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

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.

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 result^{5}. 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.