Richard Cleve

1. DENSITY MATRICES, traces, Operators and Measurements Richard Cleve Lectures 10 ,11 and 12 Michael A. Nielsen Sources: Michele Mosca

2. Review: Density matrices of pure states We have represented quantum states as vectors (e.g. ψ, and all such states are called pure states) An alternative way of representing quantum states is in terms of density matrices(a.k.a. density operators) The density matrix of a pure state ψ is the matrix  =ψψ Example: the density matrix of 0 +1 is

3. Reminder: Trace of a matrix The trace of a matrix is the sum of its diagonal elements e.g. Some properties: Orthonormal basis { }

4. Example: Notation of Density Matrices and traces Notice that 0=0|, and 1=1|. So the probability of getting 0 when measuring | is: where  = || is called the density matrix for the state |

5. Review: Mixture of pure states A state described by a state vector | is called a pure state. What if we have a qubit which is known to be in the pure state |1 with probability p1, and in |2 with probability p2 ? More generally, consider probabilistic mixturesof pure states (called mixed states):

6. Density matrices of mixed states A probability distribution on pure states is called a mixed state: ( (ψ1, p1), (ψ2, p2), …, (ψd, pd)) The density matrix associated with such a mixed state is: Example: the density matrix for ((0, ½ ), (1, ½ ))is: Question: what is the density matrix of ((0+1, ½ ), (0− 1, ½ )) ?

7. where is the density matrixfor the mixed state  Density matrix of a mixed state (use of trace) …then the probability of measuring 0 is given by conditional probability: Density matrices contain all the useful information about an arbitrary quantum state.

8. Recap: operationally indistinguishable states Since these are expressible in terms of density matrices alone (independent of any specific probabilistic mixtures), states with identical density matrices areoperationally indistinguishable

9. Applying Unitary Operator to a Density Matrix of a pure state If we apply the unitary operation U to the resulting state is with density matrix

10. Applying Unitary Operator to a Density Matrix of a mixed state Density Matrix How do quantum operations work for these mixed states? If we apply the unitary operation U to the resulting state is with density matrix

11. Operators on Density matrices of mixed states. Thus this is true always Effect of a unitary operation on a density matrix: applying U to still yieldsUU† Effect of a measurement on a density matrix: measuring state  with respect to the basis1, 2,..., d, still yields the kthoutcome with probability kk Why?

12. How do quantum operations work using density matrices? Effect of a measurement on a density matrix: measuring state  with respect to the basis1, 2,..., d,yields the kthoutcome with probability kk (this is because kk=kψψk=kψ2) —and thestate collapses tokk

13. 6. 0 with prob. ¼ 1 with prob. ¼ 0 +1 with prob. ¼ 0 − 1 with prob. ¼ 3. 0 with prob. ½ 1 with prob. ½ 4. 0 +1 with prob. ½ 0 −1 with prob. ½ More examples of density matrices The density matrix of the mixed state ((ψ1, p1), (ψ2, p2), …,(ψd, pd)) is: Examples (from previous lecture): 1. & 2. 0 +1 and −0 −1 both have

14. More examples of density matrices 5. 0 with prob. ½ 0 +1 with prob. ½ Examples (continued): has: ...?(later) 7. The first qubit of 01 −10

15. To Remember: Three Properties of Density Matrices • Three properties of  : • Tr= 1(TrM = M11+M22 + ... +Mdd ) • =† (i.e.is Hermitian) •  0, for all states  Moreover, for any matrix  satisfying the above properties, there exists a probabilistic mixture whose density matrix is  Exercise: show this

16. Use of Density Matrix and Trace to Calculate the probability of obtaining state in measurement If we perform a Von Neumann measurement of the state wrt a basis containing , the probability of obtaining is This is for a pure state. How it would be for a mixed state?

17. Use of Density Matrix and Trace to Calculate the probability of obtaining state in measurement (now for measuring a mixed state) Density Matrix If we perform a Von Neumann measurement of the state wrt a basis containing the probability of obtaining is The same state

18. Conclusion: Density Matrix HasComplete Information In other words, the density matrix contains all the information necessary to compute the probability of any outcome in any future measurement.

19. Spectral decomposition can be used to represent a useful form of density matrix • Often it is convenient to rewrite the density matrix as a mixture of its eigenvectors • Recall that eigenvectors with distinct eigenvalues are orthogonal; • for the subspace of eigenvectors with a common eigenvalue (“degeneracies”), we can select an orthonormal basis

20. Continue - Spectral decomposition used to diagonalize the density matrix • In other words, we can always “diagonalize” a density matrix so that it is written as where is an eigenvector with eigenvalue and forms an orthonormal basis

21. Taxonomy of various normal matrices

22. eigenvectors: Normal matrices Definition: A matrix M is normal if M†M = MM† Theorem:M is normal iff there exists a unitary U such that M = U†DU, where D is diagonal (i.e. unitarily diagonalizable) Examples of abnormal matrices: is not even diagonalizable is diagonalizable, but not unitarily

23. Unitary and Hermitian matrices Normal: with respect to some orthonormal basis Unitary:M†M = I which implies |k |2= 1, for all k Hermitian:M = M† which implies k  R, for all k Question: which matrices areboth unitary and Hermitian? Answer: reflections (k  {+1,1}, for all k)

24. Positive semidefinite matrices Positive semidefinite: Hermitian and k  0, for all k Theorem:M is positive semidefinite iff M is Hermitian and, for all , M 0 (Positive definite:k > 0, for all k)

25. Projectors and density matrices Projector: Hermitian and M2 = M, which implies that M is positive semidefinite and k  {0,1}, for all k Density matrix: positive semidefinite and TrM=1, so Question: which matrices areboth projectors and density matrices? Answer:rank-one projectors (k = 1 if k = k0 and k = 0 ifk k0)

26. normal unitary Hermitian reflection positive semidefinite projector density matrix rank one projector Taxonomy of normal matrices If Hermitian then normal

27. Review: Bloch sphere for qubits Consider the set of all 2x2 density matrices  They have a nice representation in terms of the Pauli matrices: Note that these matrices—combined with I—form a basis for the vector space of all 2x2 matrices We will express density matrices  in this basis Note that the coefficient of I is ½, since X, Y, Z have trace zero

28. Bloch sphere for qubits: polar coordinates We will express First consider the case of pure states , where, without loss of generality,  =cos()0 +e2isin()1 (,R) Therefore cz= cos(2),cx= cos(2)sin(2),cy= sin(2)sin(2) These are polar coordinates of a unit vector (cx ,cy ,cz)R3

29. 0 + = 0 +1 –i – = 0 – 1 +i = 0 +i1 + – –i = 0 –i1 +i 1 Bloch sphere for qubits: location of pure and mixed states Note that orthogonal corresponds to antipodal here Pure states are on the surface, and mixed states are inside (being weighted averages of pure states)

30. General quantum operations Decoherence, partial traces, measurements.

31. Then the mapping is a general quantum operator General quantum operations (I) General quantum operations are also called “completely positive trace preserving maps”,or “admissible operations” condition LetA1, A2 , …, Am be matrices satisfying Example 1 (unitary op): applying U to yieldsUU†

32. After looking at the outcome,  becomes 00 with prob. ||2 11 with prob. ||2 General quantum operations: Decoherence Operations Example 2 (decoherence):letA0 =00andA1 =11 This quantum op maps  to 0000 + 1111 For ψ =0 +1, Corresponds to measuring  “without looking at the outcome”

33. DefineA0 =2/300 A1=2/311 A2=2/322 General quantum operations: measurement operations Example 3 (trine state“measurement”): Let 0 = 0, 1 =1/20 + 3/21, 2 =1/20 3/21 Then Condition satisfied We apply the general quantum mapping operator • The probability that state k results in “outcome” state Ak is 2/3. • This can be adapted to actually yield the value of kwith this success probability

34. General quantum operations: Partial trace discards the second of two qubits Example 4 (discarding the second of two qubits): LetA0 =I0andA1 =I1 We apply the general quantum mapping operator State becomes  State becomes Note 1: it’s the same density matrix as for ((0, ½), (1, ½)) Note 2: the operation is the partial trace Tr2

35. Distinguishing mixed states Several mixed states can have the same density matrix – we cannot distinguish between them. How to distinguish by two different density matrices? Try to find an orthonormal basis 0,1 in which both density matrices are diagonal:

36. 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: /8=180/8=22.5

37. Distinguishing mixed states (II) Density matrices 1 and 2 are simultaneously diagonalizable 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?

38. Reminder: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,

39. Partial Trace • How can we compute probabilities for a partial system? • E.g. Partial measurement

40. Partial Trace • If the 2nd system is taken away and never again (directly or indirectly) interacts with the 1st system, then we can treat the first system as the following mixture • E.g. From previous slide

41. Partial Trace: we derived an important formula to use partial trace Derived in previous slide

42. Why? • the probability of measuring e.g. in the first register depends only on

43. Partial Trace can be calculated in arbitrary basis • Notice that it doesn’t matter in which orthonormal basis we “trace out” the 2nd system, e.g. • In a different basis

44. (cont) Partial Trace can be calculated in arbitrary basis Partial Trace Which is the same as in previous slide for other base

45. Methods to calculate the Partial Trace • Partial Trace is a linear map that takes bipartite states to single system states. • We can also trace out the first system • We can compute the partial trace directly from the density matrix description

46. Partial Trace using matrices • Tracing out the 2nd system Tr 2

47. index means 2nd system traced out Tr2( )= Examples: Partial trace (I) Two quantum registers (e.g. two qubits) in states and (respectively) are independentif 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

48. Examples: 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 As we see in last slide, partial trace is a matrix. How to calculate this matrix of partial trace?

49. Examples: Partial trace (III): calculating matrices of partial traces For 2-qubit systems, the partial trace is explicitly and

50. Unitary transformations don’t change the local density matrix • A unitary transformation on the system that is traced outdoes not affect the result of the partial trace • I.e.