# Questions tagged [partial-trace]

For questions centred around or involving the notion of partial trace. The partial trace is a generalization of the notion of trace defined for multipartite systems.

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### If the partial traces $\rho_A,\rho_B$ are pure, does it imply that $\rho$ is a product state?

Suppose $\rho$ is some bipartite state such that the partial traces $\rho_A={\rm Tr}_B\rho$ and $\rho_B={\rm Tr}_A\rho$ are both pure. Does this necessarily imply that $\rho$ is a product state? This ...
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### Permutation invariant states have permutation invariant purifications - proof?

I don't remember where I came across the statement but I'm pretty sure it is true and am interested in understanding why it holds. For any $n-$ register state $\rho^n \in H^{\otimes n}$ that is ...
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### Does the entangibility of density operators rely on what component spaces are being specified?

Is the entangibility of density operators relied on what component spaces are being specified? More precisely, let $H$ be a Hilbert space, $\rho$ be a density operator on $H$. Suppose we were not ...
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### How do I trace out the second qubit to find the reduced density operator? [duplicate]

I'm doing an exercise to trace out the second qubit to find the reduced density operator for the first qubit: $tr_2|11\rangle\langle00| = |1\rangle\langle0|\langle0|1\rangle$ I'm just wondering if I ...
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### Calculating the entropy of a quantum state

Let $\rho_{AR}$ be some $d-$dimensional pure quantum state. Consider a channel $N_{A\rightarrow B}$ that outputs a constant state in $B$. We now consider the Stinespring dilation of this channel with ...
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### Partial Trace of Werner State

I am trying to trace out the second qubit of the Werner State: \begin{align} W &=\frac{1-s}{4}I_{4}+\frac{s}{2}(|00\rangle\langle{00}|+|11\rangle\langle11|+|11\rangle \langle00|+|00\rangle \langle ...
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### Positive semidefinite relationship after partial trace

Let $\rho_{ABC}$ and $\sigma_{C}$ be arbitrary quantum states and $\lambda\in \mathbb{R}$ be minimal such that $$\rho_{ABC}\leq \lambda \rho_{AB}\otimes\sigma_C$$ We assume there are no issues with ...
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Given two states $\rho_A, \sigma_A$, Uhlmann's theorem states that the fidelity between them is achieved in the following way $$F(\rho_A, \sigma_A) = \max_{U_{R'}}F(\rho_{AR'}, (I\otimes U_{R'})\... 2answers 145 views ### Is the trace distance between multipartite states invariant under permutations? Consider two multipartite states \rho_{A_1A_2..A_L} and \sigma_{A_1A_2..A_L} in \mathcal{H}_{A_1} \otimes\mathcal{H}_{A_2} \otimes...\mathcal{H}_{A_L} . For an arbitrary permutation \pi over \... 1answer 75 views ### How can we upper bound the norm of a partial trace? Suppose we have the normalised states |\phi_{1}\rangle,|\phi_{2}\rangle \in A \otimes B where A and B are d-dimensional complex vector spaces. Suppose |\langle\phi_{2}|\phi_{1}\rangle| < ... 1answer 72 views ### Trace distance of two classical-quantum state with hashing Let's say I have a classical-quantum(cq) state \rho_{XE}, where the classical part (X) is orthogonal. It's trace distance from another uniform density operator is defined to be:$$ \frac{1}{2}||\...
Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.} I know that the trace distance ...