Bell measurement

 |0⟩ -----------------------------------MZ  +1
                                              
 |0⟩ -----------------------------------MZ  +1

 |0⟩ --H--.----------- -----------.--H--MZ  +1
          |                       |           
 |0⟩ -----X----------- -----------X-----MZ  +1

 |0⟩ --H--.------           ------.--H--MZ  +1
          |      |00⟩ + |11⟩      |           
 |0⟩ -----X------           ------X-----MZ  +1

the outcome says which Bell state is measured

 |0⟩ --H--.------           ------.--H--MZ  +1
          |      |00⟩ + |11⟩      |           
 |0⟩ -----X------           ------X-----MZ  +1

 |0⟩ --H--.------           ------.--H--MZ  +1
          |      |00⟩ + |11⟩      |           
 |0⟩ -----X------           ------X-----MZ  +1

 |0⟩ --H--.------           ------.--H--MZ  +1
          |      |01⟩ + |10⟩      |           
 |1⟩ -----X------           ------X-----MZ  -1

 |1⟩ --H--.------           ------.--H--MZ  -1
          |      |00⟩ - |11⟩      |           
 |0⟩ -----X------           ------X-----MZ  +1

 |1⟩ --H--.------           ------.--H--MZ  -1
          |      |01⟩ - |10⟩      |           
 |1⟩ -----X------           ------X-----MZ  -1

 |1⟩ --H--.------           ------.--H--MZ  -1
          |      |01⟩ - |10⟩      |           
 |1⟩ -----X------           ------X-----MZ  -1

 |1⟩ --H--.----------- -----------.--H--MZ  -1
          |                       |           
 |1⟩ -----X----------- -----------X-----MZ  -1

 |1⟩ -----------------------------------MZ  -1
                                              
 |1⟩ -----------------------------------MZ  -1

we can change the measurement basis to {±XZ,±ZX}

 |0⟩ --H--.-----             -----.--H--MZ    
          |      |00⟩ + |11⟩      |           
 |0⟩ -----X-----             -----X-----MZ    

 |0⟩ --H--.-----             -----.--H--MZ  +1
          |      |0+⟩ + |1-⟩      |           
 |0⟩ -----X--H--             --H--X-----MZ  +1

 |0⟩ --H--.-----             -----.--H--MZ  +1
          |      |0+⟩ + |1-⟩      |           
 |0⟩ -----X--H--             --H--X-----MZ  +1

|0+⟩ + |1-⟩ is stabilized by {+XZ,+ZX}

 |.⟩ --H--.-----             -----.--H--MZ  ±1
          |          ...          |           
 |.⟩ -----X--H--             --H--X-----MZ  ±1

and so on ..

---

Applications: Quantum teleportation;

• Can also do a Bell measurement as a Multi-qubit measurement using an ancilla;

• Hardware implementation: Bell measurement with linear optics;