3.4. Quantum mechanics

Again I’m assuming a reasonable knowledge of quantum mechanics, although we will need surprisingly little actual physics. My favourite interpretation of quantum mechanics is credited to Scott Aaronson, who in Quantum computing since Democritus describes it succinctly as “probability with a 2-norm instead of a 1-norm”. Admittedly this is unlikely to be very illuminating unless you already have some familiarity with the material. By making some reasonable assumptions about what reality should do and what a bit is, Aaronson argues that quantum mechanics should in principle be discoverbale to mathematicians even without experiment to confirm it. Good book, recommended.

The most important quantum-mechanical concepts for us are the following:

  1. Superposition: a general quantum system exists in a superposition of all its basic states. For example, an atom exists in a superposition of its ground state and all of its electrically excited states simultaneously, and only upon measurement does the system ‘choose’ a state to reveal. This is subtly different to the superposition of classical wave mechanics, because in quantum mechanics the wave function is not an observable. If base states are labelled \(|x_i\rangle\) and probability amplitudes are labelled \(\alpha_i\) then the state \(|\psi\rangle\) of a general \(N\)-dimensional system is written as$$|\psi\rangle=\sum_{i=0}^{N-1}\alpha_i|x_i\rangle.$$
  2. Entanglement: two (or more) quantum systems can exhibit non-classical correlation, so that the measurement of some property of one system indicates with certainty what a later measurement of a correlated system will give. This is despite the fact that the result of the first measurement was not pre-determined, and pays no respect to the spatial separation of the correlated systems. This is not (by itself) useful for long-distance communication as the result of the first measurement is essentially random in nature. Without some other way to communicate, both parties will measure apparently random noise. The famous Bell inequalities more or less amount to a proofThere are some loopholes that need closing, but they have almost all been experimentally closed and I don’t know of many or any serious physicists who expect any failings on that front. that quantum mechanics is not compatible with local realismMeaning quantum mechanics is non-local or quantum systems do not have predetermined states., and this is thanks to entanglement.
  3. No cloning: it is not possible to create a perfect copy of an arbitrary unknown state without destroying the original. This theorem has profound implications for quantum information processing and is surprisingly easy to prove.

We’re also going to be ever-concerned with the measurement problem. This is the name given to the fact that measurement of a quantum state causes it to collapse to one of the basis states of the measurement, and so in general we cannot access the information contained in a quantum state, which is encoded in the complex probability amplitudes. It is not possible even in principle to discover the exact value of the amplitudes for an arbitrary state, for the same reasons we can never decide if a coin is perfectly fair; it would require an infinite number of trials. The measurement problem will prove to be one of the key obstacles in the pursuit of quantum computation, and much of algorithm design comes down to finding clever ways to evade the problem rather than solve it.

Proof of the no cloning theorem

I said it was easy to prove so we may as well do it! But it might be a good idea to read the section on quantum circuits first.

The no cloning theorem states that it is not possible to copy an arbitrary quantum state while preseving the original. Let’s say we want to write some function \(U_\textrm{copy}\) which will copy an arbitrary quantum state. The function must have the action$$U_\textrm{copy}:|\psi\rangle|0\rangle\mapsto|\psi\rangle|\psi\rangle$$for any state \(|\psi\rangle\). It can be represented by the circuit shown in figure 1. Since the function is a quantum mechanical evolution without a measurement it must be unitary. If we act \(U_\textrm{copy}\) on a superposition of two arbitrary states then we would want the result to be$$
U_\textrm{copy}=(\alpha_1|\psi_1\rangle+\alpha_2|\psi_2\rangle)\otimes|0\rangle
=
(\alpha_1|\psi_1\rangle+\alpha_2|\psi\rangle)\otimes(\alpha_1|\psi_1\rangle+\alpha_2|\psi\rangle)\\
=
\alpha_1^2|psi_1\rangle|\psi_1\rangle+\alpha_2|\psi_2\rangle|\psi_2\rangle+\alpha_1\alpha_2|\psi_1\rangle|\psi_2\rangle+\alpha_1\alpha_2|\psi_2\rangle|\psi_1\rangle.$$
The first line is the desired copy operation, and the second line follows directly from the linearity of the Kronecker product. But this result is in direct conflict with the linearity of quantum mechanical operators, which requires that$$
U_\textrm{copy}(\alpha_1|\psi_1\rangle+\alpha_2|\psi_2\rangle)|0\rangle=(\alpha_1 U_\textrm{copy}|\psi_1\rangle|0\rangle+\alpha_2 U_\textrm{copy}|\psi_2\rangle|0\rangle)\\
=\alpha_1|\psi_1\rangle|\psi_1\rangle+\alpha_2|\psi_2\rangle|\psi_2\rangle.$$
Clearly these two expressions are not equal. We have a contradiction, and since we only made one assumption which might be suspect, namely that the copying procedure is possible at all, we have have proven by contradiction that the assumption is not valid. Copying arbitrary states is not possible.