# Tag Info

## Hot answers tagged partial-transpose

8 votes

### Is acting with a positive map on a state not part of a larger system allowed?

Any map which is not Completely Positive, Trace Preserving (CPTP), is not possible as an "allowed operation" (a more-or-less complete account of how some system transforms) in quantum mechanics, ...
• 12.1k
5 votes
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• 5,907
5 votes
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### For 2x2 and 2x3 systems, is the partial transpose the only positive but not CP operation?

The partial transpose is not the only positive but not completely positive operation that is possible on 2x2 and 2x3 systems. Trivially, any completely positive operation (such as a local unitary) ...
• 58.1k
4 votes
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### Why is $\rho$ NPT if and only if $\rho^{\otimes N}$ is NPT?

The short answer is that $(\rho^{\otimes N})^{T_B}=(\rho^{T_B})^{\otimes N}$. More explicitly, if $\rho=\sum_{ii'jj'}\rho_{ii',jj'}|i\rangle\!\langle i'|\otimes |j\rangle\!\langle j'|$, then we can ...
• 24.9k
4 votes

• 7,289
3 votes

### Does a partial transpose always have real eigenvalues?

Yes, the partial transpose takes Hermitian matrices to Hermitian matrices and consequently the eigenvalues of the partial transpose of a Hermitian matrix are all real. Proof For concreteness, suppose ...
• 22.4k
3 votes
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### Is there an easy way to calculate the eigenvalues of the partial transpose of a given matrix?

In this specific case, absolutely! Note that $$|\psi\rangle=|\phi_A\rangle|\phi_B\rangle,$$ such that $$\rho=|\phi_A\rangle\langle\phi_A|\otimes |\phi_B\rangle\langle\phi_B|.$$ Now, it is the case ...
• 58.1k
3 votes
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### How is the expression $\frac{\|\rho^{T_B}\|-1}{2}$ obtained from the definition of negativity?

Firstly $$\mathrm{Tr}[\rho_{AB}^{T_B}] = 1$$ as the transpose map is trace-preserving. As the trace of $\rho_{AB}^{T_B}$ is equal to the sum of its eigenvalues we have $$\sum_i \lambda_i = 1$$ ...
• 5,773
3 votes
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### If $\rho_{AB}$ is a separable then the partial transpose w.r.t to A is PSD

The main result that you need to complete all the steps that you mention is that if $\rho$ is a density matrix, then $\rho^T$ is also a density matrix. So, what are the key properties of a density ...
• 58.1k
2 votes
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• 22.4k
2 votes

### Defining dimension of an operator in qutip

From the official documentation: Q.dims: List keeping track of shapes for individual components of a multipartite system (for tensor products and partial traces). ...
• 24.9k
1 vote
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### Why does the entanglement negativity equal (in magnitude) the sum of the negative eigenvalues?

This is also discussed in the paper linked above. The trace norm of $X$ is defined as the sum of the absolute values of the eigenvalues of $X$: $\|X\|_1=\sum_i \lvert\lambda_i\rvert$. \$\newcommand{\tr}...
• 24.9k

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