Density matrix and von Neumann entropy - why does basis matter?

In summary, the conversation discusses the computation of the von Neumann entropy for a pure state, specifically the state |psi⟩=(|0⟩+|1⟩)/√2. The confusion arises when evaluating the entropy in different bases, with the assumption that the logarithm of a matrix is performed elementwise. However, this is only true for diagonal matrices, not general ones. Thus, the error in the reasoning is assuming the same result in both bases. The issue is resolved by realizing that the logarithm cannot be performed elementwise and using the correct calculation.
  • #1
Eigenentity
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Density matrix and von Neumann entropy -- why does basis matter?

I'm very confused by why I'm unable to correctly compute the von Neumann entropy

[tex]S = - \mathrm{Tr}(\rho \log_2{\rho})[/tex]

for the pure state

[tex]| \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2[/tex]

Now, clearly the simplest thing to do is to express [tex]|\psi\rangle[/tex] in the [tex]|+\rangle,|-\rangle[/tex] basis, where it's clear that

[tex]\rho = |+\rangle\langle+|[/tex]

In this basis, [tex]S = 0[/tex] as we'd expect for a pure state.

What I can't fathom (and I'm sure I'm missing something really obvious) is why if I evaluate the entropy in the [tex]|0\rangle,|1\rangle[/tex] basis,

[tex]\rho = \begin{pmatrix}
1/2 & 1/2 \\
1/2 & 1/2
\end{pmatrix}[/tex]

and thus

[tex]
S = - \mathrm{Tr} \left( \begin{pmatrix}
1/2 & 1/2 \\
1/2 & 1/2
\end{pmatrix} \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix} \right) = - \mathrm{Tr} \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix} = 2
[/tex]

Do I have to use the diagonal basis!? If not, what idiotic mistake am I making?
 
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  • #2


To anyone wondering. No, basis does not matter. The error in the above reasoning is the assumption that the logarithm of a matrix is performed elementwise. This is true for diagonal matrices, but not general ones.

More concretely:

[tex]\log_2 \begin{pmatrix}
1/2 & 1/2 \\
1/2 & 1/2
\end{pmatrix} \neq \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix}
[/tex]

Gah! Well, glad that's sorted.
 
Last edited:

1. What is a density matrix and how is it related to von Neumann entropy?

A density matrix is a mathematical representation of a quantum state in a mixed state, meaning it contains both classical and quantum information. It is related to von Neumann entropy, which measures the amount of uncertainty or randomness in a quantum system. The density matrix is used to calculate the von Neumann entropy, which can provide information about the purity of the quantum state.

2. How does the basis affect the density matrix and von Neumann entropy?

The basis refers to the set of vectors used to represent a quantum state. The density matrix and von Neumann entropy are dependent on the basis chosen, as different bases can result in different representations of the same quantum state. This means that the density matrix and von Neumann entropy will also vary depending on the basis used.

3. Why does the basis matter in the calculation of von Neumann entropy?

The basis matters because it determines the set of observables that can be measured in a quantum system. The von Neumann entropy is calculated based on the measurements of these observables, so a different basis can result in different measurements and therefore a different value for the von Neumann entropy.

4. How does the choice of basis affect the information obtained from the density matrix and von Neumann entropy?

The choice of basis can greatly impact the information obtained from the density matrix and von Neumann entropy. Different bases can reveal different aspects of a quantum system, so choosing the appropriate basis is important for obtaining accurate and relevant information.

5. Can the density matrix and von Neumann entropy be used to compare different quantum systems?

Yes, the density matrix and von Neumann entropy can be used to compare different quantum systems. By calculating the von Neumann entropy for each system, it is possible to compare the amount of uncertainty or randomness present in each system. This can provide insights into the similarities and differences between the systems, and can also be used to quantify the entanglement between them.

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