# Weak Coin Flipping

- [[quantum-computing]]
- [[cryptography]]
- Alice wants 1, Bob wants 0
- they both know which outcome the other wants
- first protocol
  - Alice generates a Bell-ish state $`\sqrt{\alpha}\ket{00} + \sqrt{1-\alpha}\ket{11}`$
  - Alice sends one of these qubits to Bob
  - Bob attaches a qubit by mapping $`\ket{0} \mapsto \sqrt{a}\ket{0}\ket{b=0} + \sqrt{1-\alpha}\ket{0}\ket{b=1}`$ and $`\ket{1}\mapsto\ket{1}\ket{b=1}`$
  - a whole view of our state is $`\sqrt{\alpha}\sqrt{\alpha} \ket{00,b=0} + \sqrt{1-a}\sqrt{a}\ket{00,b=1} + \sqrt{1-a}\ket{11,b=1}`$
    - if we measure $`b=0`$ we result in the state $`\ket{00}`$, but if we measure $`b=1`$ we result in the state $`\sqrt{\alpha}\ket{00} + \ket{11}`$
  - if Bob measures $`b=0`$, Bob sends back his qubit, so Alice should be able to verify that she has $`\ket{00}`$
  - if Bob measures $`b=1`$, Bob asks Alice to send her qubit, and Bob verifies that he has $`\sqrt{\alpha}\ket{00} + \ket{11}`$
  - tldr the person who is about to lose verifies that they weren't tricked
  - if we choose $`\alpha = \frac{1}{\sqrt{2}}`$, they can cheat with only $`71\%`$ probability: this differs from [[strong-coin-flipping]] because we already know the output both parties desire
- a better protocol requires higher dimensions
  - the idea is that we do weak measurements and pass information back and forth
  - we can change the "skew" of a superposition this way
- we can get arbitrarily close to $`\frac{1}{2}`$ probability of cheating: gain over classical where weak coin flipping is completely impossible