lecturenotes13 - CSE 599d - Quantum Computing Mixed Quantum...

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Unformatted text preview: CSE 599d - Quantum Computing Mixed Quantum States and Open Quantum Systems Dave Bacon Department of Computer Science & Engineering, University of Washington So far we have been dealing with quantum states which are what are known as pure quantum states. Here pure refers to the fact that our description of the system is entirely quantum mechanical. But we saw when we were discussing our information processing machines, that there were these equally valid probabilistic machines that had their own equally valid formulation. Is there a way to include the latter within the confines of the former and in particular to mix quantum descriptions with classical descriptions? This leads us to what are known as mixed quantum states which we will discuss in this lecture. Another reason to care about mixed quantum states is to deal with the case where we have only part of a quantum system. Thus, for instance, we may have the entangle two qubit state 1 √ 2 ( | 00 i + | 11 i ). Now of course we can always figure out what quantum theory predict for our half of this quantum system by just acknowledging that this is the true description of the quantum system. But often it is convenient to discuss just one half of this quantum state, i.e. we are looking for a description of one of the two qubits for this state. It is important to realize that we must always use such descriptions appropriately: just because our description of one half is different than that of the whole does not mean that the state has changed in any way! It will turn out that the appropriate way to discuss quantum systems like this is to again consider mixed quantum states. I. MIXED STATES AND DENSITY OPERATORS Suppose I set up the following situation. With probability p I prepare a quantum system into a state with the description | ψ i and with probability 1- p I prepare a quantum system into a state with the description | ψ 1 i . Now of course we could always keep around the above words describing this situation. But this seems like a lot of work. Is there a better way to keep track of all future predictions we can make on such a system? More generally suppose that I prepare state | ψ i i with probability p i . Certainly I could keep those words as my description of the quantum system, but there is a better way. We call a setting where we prepare state | ψ i i with probability p i an ensemble of pure states. Such ensembles are described by density operator (often called a density operator.) In particular for the above ensemble, the density operator is given by ρ = X i p i | ψ i ih ψ i | (1) where p i ≥ 0 and ∑ i p i = 1. Notice that if we have a single pure state, | ψ i , then ρ = | ψ ih ψ | is just a projector onto this state. Further note that ρ is Hermitian and is also positive. Finally we note that the trace of ρ is unity: Tr[ ρ ] = X x h x | ρ | x i = X x h x | X i p i | ψ i ih ψ i || x i = X i p i X x |h ψ i | x i| 2 = X i p i = 1 (2) In fact we can show that any matrix which is positive (and hence necessarily hermitian) and has trace unity represents...
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This note was uploaded on 11/06/2011 for the course CSE 599 taught by Professor Staff during the Fall '08 term at University of Washington.

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lecturenotes13 - CSE 599d - Quantum Computing Mixed Quantum...

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