiment replaces the two classical bit states of each bit in the ensemble by the two basis vectors of a two-dimensional Hilbert space known as a ***quantum bit,*** or ***qbit,*** and then tensors them all together. Therefore we are dealing with a spatial ensemble of ***N*** qbits, technically described as a ***quantum register Q*^{N},

a Hilbert space defined as the tensor product of all of the two-dimensional qbit Hilbert spaces. Such a register has dimension 2^{N}, which means that for the sort of *N* encountered with macroscopic scale SUOs, we would be faced with an intractable problem. An ensemble of even a hundred qbits would be far too complicated for us to deal with in detail, being a complex vector space of 2^{100} = 1,267,650,600,228,229,401,496,703,205,376 dimensions. Macroscopic SUOs could easily require a quantum register of 10^{20} qbits,

We will assume for simplicity that the qbits in the quantum register do not interact with each other, but can interact with other elements of their environment. This assumption means that we can meaningfully discuss the evolution of a single bit and then take what amounts to a temporal ensemble average.^{82} This is similar to the approach taken in the theory of nuclear magnetic resonance (NMR) [Abragam, 1961]. In NMR, the individual nuclear spins in a sample behave analogously to the qbits we are discussing, each nuclear spin interacting with its local environment only. This consists of externally imposed magnetic fields plus local spin-coupling to neighbouring spins. Provided the number of such neighbours is not excessive, then it can be reasonably assumed that we are not dealing with an intractable quantum register. The irreversible environmental effects acting to dephase the Larmor rotations of individual spins can then be parametrized in terms of various relaxation times [Jaroszkiewicz & Strange, 1985]. These give information about that environment that can then be analysed to give visual images of that environment, the basis of magnetic imaging in medicine [Stehling *et al.,* 1987].

The theory of measurement and observation in QM is still controversial. We shall discuss temporal correlations from the perspective of the so-called *strong *measurement assumption. In a strong measurement of an observable A, a previously prepared quantum state Ф collapses to one of the eigenstates | *a)* of the observable and the observer registers a measured value *X _{a},* the corresponding eigenvalue of that operator. Specifically, according to the von Neumann projection postulate [von Neumann, 1955; Luders, 1951], the initial state reduces to

We shall follow the evolution of a typical single qbit in a non-interacting ensemble from initial time t_{0} to intermediate time t_{1} and then to final time t_{2}. Without loss of generality, suppose the initial qbit state | Ф, t_{0}) is a normalized pure state given by

^{82} An average over many repetitions over time of the same experiment: this requires only one copy of the SUO, but it is used repeatedly.

where | *ty _{up}* |

^{2}+|

*^*|

_{dowm}^{2}= 1. If a strong measurement of the

*up-down*observable

*Q*is made just after time

*t*the result is a mixed state given by the density matrix

_{0},^{83}

We shall discuss the correlation C_{01} of two measurements of Q, at time t_{0} and time t_{1}. There are several ways to calculate C_{01}. We shall use the density matrix approach in the Heisenberg picture, discussed in Chapter 24. First, we need to specify the quantum evolution operator *U** _{10}* taking states from time t

_{0}time t

_{1}. The most general unitary operator up to an arbitrary phase is of the form

where | *a** _{1}*1

^{2}+ |

*b*

*1*

_{1}^{2}= 1. Then the correlation C

_{01}is given by [Wang, 2002] where

We find

We note this is independent of the initial state of the qbit.

Now consider a further evolution from time *t*1 to time *t*2 with evolution operator

where | *a** _{2}*1

^{2}+ |

*b*

*1*

_{2}^{2}= 1. Then C

_{12}= 2| a

_{2}1

^{2}- 1.

According to *QM* principles, evolution from t_{0} to t_{2} is given by the evolution operator *U** _{20}* =

*U*

_{21}*U*

_{01}. We find

from which we deduce

^{83} This density matrix is the observer’s quantum theory *prediction,* at initial time t_{0}, of the quantum state of the SUO as it *should* be just after time t_{0}.