13.3: Model 2- Superposition of States (the Qubit)

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Page ID: 50238

Paul Penfield, Jr.
Massachusetts Institute of Technology via MIT OpenCourseWare

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The second model makes use of the fact that the states in quantum mechanics can be expressed in terms of wave functions which obey the Schrödinger equation. Since the Schrödinger equation is linear, any linear combination of wave functions that obey it also obeys it. Thus, if we associate the logical value 0 with the wave function \(\psi_0\) and the logical value 1 with the wave function \(\psi_1\) then any linear combination of the form

\(\psi = \alpha \psi_0 + \beta \psi_1 \tag{13.1}\)

where \(\alpha\) and \(\beta\) are complex constants with \(| \alpha |^2 + |\beta |^2 |\)= 1, is a valid wave function for the system. Then the probability that a measurement returns the value 0 is \(| \alpha |^2\) and the probability that a measurement returns the value 1 is \(| \beta |^2\). When a measurement is made, the values of \(\alpha\) and \(\beta\) change so that one of them is 1 and the other is 0, consistent with what the measurement returns.

It might seem that a qubit defined in this way could carry a lot of information because both \(\alpha\) and \(\beta\) can take on many possible values. However, the fact that a measurement will return only 0 or 1 along with the fact that these coefficients are destroyed by a measurement, means that only one bit of information can be read from a single qubit, no matter how much care was exerted in originally specifying \(\alpha\) and \(\beta\) precisely.