 ...@@footnote^{1}

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ...

The original BB84 talked about photons and their polarizations. As I am not
addressing the actual physical implementations, I will speak of ``qubits'',
``photons'' or ``particles'' as if they were synonyms.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... Protocol^{2}

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ...b^{3}

Crépeau suggested [8] that quantum transmition of photons is prone to
duplication. I.e., sometimes Alice will inadvertedly send two photons (with
identical polarization) instead of just one. In this lucky situation, Bob should
measure both particles with respect to the same basis. If he happens to obtain
two different answers, he will know that the photons were not aligned with that
base, which means, the other base was used to encode them. b is revealed.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ...^{4}

A tedious but straightforward calculation shows that we can restrict ourselves
to realvalued
pairs only (thanks to I. Armendariz at
NYU).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... result^{5}

Quantum measurements are symmetric, so it is the same result if we think that
Bob measured first.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ...

In fact, tex2html_wrap_inline A applies a smaller tex2html_wrap_inline U_A_i , a transformation
of tex2html_wrap_inline H_AC , and the bigger state space is modified by tex2html_wrap_inline U_A_i=U_A_iI_B ,
(tex2html_wrap_inline I_B being the identity over tex2html_wrap_inline H_B )
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... issue.^{6}

Allah knows better.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... unitary^{7}

of norm one.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... Hilbert^{8}

Complex Euclidean vector space of finite or infinite dimension.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... transformation^{9}

A unitary transformation is one that preserves orthogonality and norm, such
as a rotation for example. Quantum physics states that the evolution of quantum
states occurs only by unitary operations. The quantum computing formalism indicates
that: (1) Any unitary transformation can in principle be built (from elementary
quantum gates) and (2) any classical computation or algorithm can be ``compiled''
into a quantum operation, replacing boolean gates with quantum gates.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... is^{10}

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
 ... yields^{11}

This is the definition of the Schmidt decomposition.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.