Werner state

A Werner state

{{cite journal

|author=Reinhard F. Werner

|year=1989

|title=Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model

|journal=Physical Review A

|volume=40 |issue=8 |pages=4277–4281

|doi=10.1103/PhysRevA.40.4277

|bibcode=1989PhRvA..40.4277W

|pmid=9902666

}} is a {{nowrap| $d^2$ × $d^2$ }}-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form $U \otimes U$ . That is, it is a bipartite quantum state $\rho_{AB}$ that satisfies

: $\rho_{AB} = (U \otimes U) \rho_{AB} (U^\dagger \otimes U^\dagger)$

for all unitary operators U acting on d-dimensional Hilbert space. These states were first developed by Reinhard F. Werner in 1989.

General definition

Every Werner state $W_{AB}^{(p,d)}$ is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight $p \in [0,1]$ being the main parameter that defines the state, in addition to the dimension $d \geq 2$ :

: $W_{AB}^{(p,d)} = p \frac{2}{d(d+1)} P^\text{sym}_{AB} + (1-p) \frac{2}{d(d-1)} P^\text{as}_{AB},$

where

: $P^\text{sym}_{AB} = \frac{1}{2}(I_{AB}+F_{AB}),$

: $P^\text{as}_{AB} = \frac{1}{2}(I_{AB}-F_{AB}),$

are the projectors and

: $F_{AB} = \sum_{ij} |i\rangle \langle j|_A \otimes |j\rangle \langle i|_B$

is the permutation or flip operator that exchanges the two subsystems A and B.

Werner states are separable for p ≥ {{frac|1|2}} and entangled for p < {{frac|1|2}}. All entangled Werner states violate the PPT separability criterion, but for d ≥ 3 no Werner state violates the weaker reduction criterion. Werner states can be parametrized in different ways. One way of writing them is

: $\rho_{AB} = \frac{1}{d^2-d \alpha}(I_{AB} - \alpha F_{AB}),$

where the new parameter α varies between −1 and 1 and relates to p as

: $\alpha = ((1-2p)d+1)/(1-2p+d) .$

Two-qubit example

Two-qubit Werner states, corresponding to $d=2$ above, can be written explicitly in matrix form as $W_{AB}^{(p,2)} = \frac{p}{6} \begin{pmatrix}2 & 0 & 0 & 0 \\ 0&1 & 1 &0 \\0&1&1&0\\0&0&0&2\end{pmatrix}
+ \frac{(1-p)}{2}
\begin{pmatrix}0 & 0 & 0 & 0 \\ 0&1 & -1 &0 \\0&-1&1&0\\0&0&0&0\end{pmatrix}
= \begin{pmatrix}
\frac{p}{3} & 0 & 0 & 0 \\
0 & \frac{3-2p}{6} & \frac{-3+4p}{6} & 0 \\
0 & \frac{-3+4p}{6} & \frac{3-2p}{6} & 0\\
0 & 0 & 0 & \frac{p}{3}
\end{pmatrix}.$ Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state: $W_{AB}^{(\lambda,2)} =
\lambda |\Psi^-\rangle\!\langle\Psi^-| + \frac{1-\lambda}{4}I_{AB},
\qquad |\Psi^-\rangle\equiv \frac{1}{\sqrt2}(|01\rangle-|10\rangle),$ where $\lambda\in[-1/3,1]$ (or, confining oneself to positive values, $\lambda\in[0,1]$ ) is related to $p$ by $\lambda=(3-4p)/3$ . Then, two-qubit Werner states are separable for $\lambda \leq 1/3$ and entangled for $\lambda > 1/3$ .

Werner-Holevo channels

A Werner-Holevo quantum channel $\mathcal{W}_{A\rightarrow B}^{\left(
p,d\right) }$ with parameters $p\in\left[ 0,1\right]$ and integer $d\geq2$

is defined as

{{cite journal

|author=Reinhard F. Werner and Alexander S. Holevo

|year=2002

|title=Counterexample to an additivity conjecture for output purity of quantum channels

|journal=Journal of Mathematical Physics

|volume=43 |issue=9 |pages=4353–4357

|doi=10.1063/1.1498491

|arxiv=quant-ph/0203003

|bibcode=2002JMP....43.4353W

|s2cid=42832247

}}

{{cite journal

|year=2004

|title=Additivity of minimal entropy out- put for a class of covariant channels

|journal=unpublished

|arxiv=quant-ph/0410195

|bibcode=2004quant.ph.10195F

}}

{{cite journal

|author=Debbie Leung and William Matthews

|year=2015

|title=On the power of PPT-preserving and non-signalling codes

|journal=IEEE Transactions on Information Theory

|volume=61 |issue=8 |pages=4486–4499

|doi=10.1109/TIT.2015.2439953

|arxiv=1406.7142

|s2cid=14083225

}}

: $\mathcal{W}_{A\rightarrow B}^{\left( p,d\right) } = p
\mathcal{W}_{A\rightarrow B}^{\text{sym} }+\left( 1-p\right)\mathcal{W}_{A\rightarrow
B}^{\text{as} },$

where the quantum channels $\mathcal{W}_{A\rightarrow B}^{\text{sym} }$ and

$\mathcal{W}_{A\rightarrow B}^{\text{as} }$ are defined as

: $\mathcal{W}_{A\rightarrow B}^{\text{sym} }(X_{A}) =
\frac{1}{d+1}\left[\operatorname{Tr}[X_{A}]I_{B}+\operatorname{id}_{A\rightarrow B}
(T_{A}(X_{A}))\right],$

: $\mathcal{W}_{A\rightarrow B}^{\text{as} }(X_{A}) =
\frac{1}{d-1}\left[\operatorname{Tr}[X_{A}]I_{B}-\operatorname{id}_{A\rightarrow B}
(T_{A}(X_{A}))\right],$

and $T_{A}$ denotes the partial transpose map on system A. Note that the

Choi state of the Werner-Holevo channel $\mathcal{W}_{A\rightarrow B}^{p,d}$

is a Werner state:

: $\mathcal{W}_{A\rightarrow B}^{\left( p,d\right) }(\Phi_{RA})=p \frac{2}{d\left( d+1\right) }P_{RB}^{\text{sym}}+
\left(
1-p\right)\frac{2}{d\left( d-1\right) }P_{RB}^{\text{as}},$

where $\Phi_{RA} = \frac{1}{d} \sum_{i,j} |i\rangle \langle j|_R \otimes |i\rangle \langle j|_A$ .

Multipartite Werner states

Werner states can be generalized to the multipartite case.{{cite journal |last1=Eggeling |first1=Tilo |last2=Werner |first2=Reinhard |title=Separability properties of tripartite states with UxUxU-symmetry |journal=Physical Review A |date=2001 |volume=63 |page=042111 |doi=10.1103/PhysRevA.63.042111|arxiv=quant-ph/0010096|s2cid=119350302 }} An N-party Werner state is a state that is invariant under $U \otimes U \otimes \cdots \otimes U$ for any unitary U on a single subsystem. The Werner state is no longer described by a single parameter, but by N! − 1 parameters, and is a linear combination of the N! different permutations on N systems.

References

Category:Quantum states