# Tag Info

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 $\mu=2$.
• 4,598
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 ...
• 14.8k
Accepted

### Partial trace over a product of matrices - one factor is in tensor product form

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 ...
• 4,598

### Partial trace over a product of matrices - one factor is in tensor product form

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 ...

• 48.2k

### How many Kraus operators are required to characterise a channel with different start and end dimensions?

The answer is yes, you need $d_1 d_2$ operators, as already pointed out in the other answer. Here I'll show explicitly why this is the case. Let $\Phi\in\mathrm{T}(\mathcal X,\mathcal Y)$ be a CPTP ...
• 19.7k
Accepted

### How is the partial trace related to the operator sum representation?

I think it helps here to write things explicitly. Suppose $\mathcal E(\rho)=\operatorname{Tr}_E[U(\rho\otimes|\mathbf e_0\rangle\!\langle\mathbf e_0|)U^\dagger]$. Pick a basis for the environment in ...
• 19.7k
Accepted

### Does the trace distance between marginals bound the distance between the overall states?

No. Just take two Bell states. They have identical reduced density matrices yet are orthogonal, that is, as distant from each other as it gets.
• 5,107

• 4,252
Accepted

### Which simple property of partial trace are we using here?

mixed-product property of the Kronecker product: $(A \otimes B)(C \otimes D) = (AC \otimes BD)$ definition of the partial trace: $\text{Tr}_1(A\otimes B) = \text{Tr}(A)B$ cyclic property of the ...
• 6,013
### How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?