1. Introduction to Quantum Information Processing Lecture 4 Michele Mosca

2. Overview l Von Neumann measurements l General measurements l Traces and density matrices and partial traces

3. “Von Neumann measurement in the computational basis” l Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis l If we measure we get with probability

4. In section 2.2.5, this is described as follows l We have the projection operators and satisfying l We consider the projection operator or “observable” l Note that 0 and 1 are the eigenvalues l When we measure this observable M, the probability of getting the eigenvalue is and we are in that case left with the state

5. “Expected value” of an observable If we associate with outcome the eigenvalue then the expected outcome is

6. “Von Neumann measurement in the computational basis” l Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis l Say we have the state l If we measure all n qubits, then we obtain with probability l Notice that this means that probability of measuring a in the first qubit equals

7. Partial measurements l (This is similar to Bayes Theorem) l If we only measure the first qubit and leave the rest alone, then we still get with probability l The remaining n-1 qubits are then in the renormalized state

8. In section 2.2.5 l This partial measurement corresponds to measuring the observable

9. Von Neumann Measurements l A Von Neumann measurement is a type of projective measurement. Given an orthonormal basis, if we perform a Von Neumann measurement with respect to of the state then we measure with probability

10. Von Neumann Measurements l E.x. Consider Von Neumann measurement of the state with respect to the orthonormal basis l Note that l We therefore getwith probability

11. Von Neumann Measurements l Note that

12. How do we implement Von Neumann measurements? l If we have access to a universal set of gates and bit-wise measurements in the computational basis, we can implement Von Neumann measurements with respect to an arbitrary orthonormal basisas follows.

13. How do we implement Von Neumann measurements? l Construct a quantum network that implements the unitary transformation l Then “conjugate” the measurement operation with the operation

14. Another approach

15. Ex. Bell basis change l Consider the orthonormal basis consisting of the “Bell” states l Note that

16. Bell measurement l We can “destructively” measure l Or non-destructively project

17. Most general measurement

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

19. Density Matrices 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 | 

20. 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 p 1, and in |  2  with probability p 2 ? More generally, consider probabilistic mixtures of pure states (called mixed states):

21. Density matrix of a mixed state …then the probability of measuring 0 is given by conditional probability: whereis the density matrix for the mixed state  Density matrices contain all the useful information about an arbitrary quantum state.

22. Density Matrix If we apply the unitary operation U to the resulting state is with density matrix

23. Density Matrix If we apply the unitary operation U to the resulting state is with density matrix

24. Density Matrix If we perform a Von Neumann measurement of the state wrt a basis containing, the probability of obtaining is

25. Density Matrix If we perform a Von Neumann measurement of the state wrt a basis containing the probability of obtaining is

26. Density Matrix In other words, the density matrix contains all the information necessary to compute the probability of any outcome in any future measurement.

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

28. Spectral decomposition l 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

29. Partial Trace l How can we compute probabilities for a partial system? l E.g.

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

31. Partial Trace

32. Why? l the probability of measuring e.g. in the first register depends only on

33. Partial Trace l Notice that it doesn’t matter in which orthonormal basis we “trace out” the 2 nd system, e.g. l In a different basis

34. Partial Trace

35. Distant transformations don’t change the local density matrix l Notice that the previous observation implies that a unitary transformation on the system that is traced out does not affect the result of the partial trace l I.e.

36. Distant transformations don’t change the local density matrix l In fact, any legal quantum transformation on the traced out system, including measurement (without communicating back the answer) does not affect the partial trace l I.e.

37. Why?? l Operations on the 2 nd system should not affect the statistics of any outcomes of measurements on the first system l Otherwise a party in control of the 2 nd system could instantaneously communicate information to a party controlling the 1 st system.

38. Principle of implicit measurement l If some qubits in a computation are never used again, you can assume (if you like) that they have been measured (and the result ignored) l The “reduced density matrix” of the remaining qubits is the same

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

40. Partial Trace using matrices l Tracing out the 2 nd system

41. Most general measurement