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 QN,
a Hilbert space defined as the tensor product of all of the two-dimensional qbit Hilbert spaces. Such a register has dimension 2N, 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 2100 = 1,267,650,600,228,229,401,496,703,205,376 dimensions. Macroscopic SUOs could easily require a quantum register of 1020 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 Xa, 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 t0 to intermediate time t1 and then to final time t2. Without loss of generality, suppose the initial qbit state | Ф, t0) 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 | tyup |2 +| ^dowm |2 = 1. If a strong measurement of the up-down observable Q is made just after time t0, the result is a mixed state given by the density matrix83
We shall discuss the correlation C01 of two measurements of Q, at time t0 and time t1. There are several ways to calculate C01. We shall use the density matrix approach in the Heisenberg picture, discussed in Chapter 24. First, we need to specify the quantum evolution operator U10 taking states from time t0 time t1. The most general unitary operator up to an arbitrary phase is of the form
where | a112 + | b112 = 1. Then the correlation C01 is given by [Wang, 2002] where
We note this is independent of the initial state of the qbit.
Now consider a further evolution from time t1 to time t2 with evolution operator
where | a212 + | b212 = 1. Then C12 = 2| a212 - 1.
According to QM principles, evolution from t0 to t2 is given by the evolution operator U20 = U21U01. We find
from which we deduce
83 This density matrix is the observer’s quantum theory prediction, at initial time t0, of the quantum state of the SUO as it should be just after time t0.