# Tag Info

Accepted

### Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

The equation at the top of the question is not correct: there is a missing factor of $1/d$ on the right-hand side. Let's eliminate this factor from the left-hand side to make it simpler, so that the ...
• 4,853
Accepted

### Positive semidefinite relationship after partial trace

No, not necessarily. For example, take $\rho$ to be a GHZ state and let $\sigma$ be the completely mixed state of one qubit. We then have $\lambda=4$ and $\mu=2$.
• 4,853
Accepted

### What's the 'physical consistency' in the partial trace scenario?

Measurement average Measurement average $\langle M \rangle_\rho$ of observable (a Hermitian operator) $M$ on the state $\rho$ is the average of measurement outcomes $m$ in the limit of infinite number ...
• 18.2k

### Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

Here the important fact is that the maximally mixed state is in fact an identity matrix. Let me rewrite the expression on the left in index notation (the summation sign is omitted according to the ...

• 13.1k

### What's the 'physical consistency' in the partial trace scenario?

The point of physical consistency is about how we can define the operator $M$ as acting on system $\rho^A$, or we can define the operator $M \otimes \mathbb{1}_B$ as acting on the system $\rho^{AB}$, ...
Accepted

### Can we express $\mathrm{tr}_A((A\otimes B)\rho_{AB})$ in terms of $A$, $B$, $\rho_A$ and $\rho_B$?

In general, the knowledge of the marginals $\rho_A$ and $\rho_B$ and the operators $A$ and $B$ is insufficient to compute $\mathrm{tr}_A((A\otimes B)\rho_{AB})$. Indeed, we can find two different ...
• 18.2k
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### Prove that a channel is close to acting on only one system

I suppose you're asking the following: for any $\epsilon\ge0$ and $\Phi$ that satisfy conditions, is there $\delta_\epsilon \rightarrow 0$ when $\epsilon \rightarrow 0$, such that there exists a ...
• 6,353
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• 51.1k
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### Partial trace and SWAP in the basis of subsystems

Using the fact $e^{i\theta S} = \text{cos}(\theta) \cdot I + i \cdot \text{sin}(\theta) \cdot S$, we calculate \begin{align*} e^{-i\theta S} (\rho \otimes \sigma) e^{i\theta S} & = (\text{cos}(...
• 1,366