Editing Quantum Strong Coin Flipping (section)

==Protocol Description==

Two single qubit quantum states <math>\psi(0) = c|0\rangle + s|1\rangle</math> and <math>\psi(1) = c|0\rangle - s|1\rangle</math>, where <math>c,s \in \mathbb{R}</math> are defined such that the angle between them is <math>\theta</math>.
<math>\theta</math> is proposed to be <math>\pi/9</math>.
Thus, <math>\Phi(0)=\otimes^n_{k=1}\psi(0)</math> and <math>\Phi(1)=\otimes^n_{k=1}\psi(1)</math>.


# For <math>j = 1,2,...,m</math>:
## First party picks at random a bit <math>a_j</math>.
## Second party picks at random a bit <math>b_j</math>.
# For <math>i = 1,2,...n,</math>:
## For <math>j = 1,2,...,m</math>:
### The first party picks a random bit <math>c_{ij}</math> and sends a pair of qubits in the state <math>\psi(c_{ij})\otimes\psi(\bar{c}_{ij})</math> to the second party.
### The second party picks a random bit <math>d_{ij}</math> and sends a pair of qubits in the state <math>\psi(d_{ij})\otimes\psi(\bar{d}_{ij})</math> to the first party.
# For <math>i = 1,2,...n,</math>:
## For <math>j = 1,2,...,m</math>:
### The first party announces <math>e_{ij} = a_j \oplus c_{ij}</math>.
### The second party returns the second particle if <math>e_{ij}=0</math> and first particle otherwise.
### The second party announces <math>f_{ij} = b_j \oplus d_{ij}</math>.
### The first party returns the second particle if <math>f_{ij}=0</math> and first particle otherwise. <br/> At this stage, for every <math>j</math>, the <math>n</math> qubits sent by the first party and not returned by the second party are in the state <math>\Phi(a_j)</math> and the <math>n</math> qubits sent by the second party and not returned by the first party are in the state <math>\Phi(b_j)</math>.<br/>Also, for every j, the n qubits returned by the first party are in the state <math>\Phi(\bar{b}_j)</math> and the n qubits returned by the second party are in the state <math>\Phi(\bar{a}_j)</math>.
# For <math>j = 1,2,...,m</math>:
## First party announces <math>a_j</math>.
## Second party announces <math>b_j</math>.
## Second party executes POVM <math>(E_{a_j},E_{a_j}^\perp)</math> on <math>\Phi(a_j)</math> and notes the outcome <math>\tilde{a}</math>.
## First party executes POVM <math>(E_{b_j},E_{b_j}^\perp)</math> on <math>\Phi(b_j)</math> and notes the outcome <math>\tilde{b}</math>.<br/>If either <math>\tilde{a} = \perp</math> or <math>\tilde{b} = \perp</math>, the protocol aborts.   
# For <math>j = 1,2,...,m</math>:
## First party measures the returned state <math>\Phi(\bar{a}_j)</math> with POVM <math>(E_{\bar{a}_j},E_{\bar{a}_j}^\perp)</math>.
## Second party measures the returned state <math>\Phi(\bar{b}_j)</math> with POVM <math>(E_{\bar{b}_j},E_{\bar{b}_j}^\perp)</math>. 
If either of the outcomes is <math>\perp</math>, the protocol aborts.

The final bit of the first party is <math>A\oplus\tilde{B}</math> where <math>A=\oplus_ja_j</math> and <math>\tilde{B}=\oplus_j\tilde{b}_j</math>.
The second party’s final bit is <math>\tilde{A}\oplus B</math> where <math>\tilde{A}=\oplus_j\tilde{a}_j</math> and <math>B=\oplus_jb_j</math>.