Questions tagged [partial-transpose]
For questions about partial transpose, i.e. the transpose limited to a subsystem of a composite system.
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General Bell state expression: What condition for mixture of Bell states to be entangled?
Convention: $|qubit_{A}, qubit_{B}\rangle$
The general Bell state equation: $|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k\oplus b\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\...
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Finding entanglement in matrix that is a sum of 4 Bell states
A general Bell state:
$|\beta(a,b)\rangle = \frac{1}{\sqrt{2}}[|0,0 \oplus b\rangle + (-1)^{a}|1,1 \oplus b \rangle]$
$|\beta(0,0)\rangle = \frac{1}{2}[|00\rangle \langle 00| + |00\rangle \langle 11| +...
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how to obtain partial transpose of a Tripartite operator?
i know for a bipartite system with elements
|ij><kl|
elements of its partial transpose are
|kj><il|
now suppose a ...
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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 ...
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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
$$
M^{T_A} := \left(\sum_{abcd} M^{ab}_{cd} \underbrace{|a\rangle \langle b| }_{A}\otimes \underbrace{|c \rangle \...
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What are examples of zero capacity quantum channels with Choi rank less than $d$?
All the currently known examples of quantum channels with zero quantum capacity are either PPT or anti-degradable. These notions can be conveniently defined in terms of the Choi matrix of the given ...
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Is there an easy way to calculate the eigenvalues of the partial transpose of a given matrix? [duplicate]
Consider the state
$$|\psi\rangle=(\cos\theta_A|0\rangle+\sin\theta_A|1\rangle)\otimes(\cos\theta_B|0\rangle+e^{i\phi_B}\sin\theta_B|1\rangle).$$
To calculate the $\rho^{T_B}$ I first calculate the $\...
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How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?
How can one argue that the partial transpose $\rho^{T_B}$ of a general separable state is positive?
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How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?
How can I calculate the eigenvalues of $\rho^{T_{B}}$ (PPT) of the following state
$$
\rho =\frac{1}{2}|0\rangle\langle0|\otimes|+\rangle\langle+|+\frac{1}{2}|+\rangle\langle+|\otimes|1\rangle\langle1|...
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In a bipartite system $AB$, why does the entanglement negativity $\mathcal{N}(\rho^{T_A})$ measure the entanglement between $A$ and $B$?
Consider a system composed of two subsystems $A$ and $B$ living in $\mathcal{H}=\mathcal{H}_A \otimes \mathcal{H}_B$. The density matrix of the system $AB$ is defined to be $\rho$. The entanglement ...
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Defining dimension of an operator in qutip
My main question: Can someone please explain to me how the list of array is used to define the dimension in qutip?
Context:
If I have my density operator ...
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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 ...
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Compute the negativity of maximally entangled bipartite states
The entanglement negativity $\mathcal N(\rho)$ of a (bipartite) state $\rho$ is defined as the absolute value of the sum of the negative eigenvalues of the partial transpose of a state, or ...
glS♦
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Why does the entanglement negativity equal (in magnitude) the sum of the negative eigenvalues?
The entanglement negativity, introduced in (Vidal and Werner 2002), is defined as
$$\mathcal N(\rho) \equiv \frac{\|\rho^{T_B}\|_1-1}{2}.$$
It is mentioned there that this equals the sum of the ...
glS♦
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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 ...
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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 $\rho^{\otimes N}$ ...
glS♦
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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, ...
glS♦
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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_{...
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Equivalent determinant condition for Peres-Horodecki criteria
The Peres-Horodecki criteria for a 2*2 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 (...
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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 ...
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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} ...
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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: The criteria for detecting if a state $\rho$ is ...
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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 ...
