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The image is taken from this link.

enter image description here Here Alice is using random bases to encode 0 or 1. After the process is completed, Bob has similarly polarized photons as Alice. These polarization can be any of the $\lvert 0 \rangle , \lvert 1 \rangle, \lvert + \rangle$ or $\lvert - \rangle$. However, how would Bob know what Alice meant for which two of these bases? Meaning, Alice might choose ${\lvert 0 \rangle, \lvert + \rangle}$ to encode a 0 and ${\lvert 1 \rangle, \lvert - \rangle}$ to encode a 1 or vice versa. How do they determine which polarization encodes which bits?

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That’s the public discussion stage: Alice and Bob can both announce which basis they chose for each round. If they happened to pick the same basis on a given round, they know that (in a perfect world) their answers were the same, so they can translate them into a 0/1 value that nobody else knows. That translation is arbitrary, and they’ve probably agreed it in advance.

The natural way to do this is to associate an operator with each measurement basis, e.g. X or Z (the Pauli matrices). The measurement answers are then e.g. $(\mathbb{I}+(-1)^xX)/2$ where x is a bit value which we use as the translation.

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  • $\begingroup$ But the protocol says they only reveal the choice of basis at each iteration, not which basis was used to encode a bit. Isn't it? $\endgroup$ – Hasan Iqbal May 28 '18 at 19:29
  • $\begingroup$ There's no difference. Whatever basis Alice chose, she encodes a bit in the state that she prepares. Later, Alice and Bob only use the bits from the cases where Bob measured in the same basis as Alice prepared, and discard the rest. $\endgroup$ – DaftWullie May 28 '18 at 19:40

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