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Use this tag when having questions concerning expressions with the trace of a matrix/operator.

2 votes
Accepted

Trace of Dirac matrices

.$$ 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. …
Tobias Fünke's user avatar
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 …
Tobias Fünke's user avatar
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
Tobias Fünke's user avatar
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 $$\mathrm{Tr}(\gamma^{\mu}\,\gamma^{\nu}\,\gamma^5) = -\mathrm{Tr}(\gamma^{\mu}\,\gamma^{\nu}\,\gamma^5) \quad . $$ …
Tobias Fünke's user avatar
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 …
Tobias Fünke's user avatar
5 votes
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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 $O:= O_A\otimes \mathbb I_B$. …
Tobias Fünke's user avatar
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 …
Tobias Fünke's user avatar
6 votes
Accepted

Unitary evolution and von Neumann entropy

.$$ The cyclic properties of the trace then yield the desired result, i.e. $S[\rho(t)]=S[\rho(0)]$. …
Tobias Fünke's user avatar