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Part 3 of 3 Part Series

The Essence of Quantum Computing
Part 3 of 3 Part Series

The measurement device is modeled as follows. Let Δtm be the interval during which the measuring device measures, where Δtm is only assumed to be orders of magnitude greater than Planck time. During Δtm, at some random instant, the device measures |ψ〉. This avoids temporal bias. Thus, the source of indeterminism built in postulate 3 is posited as the classical measuring device’s inability to measure with temporal precision. The measurement concurrently induces the state vector to collapse to the measured state.

If the measurement basis differs from {|0〉, |1〉}, say, it is {|x〉, |y〉} obtained by rotating the basis {|0〉, |1〉} anti-clockwise by the angle θ, then |0〉 = cos θ|x〉 – sin θ|y〉 and |1〉 = sin θ|x〉 + cos θ|y〉 and conversely
|x〉 = cos θ|0〉 – sin θ|1〉 and |y〉 = sin θ|0〉 + cos θ|1〉. Figure 13.2 shows how the vectors |0〉, |1〉 will be observed by a measuring device in the basis {|x〉, |y〉} and the probabilities with which it will measure |x〉 or |y〉. Choosing a basis different from {|0〉, |1〉} means changing the values of t0 and t1 to tx and ty and correspondingly re-labeling the eigenstates to |x〉 and |y〉.

Thus, we can easily verify that

eq6

Further, Tx and Ty corresponding to the time durations the system will be in state |x〉 and |y〉, respectively, with respect to Tc in {|0〉, |1〉} basis is given by

eq7

img1

Figure 13.2 Projection of a single particle system to {|0〉, |1〉} and {|x〉, |y〉} bases.

Thus, if the measurement basis is {|x〉, |y〉}, the system, when measured, will randomly collapse to |x〉 or |y〉 with probability Tx/Tc = |α cos θ + β sin θ|2 or Ty/Tc = |β cos θα sin θ|2, respectively.

Finally, the posited model of entanglement between qubits requires that any unitary operation that causes entanglement, say, between two qubits, also synchronizes their sub-Planck level oscillations. This is shown in Figure 13.3 for the two-qubit Bell states,

eq8

img3

Figure 13.3 Two-particle entangled systems;|ψ1〉 (left), |ψ2〉 (right).

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