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Results tagged with trace
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user 232314
Use this tag when having questions concerning expressions with the trace of a matrix/operator.
2
votes
Accepted
Trace of Dirac matrices
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In particular, since $\eta^{\mu\nu}$ is a number, you can pull it out of the trace operation. In the last step, we used that the trace of the $n\times n$ identity matrix is $n$. … Also note that Wikipedia contains some proofs of (trace) identities involving the gamma matrices. …
1
vote
Accepted
How do I trace out the second qubit to find the reduced density operator?
Regarding your results: Both coincide, since $\langle 0|1\rangle \in \mathbb{C}$.
Edit: In fact, the result is zero, because both states are orthogonal, which is also used in the calculation performed …
1
vote
Prove that the partial trace preserves density operators
mathrm{Tr}_{\mathrm{B}}(\rho) = \sum\limits_k \lambda_k \, \mathrm{Tr}_{\mathrm{B}}(|\lambda_k\rangle \langle \lambda_k|) \quad , $$
where the second equality follows from the linearity of the partial trace …
3
votes
Accepted
Proof of $\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=0$
\end{align}
The cyclic properties of the trace then eventually yield
$$\mathrm{Tr}(\gamma^{\mu}\,\gamma^{\nu}\,\gamma^5) = -\mathrm{Tr}(\gamma^{\mu}\,\gamma^{\nu}\,\gamma^5) \quad . $$ …
2
votes
Confusion about $\partial_\mu x^\mu = 4$
Note that $ \partial_{\mu} x^{\nu} = \frac{\partial x^{\nu}}{\partial x^{\mu}} = \delta^{\nu}_{\mu}$ and hence
$$\partial_{\mu} x^{\mu} = \frac{\partial x^{\mu}}{\partial x^{\mu}} =\frac{\partial x^0 …
5
votes
Accepted
How to derive the reduced density matrix in a mathematically precise and correct way
\tag{2}$$
For a density operator $\rho$ on $H$ we define the partial trace of $\rho$ as
$$\rho_A:=\mathrm{Tr_B}\,\rho := \sum\limits_{k \in K} \left(\mathbb I_A \otimes \langle \psi_k| \right) \rho\left … \end{align}
Here, $\{|\varphi_j\rangle\}_{j\in J}$ denotes an orthonormal basis in $H_A$, $\mathrm{Tr}^{(A)}$ the trace operation on $H_A$ and $O:= O_A\otimes \mathbb I_B$. …
2
votes
Accepted
Inequality on tensor product in trace
to show that
$$ N\, \mathrm{Tr}_{H} \rho \log \sigma = \mathrm{Tr}_{H_N} \rho_N\log \sigma_N\tag{3} \quad , $$
for $\rho$ and $\sigma$ density matrices, i.e. positive semi-definite operators of unit trace … \tag{5}$$
Finally, by applying the trace operation $\mathrm{Tr}_{H_N}$ on $(5)$, using the very basic properties of the trace on tensor product spaces and the fact that $\rho$ is of unit trace, we arrive …
6
votes
Accepted
Unitary evolution and von Neumann entropy
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The cyclic properties of the trace then yield the desired result, i.e. $S[\rho(t)]=S[\rho(0)]$. …