Density Operators, Trace and Partial Trace

In summary: Your summary is excellent and very clear. In summary, the trace operation is used to calculate useful results such as the expectation value of an operator over a density operator and the norm of a density operator. It is also used to determine if a density operator represents a pure state or a mixed state. The trace does not change when the basis of the density operator is changed.
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skynelson
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I have some math questions about quantum theory that have been bugging me for a while, and I haven't found a suitable answer in my own resources. I'll start with the Trace operation.

Question A) My understanding is that if we take system A and perform the partial trace over system B, we essentially remove any dependence or correlation of the system A on system B. For example, I have heard reference in the literature to 'performing a partial trace over the environment' in order to essentially remove the influence of the environment on the system being analyzed.

Is this intuitive understanding correct?

Can anyone provide a more concrete understanding of why the trace is interesting? Since it is simply the sum of the diagonal elements, can you explain why it is interesting/useful in the following two cases, and what it represents?
1) A diagonalized matrix (the diagonals are the eigenvalues of the system)
2) A Hermitian but non-diagonalized matrix

Question B)
Furthermore, let's say we have a density operator, which may have non-diagonal elements, but has trace equal to 1 (because it is normalized). (Am I correct so far?)

1) What is the meaning of taking the trace, and disregarding the off-diagonal elements? Why are we allowed to disregard the rest of the matrix?
2) Will the trace change if we change the basis of the density operator?


You may find that my questions belie misunderstandings in the nature of some of these concepts, which is why I appreciate any help that can be provided!
 
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  • #2
skynelson said:
Question A) My understanding is that if we take system A and perform the partial trace over system B, we essentially remove any dependence or correlation of the system A on system B. For example, I have heard reference in the literature to 'performing a partial trace over the environment' in order to essentially remove the influence of the environment on the system being analyzed.

Is this intuitive understanding correct?
Yes.

skynelson said:
Can anyone provide a more concrete understanding of why the trace is interesting? Since it is simply the sum of the diagonal elements, can you explain why it is interesting/useful in the following two cases, and what it represents?
1) A diagonalized matrix (the diagonals are the eigenvalues of the system)
2) A Hermitian but non-diagonalized matrix
The trace is interesting because it is the way to calculate useful results. For instance, the expectation value of operator ##A## over the density operator ##\rho## is ##Tr(A \rho)##. It is also used to calculate the norm of a density operator, ##Tr(\rho)##, and to check if it represents a pure state, ##Tr(\rho^2) = 1##, or a mixed state, ##Tr(\rho^2) < 1##.

skynelson said:
Question B)
Furthermore, let's say we have a density operator, which may have non-diagonal elements, but has trace equal to 1 (because it is normalized). (Am I correct so far?)
Correct.

skynelson said:
1) What is the meaning of taking the trace, and disregarding the off-diagonal elements? Why are we allowed to disregard the rest of the matrix?
It gives the norm. It is not that the off-diagonal elements are disregarded, it is that they are not relevant to the norm.

skynelson said:
2) Will the trace change if we change the basis of the density operator?
No. Note that the above examples I gave are in terms of the density operator, without reference to a particular representation (which would give a density matrix).
 
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Thank you Dr. Claude!
 
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What is a density operator?

A density operator, also known as a density matrix, is a mathematical representation of the quantum state of a physical system. It is a matrix that describes the probability of finding a particle in each possible state.

How is a density operator related to trace?

The trace of a density operator is equal to the sum of the probabilities of all possible states, which is always equal to one. This is because the trace is a measure of the total probability of finding a particle in any state.

What is the significance of the trace of a density operator being equal to one?

The trace being equal to one means that the total probability of finding the particle in any state is 100%, which is necessary for a valid quantum state. It also allows for the calculation of expectation values and other important properties of the system.

What is a partial trace?

A partial trace is a mathematical operation that is used to trace out or "ignore" certain degrees of freedom in a larger system. It is commonly used when dealing with entangled states or systems with multiple particles.

How is a partial trace calculated?

The partial trace is calculated by taking the trace of the density operator of the larger system, after tracing out the degrees of freedom that are being ignored. It is important to note that the resulting density operator will be smaller in size, as it only describes the remaining degrees of freedom.

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