Tensor Products: Showing Positivity of $\rho_{AB}(t)$

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B(0). Since 1_B is the identity operator, it will not change the positivity of \rho_B(0). This means that the second part of the equation is also a positive operator.Since we have shown that both parts of the expanded equation are positive operators, then the entire equation \rho_{AB}(t) = \Gamma \rho_A(0) \otimes 1_B \rho_B(0) must also be a positive operator. This shows that \rho_{AB}(t) is a positive map, as desired.In summary, we have shown that if \rho_{AB}(0) is a positive map, then \rho_{AB}(t) = \Gamma\otimes
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Kreizhn
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Homework Statement



Given that [itex] \Gamma(\rho_A) = \displaystyle \sum_k A_k \rho_A(0) A_k^\dagger [/itex], I need to show that if [itex]\rho_{AB} (0)[/itex] is a positive map, then [itex] \rho_{AB}(t) = \Gamma\otimes 1_B ( \rho_{AB}(0) ) [/itex] is also a positive map.

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The Attempt at a Solution


This doesn't seem like it should be very hard, I'm just not very comfortable with tensor products. I'm not sure how this should be expanded out, and so if anyone could help, it would be appreciated.
 
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Sure, let's break it down step by step. First, let's define what a positive map is. A positive map is a linear map that maps positive operators to positive operators. In other words, if we have an operator A that is positive, then the map will take A to another operator B that is also positive.

Now, let's look at the given equation: \Gamma(\rho_A) = \displaystyle \sum_k A_k \rho_A(0) A_k^\dagger. This is known as the Lindblad master equation, and it represents the time evolution of a quantum system. In this equation, \rho_A(0) is the initial density matrix of the system, and A_k are a set of operators that represent the physical processes that can occur in the system.

Next, we have the equation \rho_{AB}(t) = \Gamma\otimes 1_B ( \rho_{AB}(0) ), which is the time evolution of a bipartite system, where \rho_{AB}(0) is the initial density matrix of the system and 1_B is the identity operator on system B.

To show that this is a positive map, we need to show that if \rho_{AB}(0) is a positive operator, then \rho_{AB}(t) will also be a positive operator. Let's break down the equation to see how this can be shown.

First, we can rewrite the equation as \rho_{AB}(t) = (\Gamma \otimes 1_B) \rho_{AB}(0). This means that we are applying the map \Gamma \otimes 1_B to the initial density matrix \rho_{AB}(0).

Next, we can expand the tensor product to get \rho_{AB}(t) = \Gamma \rho_A(0) \otimes 1_B \rho_B(0), where \rho_A(0) and \rho_B(0) are the initial density matrices of systems A and B, respectively.

Now, let's focus on the map \Gamma \rho_A(0). Since \Gamma is a positive map, and \rho_A(0) is a positive operator, then \Gamma \rho_A(0) will also be a positive operator. This means that the first part of our expanded equation is a positive operator.

Next, let's look at the second part of the equation, 1_B
 

Related to Tensor Products: Showing Positivity of $\rho_{AB}(t)$

What is a tensor product?

A tensor product is a mathematical operation that combines two objects to create a new object. In the context of quantum mechanics, it is used to describe the state of a composite system that is made up of two or more subsystems.

How is a tensor product used in quantum mechanics?

In quantum mechanics, the tensor product is used to describe the state of a composite system, such as two quantum particles, by combining the states of each individual subsystem. This allows for a more comprehensive understanding of the behavior of the composite system.

What is the significance of showing positivity of $\rho_{AB}(t)$?

Positivity of $\rho_{AB}(t)$ is important as it ensures that the state of the composite system is physically valid and consistent with the principles of quantum mechanics. It also allows for the calculation of meaningful quantities, such as probabilities and expectation values.

How is positivity of $\rho_{AB}(t)$ shown?

To show positivity of $\rho_{AB}(t)$, one must demonstrate that the density matrix $\rho_{AB}(t)$ satisfies certain conditions, such as being Hermitian and having non-negative eigenvalues. This can be done through mathematical proofs and calculations.

What are some applications of tensor products in quantum mechanics?

Tensor products have various applications in quantum mechanics, including in the description of entanglement, quantum information processing, and the study of many-particle systems. They are also used in the development of quantum algorithms and in quantum computing.

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