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, ...
5 votes
Accepted

Is there an identity for the partial transpose of a product of operators?

Your suspicion is correct, even when $A=B$. Consider the Hilbert space of two qubits and let $^{T_A}$ denote the partial transpose with respect to one of them. Suppose that $$ A=B=\begin{pmatrix} 1 &...
  • 14.8k
5 votes
Accepted

Are inseparable states with positive partial transpose nonlocal?

This question was solved in 2014 by Vértesi and Brunner: they found a quantum state with positive partial transposition that violated a Bell inequality. The conjecture that all states with positive ...
5 votes
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Understanding the classification of quantum states based on partial transposition: representations of the bipartite density matrix

For any orthonormal basis that you pick, call it $|e_i\rangle$, you can write a matrix in terms of that basis as $$ \rho=\sum_{i,j}\rho_{i,j}|e_i\rangle\langle e_j|. $$ When you're talking about a ...
  • 48.1k
5 votes
Accepted

Are entanglement witnesses of this form optimal?

Answer to edited question: It's still not true for qubit systems. Consider these two unit vectors, both of which are entangled: $$ |\phi\rangle = \frac{1}{\sqrt{2}} | 00\rangle + \frac{1}{\sqrt{2}} | ...
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5 votes
Accepted

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) ...
  • 48.1k
4 votes
Accepted

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 ...
  • 19.6k
4 votes

Equivalent determinant condition for Peres-Horodecki criteria

This is called Sylvester's Criterion. There's plenty of information available once you have the name. The linked wikipedia article contains a proof. Strictly, Sylvester's Criterion requires that $W_2,...
  • 48.1k
4 votes

Structural Physical Approximation of Partial Transpose

In the paper that you refer to, they are essentially asking "when can we implement the partial transpose map $\Theta=I_2\otimes\Lambda$?". So, that means the SPA of this map must be positive. What you ...
  • 48.1k
4 votes

In a bipartite system $AB$, why does the entanglement negativity $\mathcal{N}(\rho^{T_A})$ measure the entanglement between $A$ and $B$?

There is no good definition of what is an "amount of entanglement". We have some requirements, such as saying that a measure of entanglement must be convex and cannot increase under local ...
3 votes

Does a partial transpose always have real eigenvalues?

Any matrix $M$ in the tensor product of two spaces can be written as $$ M = \sum_{ijkl} a_{ij,kl} E_{ij} \otimes E_{kl}, $$ where $E_{ij}$ are matrix units and $a_{ij,kl}$ are entries of the matrix $...
  • 6,013
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 ...
  • 14.8k
3 votes
Accepted

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 ...
  • 48.1k
2 votes
Accepted

how to obtain partial transpose of a Tripartite operator?

This is basically up to you: which elements are you transposing? If you're talking about transposing just the third system, then you'd be talking about $$ |abc\rangle\langle xyz|\mapsto |abz\rangle\...
  • 48.1k
2 votes
Accepted

How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?

First observe that if $R$ and $S$ are positive (aka positive semi-definite) operators and $q\in [0, +\infty)$ then $R^T$, $qR$, $R\otimes S$ and $R+S$ are all positive operators. By definition, if $\...
  • 14.8k
2 votes
Accepted

How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?

First note that $$ \begin{align} \rho^{T_B} &= \frac{1}{2}|0\rangle\langle0|\otimes(|+\rangle\langle+|)^T+\frac{1}{2}|+\rangle\langle+|\otimes(|1\rangle\langle1|)^T \\ &= \frac{1}{2}|0\rangle\...
  • 14.8k
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). ...
  • 19.6k
1 vote
Accepted

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}...
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