Attach an extra qubit=in register C. Conditioned on |1)in register C,swap content in A and B.Here swap is the operation la川b〉→lb)la) Measure the register C in {+)-)basis,and the test is considered to pass if the outcome is|+）. To get some intuition of the test,let's consider a special case first.If)=l),then the second step has no effect,and thus the measurement in the third step passes with probability 1.In general,note that after Step 2,the state becomes a0wo+1ow》-a(他是ei6+之w） 2 =+)o〉+1p2+-)〉=1 2 2 So the measurement in Step 3 gives a -)with probability ®o≥6-e1-e60D0w6)-wIw)=2a-Kw1op) 2 Now we give a quantum protocol in private-coin SMP model for EQn.Actually,Alice and Bob do not even use any randomness. ●Alice:Send l中x)=∑li)lE(x)i) ●Bob:Send中y〉=∑1lilE6y)i Referee::Apply Swap-Test on{lψ，lp,y}.Output“x=y”if the test passed and output x≠y”if the test didn't pass. The analysis of the protocol is easy.If x =y,then)=)and the Swap-Test passes with probability 1.So Referee outputs)=ly.以Ifx≠y,then not only l中x)≠lyy〉but they are far apart: o,,=元：E=Eo)D≥1-a as guaranteed by the error correcting code.Thus the Swap-Test fails with n(1)probability. (As always,one can repeat the protocol to reduce the error probability exponentially.)⚫ Attach an extra qubit |+〉 = 1 √2 (|0〉 + |1〉) in register C. ⚫ Conditioned on |1〉 in register C, swap content in A and B. Here swap is the operation |𝑎〉|𝑏〉 → |𝑏〉|𝑎〉. ⚫ Measure the register C in {|+〉, |−〉} basis, and the test is considered to pass if the outcome is |+〉. To get some intuition of the test, let’s consider a special case first. If |𝜓〉 = |𝜙〉, then the second step has no effect, and thus the measurement in the third step passes with probability 1. In general, note that after Step 2, the state becomes 1 √2 (|0〉|𝜓〉|𝜙〉 + |1〉|𝜙〉|𝜓〉) = 1 √2 ( |+〉 + |−〉 √2 |𝜓〉|𝜙〉 + |+〉 − |−〉 √2 |𝜙〉|𝜓〉) = |+〉 |𝜓〉|𝜙〉 + |𝜙〉|𝜓〉 2 + |−〉 |𝜓〉|𝜙〉 − |𝜙〉|𝜓〉 2 So the measurement in Step 3 gives a |−〉 with probability ‖ |𝜓〉|𝜙〉 − |𝜙〉|𝜓〉 2 ‖ 2 = 1 4 (〈𝜓|〈𝜙| − 〈𝜙|〈𝜓|)(|𝜓〉|𝜙〉 − |𝜙〉|𝜓〉) = 1 2 (1 − |⟨𝜓|𝜙⟩| 2 ) Now we give a quantum protocol in private-coin SMP model for 𝐄𝐐𝑛. Actually, Alice and Bob do not even use any randomness. ⚫ Alice: Send |𝜓𝑥 〉 = ∑ |𝑖〉|𝐸(𝑥)𝑖 𝑚 〉 𝑖=1 . ⚫ Bob: Send |𝜓𝑦〉 = ∑ |𝑖〉|𝐸(𝑦)𝑖 𝑚 〉 𝑖=1 . ⚫ Referee: Apply Swap-Test on {|𝜓𝑥 〉,|𝜓𝑦 〉}. Output “𝑥 = 𝑦” if the test passed and output “𝑥 ≠ 𝑦” if the test didn’t pass. The analysis of the protocol is easy. If 𝑥 = 𝑦, then |𝜓𝑥 〉 = |𝜓𝑦〉 and the Swap-Test passes with probability 1. So Referee outputs |𝜓𝑥 〉 = |𝜓𝑦〉. If 𝑥 ≠ 𝑦, then not only |𝜓𝑥 〉 ≠ |𝜓𝑦〉 but they are far apart: |⟨𝜓𝑥 |𝜓𝑦 ⟩| = 1 𝑚 (|{𝑖: 𝐸(𝑥) 𝑖 = 𝐸(𝑦) 𝑖 }|) ≥ 1 − Ω(1) as guaranteed by the error correcting code. Thus the Swap-Test fails with Ω(1) probability. (As always, one can repeat the protocol to reduce the error probability exponentially.)