Richard Cleve DC 3524 cleve@cs.uwaterloo Course web site at:

Introduction to Quantum Information ProcessingCS 467 / CS 667Phys 467 / Phys 767C&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)

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

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

DefineA0 =2/300 A1=2/311 A2=2/322 Correction to Lecture 11: 2/3 instead of 2/3 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” Ak is 2/3, and this can be adapted to actually yield the value of kwith this success probability

0 with prob. ½ 1 with prob. ½ 0 with prob. cos2(/8) 1 with prob. sin2(/8) 0 with prob. ½ 1 with prob. ½ 1 + 0 0 with prob. ½ 0 +1 with prob. ½ 0 Distinguishing mixed states (I) What’s the best distinguishing strategy between these two mixed states? 1 also arises from this orthogonal mixture: … as does 2 from:

Distinguishing mixed states (II) We’ve effectively found an orthonormal basis 0,1 in which both density matrices are diagonal: 1 1 + 0 Rotating 0,1to 0, 1the scenario can now be examined using classical probability theory: 0 Distinguish between two classical coins, whose probabilities of “heads” are cos2(/8)and ½ respectively (details: exercise) Question: what do we do if we aren’t so lucky to get two density matrices that are simultaneously diagonalizable?

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

Recap: general quantum ops General quantum operations(a.k.a.“completely positive trace preserving maps”): LetA1, A2 , …, Am be matrices satisfying Then the mapping is a general quantum op Example: applying U to yieldsUU†

index means 2nd system traced out Tr2( )= Partial trace (I) Two quantum registers (e.g. two qubits) in states and (respectively) are independent if then the combined system is in state  =   In such circumstances, if the second register (say) is discarded then the state of the first register remains  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, Tr2 ,as the unique linear operator satisfying the identity Tr2( )=  For example, it turns out that

Partial trace (II) We’ve already seen this defined in the case of 2-qubit systems: discarding the second of two qubits LetA0 =I0andA1 =I1 For the resulting quantum operation, state  becomes  For d-dimensional registers, the operators are Ak =Ik, where0, 1, …, d1 are an orthonormal basis

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

Then the corresponding POVM is a stochastic operation on  that, with probability produces the outcome: j (classical information) (the collapsed quantum state) POVMs (I) Positive operator valued measurement (POVM): LetA1, A2 , …, Am be matrices satisfying Example 1:Aj = jj (orthogonal projectors) This reduces to our previously defined measurements …

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

DefineA0 =2/300 A1=2/311 A2=2/322 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…

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

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   quantum output input 0 j classical output 0 0

Separable states A bipartite (i.e. two register) state  is a: • product state if = • separablestate if (p1 ,…,pm 0) (i.e. a probabilistic mixture of product states) Question: which of the following states are separable?

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

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Richard Cleve DC 3524 cleve@cs.uwaterloo Course web site at: