A Werner state[1] is a
×
-dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form
. That is, it is a bipartite quantum state
that satisfies
![{\displaystyle \rho _{AB}=(U\otimes U)\rho _{AB}(U^{\dagger }\otimes U^{\dagger })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae7d98afcbc2dfa60dcc7c3865e2e9f35d011ad)
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
is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight
being the main parameter that defines the state, in addition to the dimension
:
![{\displaystyle W_{AB}^{(p,d)}=p{\frac {2}{d(d+1)}}P_{AB}^{\text{sym}}+(1-p){\frac {2}{d(d-1)}}P_{AB}^{\text{as}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d81ccf5b0f9f85999601bb2b987473c581a997b4)
where
![{\displaystyle P_{AB}^{\text{sym}}={\frac {1}{2}}(I_{AB}+F_{AB}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/691dd2a0d8d17d80f1008e2739e79b46f7d4977a)
![{\displaystyle P_{AB}^{\text{as}}={\frac {1}{2}}(I_{AB}-F_{AB}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35f33c2ec7f489de31352f356375f31c5a9cba9c)
are the projectors and
![{\displaystyle F_{AB}=\sum _{ij}|i\rangle \langle j|_{A}\otimes |j\rangle \langle i|_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb4958313e2d7e9e0e3a74e6f1d4145b724fbe8)
is the permutation or flip operator that exchanges the two subsystems A and B.
Werner states are separable for p ≥ 1⁄2 and entangled for p < 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
![{\displaystyle \rho _{AB}={\frac {1}{d^{2}-d\alpha }}(I_{AB}-\alpha F_{AB}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4909286c86bb7c31143e38d44767a83b384d31)
where the new parameter α varies between −1 and 1 and relates to p as
![{\displaystyle \alpha =((1-2p)d+1)/(1-2p+d).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/100ec2befd2e8ffcd2c4030192be74080866416c)
Two-qubit example
Two-qubit Werner states, corresponding to
above, can be written explicitly in matrix form as
Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state:
where
(or, confining oneself to positive values,
) is related to
by
. Then, two-qubit Werner states are separable for
and entangled for
.
Werner-Holevo channels
A Werner-Holevo quantum channel
with parameters
and integer
is defined as [2] [3] [4]
![{\displaystyle {\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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b53e13657d5a6f9015ac2279cbec1e67d437109)
where the quantum channels
and
are defined as
![{\displaystyle {\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6a8ec3313c85a6eb18e6e158efa8f4a206ac84)
![{\displaystyle {\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36bc2b6db13bf41371b1fef8ec6f85108323c9df)
and
denotes the partial transpose map on system A. Note that the Choi state of the Werner-Holevo channel
is a Werner state:
![{\displaystyle {\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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/488b832abadbdb268a578fccb86ab3778e0f6eab)
where
.
Multipartite Werner states
Werner states can be generalized to the multipartite case.[5] An N-party Werner state is a state that is invariant under
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
- ^ Reinhard F. Werner (1989). "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A. 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/PhysRevA.40.4277. PMID 9902666.
- ^ Reinhard F. Werner and Alexander S. Holevo (2002). "Counterexample to an additivity conjecture for output purity of quantum channels". Journal of Mathematical Physics. 43 (9): 4353–4357. arXiv:quant-ph/0203003. Bibcode:2002JMP....43.4353W. doi:10.1063/1.1498491. S2CID 42832247.
- ^ Fannes, Mark; Haegeman, B.; Mosonyi, Milan; Vanpeteghem, D. (2004). "Additivity of minimal entropy out- put for a class of covariant channels". unpublished. arXiv:quant-ph/0410195. Bibcode:2004quant.ph.10195F.
- ^ Debbie Leung and William Matthews (2015). "On the power of PPT-preserving and non-signalling codes". IEEE Transactions on Information Theory. 61 (8): 4486–4499. arXiv:1406.7142. doi:10.1109/TIT.2015.2439953. S2CID 14083225.
- ^ Eggeling, Tilo; Werner, Reinhard (2001). "Separability properties of tripartite states with UxUxU-symmetry". Physical Review A. 63: 042111. arXiv:quant-ph/0010096. doi:10.1103/PhysRevA.63.042111. S2CID 119350302.