13.4: Model 3- Multiple Qubits with Entanglement

Consider a quantum mechanical system with four states, rather than two. Let us suppose that it is possible to make two different measurements on the system, each of which returns either 0 or 1. It is natural to denote the stationary states with two subscripts, one corresponding to the first measurement and the other to the second. Thus the general wave function is of the form

\(\psi = \alpha_{00}\psi_{00} + \alpha_{01}\psi_{01} + \alpha_{10}\psi_{10} + \alpha_{11}\psi_{11} \tag{13.2}\)

where the complex coefficients obey the normalization condition

\(1 =| \alpha_{00} |^2 + | \alpha_{01} |^2 + | \alpha_{10} |^2 + | \alpha_{11} |^2 \tag{13.3}\)

You may think of this model as two qubits, one corresponding to each of the two measurements. These qubits are not independent, but rather are entangled in some way. Then it is natural to ask what happens if one of them is measured. A measurement of, for example, the first qubit will return 0 with probability \(| \alpha_{00} |^2 + | \alpha_{01} |^2\) and if it does the wave function collapses to only those stationary states that are consistent with this measured value,

\[\psi=\frac{\alpha_{00} \psi_{00}+\alpha_{01} \psi_{01}}{\sqrt{\left|\alpha_{00}\right|^{2}+\left|\alpha_{01}\right|^{2}}} \tag{13.4} \]

(note that the resulting wave function was “re-normalized” by dividing by \(\sqrt{\left|\alpha_{00}\right|^{2}+\left|\alpha_{01}\right|^{2}}\)).

There is no need for this system to be physically located in one place. In fact, one of the most interesting examples involves two qubits which are entangled in this way but where the first measurement is done in one location and the second in another. A simple case is one in which there are only two of the four possible stationary states initially, so \(\alpha_{01} = 0\) and \(\alpha_{10}\) = 0. This system has the remarkable property that as a result of one measurement the wave function is collapsed to one of the two possible stationary states and the result of this collapse can be detected by the other measurement, possibly at a remote location.

It is possible to define several interesting logic gates which act on multiple qubits. These have the property that they are reversible; this is a general property of quantum-mechanical systems.

Among the interesting applications of multiple qubits are

• Computing some algorithms (including factoring integers) faster than classical computers
• Teleportation (of the information needed to reconstruct a quantum state)
• Cryptographic systems
• Backwards information transfer (not possible classically)
• Superdense coding (two classical bits in one qubit if another qubit was sent earlier)

These applications are described in several books and papers, including these three:

• T. P. Spiller, “Quantum Information Processing: Cryptography, Computation, and Teleportation,” Proc. IEEE, vol. 84, no. 12, pp. 1719–1746; December, 1996. Although this article is now several years old, it is still an excellent introduction.
• Michael A. Nielsen and Isaac L. Chuang, “Quantum Computation and Quantum Information,” Cambridge University Press, Cambridge, UK; 2000
• Hoi-Kwong Lo, Sandu Popescu, and Tim Spiller, “Introduction to Quantum Computation and Information,” World Scientific, Singapore; 1998. The book is based on a lecture series held at Hewlett-Packard Laboratories, Bristol, UK, November 1996–April, 1997