1. 1 Introduction to Quantum Information Processing CS 467 / CS 667 Phys 467 / Phys 767 C&O 481 / C&O 681 Richard Cleve DC 3524 cleve@cs.uwaterloo.ca Course web site at: http://www.cs.uwaterloo.ca/~cleve/courses/cs467 Lecture 12 (2005)

2. 2 Contents Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

3. 3 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

4. 4 General quantum operations (III) Example 3 (trine state “measurent”): Let  0  =  0 ,  1  =  1/2  0  +  3/2  1 ,  2  =  1/2  0    3/2  1  Then The probability that state  k  results in “outcome” A k is 2/3, and this can be adapted to actually yield the value of k with this success probability Define A 0 =  2/3  0  0  A 1 =  2/3  1  1  A 2 =  2/3  2  2  Correction to Lecture 11:  2/3 instead of 2/3

5. 5 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

6. 6 Distinguishing mixed states (I)  0  with prob. ½  0  +  1  with prob. ½  0  with prob. ½  1  with prob. ½  0  with prob. cos 2 (  /8)  1  with prob. sin 2 (  /8) 00 ++ 00 11  0  with prob. ½  1  with prob. ½ What’s the best distinguishing strategy between these two mixed states?  1 also arises from this orthogonal mixture: … as does  2 from:

7. 7 Distinguishing mixed states (II) 00 ++ 00 11 We’ve effectively found an orthonormal basis   0 ,   1  in which both density matrices are diagonal: Rotating   0 ,   1  to  0 ,  1  the scenario can now be examined using classical probability theory: Question: what do we do if we aren’t so lucky to get two density matrices that are simultaneously diagonalizable? Distinguish between two classical coins, whose probabilities of “heads” are cos 2 (  /8) and ½ respectively (details: exercise) 11

8. 8 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

9. 9 Basic properties of the trace The trace of a square matrix is defined as It is easy to check that The second property implies and Calculation maneuvers worth remembering are: and Also, keep in mind that, in general,

10. 10 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

11. 11 Recap: general quantum ops Example: applying U to  yields U  U † General quantum operations (a.k.a. “completely positive trace preserving maps” ): Let A 1, A 2, …, A m be matrices satisfying Then the mappingis a general quantum op

12. 12 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

13. 13 Partial trace (I) In such circumstances, if the second register (say) is discarded then the state of the first register remains  Two quantum registers (e.g. two qubits) in states  and  (respectively) are independent if then the combined system is in state  =    In general, the state of a two-register system may not be of the form    (it may contain entanglement or correlations) We can define the partial trace, Tr 2 , as the unique linear operator satisfying the identity Tr 2 (    ) =  For example, it turns out that Tr 2 () =) = index means 2 nd system traced out

14. 14 Partial trace (II) We’ve already seen this defined in the case of 2-qubit systems: discarding the second of two qubits Let A 0 = I   0  and A 1 = I   1  For the resulting quantum operation, state    becomes  For d -dimensional registers, the operators are A k = I   k , where  0 ,  1 , …,  d  1  are an orthonormal basis

15. 15 Partial trace (III) For 2-qubit systems, the partial trace is explicitly and

16. 16 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

17. 17 POVMs (I) Positive operator valued measurement (POVM): Let A 1, A 2, …, A m be matrices satisfying Then the corresponding POVM is a stochastic operation on  that, with probability produces the outcome: j (classical information) (the collapsed quantum state) Example 1: A j =  j  j  (orthogonal projectors) This reduces to our previously defined measurements …

18. 18 POVMs (II) Moreover, When A j =  j  j  are orthogonal projectors and  = , = Tr  j  j  j  j  =  j  j  j  j  =  j  2

19. 19 POVMs (III) Example 3 (trine state “measurent”): Let  0  =  0 ,  1  =  1/2  0  +  3/2  1 ,  2  =  1/2  0    3/2  1  Then If the input itself is an unknown trine state,  k  k , then the probability that classical outcome is k is 2/3 = 0.6666… Define A 0 =  2/3  0  0  A 1 =  2/3  1  1  A 2 =  2/3  2  2 

20. 20 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

21. 21 Simulations among operations (I) Fact 1: any general quantum operation can be simulated by applying a unitary operation on a larger quantum system: U 00 00 00   Example: decoherence 00  0  +  1  output discard input

22. 22 Simulations among operations (II) Fact 2: any POVM can also be simulated by applying a unitary operation on a larger quantum system and then measuring: U 00 00 00   quantum output input classical output j

23. 23 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

24. 24 Separable states product state if  =  separable state if A bipartite (i.e. two register) state  is a: Question: which of the following states are separable? (i.e. a probabilistic mixture of product states) ( p 1,…, p m  0 )

25. 25 Correction to Lecture 11: 2/3   2/3 Distinguishing mixed states Basic properties of the trace Recap: general quantum operations Partial trace POVMs Simulations among operations Separable states Continuous-time evolution

26. 26 Continuous-time evolution Although we’ve expressed quantum operations in discrete terms, in real physical systems, the evolution is continuous 00 11 Let H be any Hermitian matrix and t  R Then e iHt is unitary – why? H = U † DU, where Therefore e iHt = U † e iDt U = (unitary)

27. 27