Multi-qubit measurement using an ancilla

Measure ZZ

|0⟩   ----H---.---.---H----MZ
              |   |
      --------.---|-----------
|ψ⟩               |         
      ------------.-----------

After H:
|+⟩(a|00⟩ + b|01⟩ + c|10⟩ + d|11⟩) 

After first CZ:
a|000⟩ + b|001⟩ + c|010⟩ + d|011⟩ +
a|100⟩ + b|101⟩ - c|110⟩ - d|111⟩

After second CZ:
a|000⟩ + b|001⟩ + c|010⟩ + d|011⟩ +
a|100⟩ - b|101⟩ - c|110⟩ + d|111⟩

After H:
a|+00⟩ + b|+01⟩ + c|+10⟩ + d|+11⟩ +
a|-00⟩ - b|-01⟩ - c|-10⟩ + d|-11⟩
=
a|000⟩ + b|101⟩ + c|110⟩ + d|011⟩

After MZ:
if +1:
a|00⟩ + d|11⟩

if -1:
b|01⟩ + c|10⟩

Measure XX

|0⟩   ----H---.---.---H----MZ
              |   |
      --------X---|-----------
|ψ⟩               |         
      ------------X-----------

After H:
|+⟩(a|00⟩ + b|01⟩ + c|10⟩ + d|11⟩)

After first CX:
a|000⟩ + b|001⟩ + c|010⟩ + d|011⟩ +
a|110⟩ + b|111⟩ + c|100⟩ + d|101⟩

After first CX:
a|000⟩ + b|001⟩ + c|010⟩ + d|011⟩ +
a|111⟩ + b|110⟩ + c|101⟩ + d|100⟩

After H:
a|+00⟩ + b|+01⟩ + c|+10⟩ + d|+11⟩ +
a|-11⟩ + b|-10⟩ + c|-01⟩ + d|-00⟩

After MZ:
if +1:
(a + d)(|00⟩ + |11⟩) + (b + c)(|10⟩ + |01⟩)

if -1:
(a - d)(|00⟩ - |11⟩) + (b - c)(|01⟩ - |10⟩)

Bell measurement using an ancilla qubit: Measure XX and ZZ. Order doesn’t matter.

Compare with a “direct” Bell measurement;

Can also do a measurement of XXX… and ZZZ…

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How to measure IX?

Any state is an eigenstate of I. If we observe +1 on the second qubit then the total state is an eigenstate of IX.