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Questions tagged [partial-transpose]

For questions about partial transpose, i.e. the transpose limited to a subsystem of a composite system.

Partial transpose swaps rows and columns in a factor within a tensor product. It plays a key role in Peres–Horodecki criterion for separability.

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

In the comments to a question I asked recently, there is a discussion between user1271772 and myself on positive operators. I know that for a positive trace-preserving operator $\Lambda$ (e.g. the partial transpose) if acting on a mixed state $\rho$…
Quantum spaghettification
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What are examples of zero capacity quantum channels with Choi rank less than $d$?

All the currently known examples of quantum channels with zero quantum capacity are either PPT or anti-degradable. These notions can be conveniently defined in terms of the Choi matrix of the given channel: A channel is said to be PPT if its Choi…
mathwizard
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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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Are inseparable states with positive partial transpose nonlocal?

In Horodecki, Horodecki and Horodecki (1998), Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?, the authors remark in the conclusions (beginning of pag. 4, emphasis mine): So, one is now faced with the problem…
glS
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Why does the partial transpose of an entangled state have at most one negative eigenvalue?

I came over this unclear claim which i wondered someone could clarify: "The partial transpose of an entangled state has at most one negative eigenvalue." I wondered if this holds for all states or just some, searched around but couldn’t find some…
Pink Elephants
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Understanding the classification of quantum states based on partial transposition: representations of the bipartite density matrix

I'm going through some slides on the PPT/NPT criteria along with Horodecki's paper, and I'm kind of stuck. Let's take this slide: Firstly, why can we write a bipartite density matrix as $\sum_{ijkl}\rho_{kl}^{ij}|i\rangle\langle j|\otimes…
Sanchayan Dutta
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Structural Physical Approximation of Partial Transpose

To make the partial transpose a complete positive and therefore physical map, one has to mix it with enough of the maximally mixed state to offset the negative eigenvalues. The most negative eigenvalue is obtained when partial transpose is applied…
Mahathi Vempati
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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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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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Are entanglement witnesses of this form optimal?

One can make an entanglement witness by taking the partial transpose of any pure entangled state. Consider $|\phi \rangle $ as any pure entangled state. Then $W = | \phi \rangle \langle \phi |^{T_2} $ is an entanglement witness. However, is there…
Mahathi Vempati
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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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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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Does a partial transpose always have real eigenvalues?

I am working with a tripartite system, but when I partially transpose the $8\times 8$ density matrix I get two complex eigenvalues. I know the criteria for the positive and negative eigenvalues, but are they always real? If they don't, what do my…
QuantumMiu
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Is there an identity for the partial transpose of a product of operators?

The partial transpose of an operator $M$ with respect to subsystem $A$ is given by $$ M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \langle d|}_B\right)= \left(\sum_{abcd} M^{ab}_{cd} …
FriendlyLagrangian
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How can I implement partial transpose on a variable in Picos (Python, trying to solve an SDP)?

I try to optimise a quantity via an SDP. I optimise over all PPT measurement operators and hence have the constraints $\Pi_k^{T_B} \succeq 0$ (PPT) for my measurement operators. The part of the code where I define the SDP and constraints…
root
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