Consider the 4-cycle graph G with edges (1,2), (2,3), (3,4), and (1,4). Construct the corresponding graph...

 

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Consider the 4-cycle graph G with edges (1,2), (2,3), (3,4), and (1,4). Construct the corresponding graph state |PG) without the second bank of Hadamard gates. Call the nodes Alice, Bob, Charlie, and Donna in that order (Donna = 4). (a) Construct the density matrix PG. Then show the result of tracing out nodes 3 and 4. Is the result a completely mixed state of two qubits? Is it pure? (b) Now let Charlie and Donna each apply a single-qubit Hadamard gate locally and then post-select on 0. Show the calculations for the state Alice and Bob are left with, as well as verifying it on Quirk. Are Alice and Bob entangled? (c) Now trace out nodes 2 and 4 instead. Are Alice and Charlie left with the completely mixed state in this case? (d) Now instead of traciong out Donna and Bob, let them each apply a single-qubit Hadamard gate locally and then post-select on 0. Are Alice and Charlie left entan- gled? (This has some of the flavor of S.W.'s presentation.) Consider the 4-cycle graph G with edges (1,2), (2,3), (3,4), and (1,4). Construct the corresponding graph state |PG) without the second bank of Hadamard gates. Call the nodes Alice, Bob, Charlie, and Donna in that order (Donna = 4). (a) Construct the density matrix PG. Then show the result of tracing out nodes 3 and 4. Is the result a completely mixed state of two qubits? Is it pure? (b) Now let Charlie and Donna each apply a single-qubit Hadamard gate locally and then post-select on 0. Show the calculations for the state Alice and Bob are left with, as well as verifying it on Quirk. Are Alice and Bob entangled? (c) Now trace out nodes 2 and 4 instead. Are Alice and Charlie left with the completely mixed state in this case? (d) Now instead of traciong out Donna and Bob, let them each apply a single-qubit Hadamard gate locally and then post-select on 0. Are Alice and Charlie left entan- gled? (This has some of the flavor of S.W.'s presentation.)