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University of California at San Diego – Department of Physics – TA: Shauna Kravec Quantum Mechanics C (Physics 130C) Winter 2014 Worksheet 9 – Solutions Please read and work on the following problems in groups of 3 to 4. Solutions will be posted on the course webpage. Announcements • The 130C web site is: http://physics.ucsd.edu/∼mcgreevy/w14/ . Please check it regularly! It contains relevant course information! • This week let’s think about some formal properties of density matrices! Problems 1. With All Your ρ’s Combined Quantum states are said to form a convex set with extremal points corresponding to pure states. What does that mean? (a) Prove that if ρ1 and ρ2 are density matrices then ρ = p ∗ ρ1 + (1 − p)ρ2 is as well where p is a probability. This implies that for every pair of points in the set the straight line that connects them is also contained in the object. For example a disk is a convex set but an annulus is not. We require ρ = ρ† , which is true by inspection, and both Tr [ρ] = 1 andTr [ρ2 ] ≤ 1 Tr [ρ] = pTr [ρ1 ] + (1 − p)Tr [ρ2 ] = p + (1 − p) = 1 It follows from the metric on m × n matrices and the Cauchy-Schwartz inequality that if A and B are positive semi-definite matrices (such as density matrices) that: 0 ≤ Tr [AB]2 ≤ Tr [A2 ]Tr [B 2 ] ≤ Tr [A]2 Tr [B]2 which for A = ρ and B = |ψihψ| implies: 0 ≤ Tr [ρ2 ] ∗ 1 ≤ Tr [ρ]2 ∗ 1 = 1 which proves the result. (b) A point is said to be extremal if it can’t be written as a (non-trivial) linear combination of other states. 1 Prove the claim that pure states are extremely in this set. P P Consider a state ρ and its spectral decomposition ρ = i pi |ψi ihψi | with i pi = 1 If this decomposition, itself a linear combination, is to be trivial then it only includes a single term: ρ = |ψihψ| which is pure 2. Scrambled ρ’s Suppose you have an operation which takes ρ → p(U1 ρU1† ) + (1 − p)(U2 ρU2† ) For some arbitrary unitary operators U1 , U2 where p again is a probability. (a) Show that there exists at least one pure state mapped to another pure state by the operation above. (Hint: Consider |ψi an eigenvector of U2† U1 ) Consider |ψi an eigenvector of U2† U1 (exists as this operator is unitary) with eigenvalue eiθ by unitarity. Consider the above operation acting on ρ = |ψihψ| We can relate: U2 |ψihψ|U2† = U2 (e−iθ U2† U1 |ψi)(hψ|U1† U2 eiθ )U2† = U1 ρU1† Using the above result we substitute: p(U1 ρU1† ) + (1 − p)(U2 ρU2† ) = p(U1 ρU1† ) + (1 − p)(U1 ρU1† ) = U1 ρU1† = |ψ 0 ihψ 0 | where |ψ 0 i = U1 |ψi proving the result P P † (b) Suppose we turn up the scrambling to ρ → m k=1 pk Uk ρUk where k pk = 1 Where this operation takes all density matrices to the maximally mixed state 21 1 Show that there must be at least 4 unitaries in the set {Uk } exactly if pk = 14 (Hint: Rule out the case of 3 unitary operators by contradiction with the previous result. Why can rule out the possibilities of 1 and 2 operators immediately?) We can rule out just one unitary trivially and two unitaries by the result above (we explicitly showed this operator takes a pure state to a pure state) so we should rule out the case of three unitary operators with the above result. Suppose we could construct this operation with three unitaries. This implies: P † † 1 = 1 k Uk ρUk and note that by U3 1U3 = 1 we can write WLOG 2 = p1 U3† U1 ρU1† U3 + p2 U3† U2 ρU2† U3 + p3 ρ Using the above result we can choose a pure input |ψihψ| such that the terms involving U1 and U2 map to a different pure state |ψ 0 ihψ 0 | letting us rewrite: 1 1 = (1 − p3 )|ψ 0 ihψ 0 | + p3 |ψihψ| which is solved by p3 = 12 2 But note there was nothing special about our choice of U3 to conjugate by. We could just as easily repeat this procedure for U1 and U2 to conclude that p1 = 12 = p2 as well which contradicts the need they sum to 1 Following this method for 4 unitaries leaves you with an extra term which allows one to explicitly construct the operator needed. 1 1 2 2