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# The EPR paradox steps in or, the quantum daemon bites its own tail.

The famous EPR (Einstein-Podolsky-Rosen) paradox describes a rather counterintuitive quality of quantum physics: the quantum state space of a set of several particles is the direct product of their individual spaces, and not the direct sum as one would expect. As a consequence, if A and B have separate particles a and b, which happen to be in an entangled state, then A may be able to gain information on B's measurement of b by measuring a.

It is possible for A to exploit this in order to cheat. Here's A's cheating scheme:

• EPR cheating scheme for BB84
1.
Commitment Phase

• A fabricates s pairs of entangled qubits and sends one particle in the pair to B while keeping the other.
2.
Disclosure Phase

• A measures her own qubits with respect to basis if she wants to disclose a zero and if she wants to disclose a one. She sends the disclosed'' bit along with the results of her measurements.
A measurement collapses the state of a quantum system into the subspace that is compatible with the value obtained. A exploits the fact that the state = is fully entangled with respect to both bases.

When Alice measures her qubit with respect to the horizontal basis, she gets either a zero or a one with 50% chance. However, this projects the entire state space of the entangled pair to either or (since the and components are zero from the start), which means that Bob will obtain the same result5. This is exactly the EPR paradox (table 4).

 qubit pair A's basis new state B's basis P(0) P(1) 0 0 1 0 + 1 1 0 0 1 = 1 0 0 + 1 1 0 1 0 1 0 1

With this procedure Alice makes sure that whenever Bob used the same basis to do his measurement, so he will get an identical result. And in the opposite case, as the angle between state and basis is , he obtains a 50%-50% distribution as expected.

Next: Brassard, Crépeau, Jozsa, Langlois: Up: Heads or Tails? Quantum Previous: Quantum Bit Commitment

1999-11-04