Quantum  Xerox  Principle

The third law of nature that plays an important role in the next lecture is the impossibility of faithfully duplicating quantum information. What it says is that a particular kind of apparatus cannot be built. We call it a Quantum Xerox Machine.^∗ It is a machine into which any system can be inserted, and which will copy that system, producing the original and a duplicate. To see why such a system is not possible, imagine that we insert a spin in the input port as in Figure 1.1. If the spin is in the up state with

Fig. 1.1. Schematic diagram of quantum Xerox machine.

respect to the z-axis, it is duplicated

 (1.1)

Similarly, if it is in the down state, it is duplicated

 (1.2)

Now suppose that the spin is inserted with its polarization along the x-axis, i.e.

 (1.3)

The general principles of quantum mechanics require the state to evolve linearly. Thus from equations 1.1 and 1.2

 (1.4)

On the other hand, a true quantum Xerox machine is required to duplicate the spin in equation 1.3

 (1.5)

The state in equation 1.4 is obviously not the same as that in equation 1.5. Thus we see that the principle of linearity forbids the existence of quantum Xerox machines. If we could construct Xeroxed quantum states, we would be able to violate the Heisenberg uncertainty principle by a set of measurements on those states.

__________________________________________________________

∗ The quantum Xerox principle is sometimes called the no-cloning principle.