For questions about the positive partial transpose (PPT) criterion, or more generally about PPT states.
Questions tagged [ppt-criterion]
13 questions
11
votes
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Estimate/determine Bures separability probabilities making use of corresponding Hilbert-Schmidt probabilities
For two-qubit states, represented by a $4\times 4$ density matrix, the generic state is described by 15 real parameters. For ease of calculation, it can help to consider restricted families of states, such as the "$X$"-states, where any matrix…
Paul B. Slater
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8
votes
2 answers
What is a separable decomposition for the Werner state?
Consider the two-qubit Werner state, defined as
$$\rho_z = z |\Psi_-\rangle\!\langle \Psi_-| + \frac{1-z}{4}I,
\quad |\Psi_-\rangle\equiv\frac{1}{\sqrt2}(|00\rangle-|11\rangle),$$
for $z\ge0$.
Using the PPT criterion, one can see that this state is…
glS
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7
votes
1 answer
Equivalent determinant condition for Peres-Horodecki criteria
The Peres-Horodecki criteria for a two-qubit state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable.
According to this paper (arXiv) page 4, left side, the following is…
Mahathi Vempati
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5
votes
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What proportions of certain sets of PPT-two-retrit states are bound entangled or separable?
For two particular (twelve-and thirteen-dimensional) sets of two-retrit states (corresponding to 9 x 9 density matrices with real off-diagonal entries), I have been able to calculate the Hilbert-Schmidt probabilities that members of the sets have…
Paul B. Slater
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5
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Finding a class $C$ of bipartite PPT states such that entanglement of $\rho \in C$ implies entanglement of $\rho + \rho^{\Gamma}$
Consider an entangled bipartite quantum state $\rho \in \mathcal{M}_d(\mathbb{C}) \otimes \mathcal{M}_{d'}(\mathbb{C})$ which is positive under partial transposition, i.e., $\rho^\Gamma \geq 0$. As separability of $\rho$ is equivalent to…
mathwizard
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4
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Why is $\rho$ NPT if and only if $\rho^{\otimes N}$ is NPT?
In Horodecki et al. (1998), to prove that distillability implies having a negative partial transpose (being NPT). The authors use the fact that "a state $\rho$ is NPT if and only if $\rho^{\otimes N}$ is".
A state $\rho$ being "NPT" means here that…
glS
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4
votes
1 answer
How are witness operators physically implemented?
Let's take an example of an entanglement witness of the form $W = | \phi \rangle \langle \phi | ^{T_2}$ where $ | \phi \rangle $ is some pure entangled state.
If I wanted to test some state $\rho$, I would have to perform $\mathrm{Tr}(W \rho)$. I…
Mahathi Vempati
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4
votes
0 answers
Finding separable decompositions of bipartite X-states using the methodology of Li and Qiao
Two recent papers of Jun-Li Li and Cong-Feng Qiao (arXiv:1607.03364 and arXiv:1708.05336) present "practical schemes for the decomposition of a bipartite mixed state into a sum of direct products of local density matrices".
I would like to know if…
Paul B. Slater
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4
votes
1 answer
For 2x2 and 2x3 systems, is the partial transpose the only positive but not CP operation?
Question: For 2x2 and 2x3 systems, is the partial transpose the only positive but not completely positive operation that is possible?
Why this came up: Entanglement detection. A state $\rho$ is separable if and only if $(I \otimes \Lambda ) \rho…
Mahathi Vempati
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4
votes
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How to show that Werner states produce correlations explainable via local hidden variable models?
Werner states can be written as
$$\rho_W=
p\frac{\Pi_+}{\binom{n+1}{2}}
+(1-p)\frac{\Pi_-}{\binom{n}{2}},
$$
with $\Pi_\pm\equiv\frac12(I\pm\mathrm{SWAP})$ projectors onto the $\pm1$ eigenspaces of the swap operator, defined as the one acting on…
glS
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3
votes
1 answer
What are the ranges of the four $q$ parameters in the magic simplex of Bell states formula?
Equation (7) in the 2012 paper, "Complementarity Reveals Bound Entanglement of Two Twisted Photons" of B. C. Hiesmayr and W. Löffler for a state $\rho_d$ in the "magic simplex" of Bell states
\begin{equation}
\rho_d= \frac{q_4 (1-\delta (d-3)) \sum…
Paul B. Slater
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3
votes
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Can an entangled symmetric mixed state be invariant to partial transpose?
Suppose we have a mixed state $\rho$ over $\mathcal H_A\otimes \mathcal H_B$, where $\mathcal H_A=\mathcal H_B=\mathbb{C}^d$ is a finite dimensional Hilbert space. Can $\rho$ be entangled if it is symmetric ($\mathsf{Swap}_{AB}\rho=\rho$) and…
FlamtapShuckle
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votes
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Are there unextendable product sets in $\mathbb{C}^2\otimes\mathbb{C}^2$?
Following the notation in Watrous' book, page 353, an unextendable product set in a bipartite space $\mathcal X\otimes \mathcal Y$ is a set of orthonormal product vectors of the form
$$\mathcal A\equiv \{u_1\otimes v_1,...,u_m\otimes v_m\},$$
with…
glS
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