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Density Matrices and Quantum Noise Michael A. Nielsen University of Queensland Goals: 1. To review a tool – the density matrix – that is used to describe noise in quantum systems. 2. To give more practical examples. Density matrices Generalization of the quantum state used to describe noisy quantum systems. Terminology: “Density matrix” = “Density operator” Ensemble pj , j Quantum subsystem Fundamental point of view What we’re going to do in this lecture, and why we’re doing it Most of the lecture will be spent mastering the density matrix. We’ve got to master a rather complex formalism. It might seem a little strange, since the density matrix is never essential for calculations – it’s a mathematical tool, introduced for convenience. Why bother with it? The density matrix seems to be a very deep abstraction – once you’ve mastered the formalism, it becomes far easier to understand many other things, including quantum noise, quantum error-correction, quantum entanglement, and quantum communication. Review: Outer product notation Let and be vectors. Define a linear operation (matrix) by Example: 1 0 0 1 1 1 Connection to matrices: If a j a j j , and b j bj j then a b k bk* a . a1 But a2 b1* b2* 1 bk* a . a1 Thus a b a2 b1* b2* b3* . As we remember, this is a matrix, we showed how to calculate it Outer product notation 1 1 Example: 0 0 1 0 0 0 0 0 Example: 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 Example: Z 0 1 1 0 Example: 0 1 0 1 0 0 0 0 Example: 1 0 1 0 1 1 0 1 Example: X 0 1 1 1 0 1 0 0 0 0 Exercise: Find an outer product representation for Y . Outer product notation One of the advantages of the outer product notation is that it provides a convenient tool with which to describe projectors, and thus quantum measurements. Recall: The projector P onto sp e1 , e2 acts as P e1 e2 e3 e1 e2 This gives us a simple explicit formula for P , since e1 e1 e2 e2 e1 e2 e3 e1 e2 More generally, the projector onto a subspace spanned by orthonormal vectors e1 ,..., em is given by P j ej ej . REMINDER: Ensemble point of view Imagine that a quantum system is in the state j with Probability of outcome k being in state j probability pj . We do a measurement described by projectors Pk . Probability of outcome k Pr k | state j pj k j Pk j pj k pj tr j j Pk k Probability of being in state j Probability of outcome k tr Pk where pj j j is the density matrix. j completely determines all measurement statistics. Qubit example REMINDER: calculate the density matrix Suppose 0 with probability 1. 1 1 0 Then 0 0 1 0 . 0 0 0 Suppose 1 with probability 1. 0 0 0 Then 1 1 0 1 . 1 0 1 0 i 1 Suppose with probability 1. 2 0 i 1 0 i 1 1 1 1 1 i Then i 1 i i 1 . 2 2 2 2 Conjugate and change kets to bras where pj j j is the density matrix. j Density matrix is a generalization of state Density matrix Qubit example: a measurement using density matrix Suppose 0 with probability p, and 1 with probability 1 p . Then p 0 0 1 p 1 1 Pr(0) 0 1 0 0 0 p p 1 p . 0 0 0 1 0 1 p Measurement in the 0 , 1 basis yields 0 1 p Pr 0 tr 0 0 1 0 0 0 1 p Similarly, Pr 1 1 p . p. Pr(1) Why work with density matrices? Answer: Simplicity! The quantum (mixed) state is: 0 with probability 0.1 1 with probability 0.1 0 1 2 0 1 2 0 i 1 2 0 i 1 2 with probability 0.15 with probability 0.15 We know the probabilities of states and we want to find or check the density matrix 21 0 1 0 2 with probability 0.25 with probability 0.25 Sum of these probabilities must be equal one Dynamics and the density matrix Suppose we have a quantum system in the state j with probability pj . The quantum system undergoes a dynamics described by the unitary matrix U . The quantum system is now in the state U j probability pj . with The initial density matrix is j pj j j . The final density matrix is ' j pjU j j U † . U † p U U j j j j . ' U U . † Initial density matrix Dynamics and the density matrix ' U U . † This way, we can calculate a new density matrix from old density matrix and unitary evolution matrix U This is analogous to calculate a new state from old state and unitary evolution matrix U. The new formalism is more powerful since it refers also to mixed states. S1 = U * S0 Single-qubit example: calculating new density matrix by operating with an inverter on old density matrix Suppose 0 with probability p, and 1 with probability 1 p . 0 p Then . 0 1 p 1 p Suppose an X gate is applied. Then ' X X 0 Suppose 0 and 1 with equal probabilities I “Completely mixed state” Then . 2 Suppose any unitary gate U is applied. I I Then ' U U † = . 2 2 0 . p 1 2 . How the density matrix changes during a measurement Worked Exercise : Suppose a measurement described by projectors Pk is performed on an ensemble giving rise to the density matrix . If the measurement gives result k show that the corresponding post-measurement density matrix is Pk Pk ' k . tr Pk Pk Characterizing the density matrix What class of matrices correspond to possible density matrices? Suppose j pj j j is a density matrix. Then tr( ) j pj tr j j j pj 1 For any vector a , a a j pj a j j a j pj a j 2 0 Trace of a density matrix is one Summary : tr =1 and is a positive matrix. Exercise : Given that tr =1 and is a positive matrix, show that there is some set of states j and probabilities pj such that = j pj j j . Summary of the ensemble point of view Definition: The density matrix for a system in state j with probability pj is pj j j . j Dynamics: ' U U † . Measurement: A measurement described by projectors Pk gives result k with probability tr Pk , and the postPk Pk ' measurement density matrix is k . tr Pk Pk Characterization: tr =1, and is a positive matrix. Conversely, given any matrix satisfying these properties, there exists a set of states j and probabilities pj such that = j pj j j . tr A j Ajj Problems to Solve 0 1 Examples: X tr X 0; 1 0 Cyclicity property: tr AB =tr BA . 1 0 I 0 1 tr I 2. Illustrate on matrices tr AB j AB jj jk Ajk Bkj jk Bkj Ajk k BA kk tr BA Exercise: Prove that tr(|aihb|) = hb|ai. Exercise: Suppose e1 ,..., ed space. Prove that I j ej † Illustrate on matrices is an orthonormal basis for state ej . Exercise: Prove that a b = b a .