Expected Value Partial Trance Density Matrix

[VVS]

Hey I am currently studying Quantum Mechanics and I have difficulty grasping a concept.

I don't understand the following step in the derivation:

$\langle X_{A} \rangle=tr\left[\left(X_{A}\otimes I_{B}\right)\rho_{AB}\right]$
$=tr_{A}\left[X_{A} tr_{B}\left[\rho_{AB}\right]\right]$

Thanks

 

[SteamKing]

I believe 'tr' stands for 'trace', not 'trance'.

 

[VVS]

yes of course

 

[WWGD]

I believe 'tr' stands for 'trace', not 'trance'.

Note if you're computing it while listening to techno music :).

 

[blue_raver22]

for your question! The concept of expected value and partial trace in quantum mechanics can be challenging to understand, but with some explanation, I hope it will become clearer for you.

First, let's define the terms in this equation. The expected value, denoted by \langle X_{A} \rangle, is a measure of the average value of a certain observable, X_{A}, in a quantum system. It represents the most probable outcome of measuring the observable in that system.

The partial trace, denoted by tr_{A}, is a mathematical operation that allows us to extract information about a subsystem in a larger composite system. In this case, we are interested in the subsystem A in the composite system AB.

Now, let's break down the equation step by step. The first step is to take the trace of the operator X_{A}\otimes I_{B} multiplied by the density matrix \rho_{AB}. This is equivalent to summing over all possible states of the composite system AB and calculating the expectation value of the observable X_{A} in each state. This is why we use the trace operation, as it is a way to calculate the expectation value in quantum mechanics.

The second step involves taking the partial trace of the composite system AB with respect to subsystem B. This means we are tracing out all the states of subsystem B and only keeping the information about subsystem A. This is why we use the subscript A for the trace operation.

Putting these steps together, we get the final equation which tells us that the expected value of observable X_{A} is equal to the trace of the observable X_{A} in the subsystem A, after tracing out the subsystem B.

I hope this explanation helps clarify the concept for you. Remember, quantum mechanics can be complex, but with practice and patience, you will grasp it. Keep studying and asking questions!