Entanglement.StaticEntTheory¶
- Entanglement.StaticEntTheory.Eeta(rho, varargin)¶
- \[E_{\eta}(\rho_{AB}) = \max - \log\|Y_{AB}^{T_{B}}\|_{\infty} \ s.t. \ -Y_{AB} \leq P^{T_{B}} \leq Y_{AB}\]
where \(P\) is the projection onto \(\operatorname{supp}(\rho_{AB})\).
- Parameters:¶
rho (
numeric) – The density matrix of the bipartite state.varargin (
numeric) – The array storing dimensions of subsystems A and B.
- Returns:¶
\(\eta\)-entanglement of \(\rho_{AB}\).
- Return type:¶
numeric
- Raises:¶
error– If either input/output dimension does not match, an error is raised.
Note
Wang, X., & Duan, R. (2017). Irreversibility of asymptotic entanglement manipulation under quantum operations completely preserving positivity of partial transpose. Physical Review Letters, 119(18), 180506.
- Entanglement.StaticEntTheory.LogFidBiNeg(rho, varargin)¶
- \[E^{1/2}_{\operatorname{N},2}(\rho_{AB}) = \log\max F(\rho, \sigma) \ s.t. \ \sigma \in \operatorname{PPT}_2(A:B)\]
- Parameters:¶
rho (
numeric) – The density matrix of the bipartite state.varargin (
numeric) – The array storing dimensions of subsystems A and B.
- Returns:¶
The logarithmic fidelity of bi-negativity of \(\rho_{AB}\).
- Return type:¶
numeric
- Raises:¶
error– If either input/output dimension does not match, an error is raised.
Note
Wang, X., Jing, M., & Zhu, C. (2023). Computable and Faithful Lower Bound for Entanglement Cost. arXiv preprint arXiv:2311.10649.
- Entanglement.StaticEntTheory.LogFidPPT(rho, varargin)¶
- \[E_{\operatorname{PPT}}(\rho_{AB}) = \log\max F(\rho, \sigma) \ s.t. \ \sigma \in \operatorname{PPT}(A:B)\]
- Parameters:¶
rho (
numeric) – The density matrix of the bipartite state.varargin (
numeric) – The array storing dimensions of subsystems A and B.
- Returns:¶
The PPT-entanglement of \(\rho_{AB}\).
- Return type:¶
numeric
- Raises:¶
error– If either input/output dimension does not match, an error is raised.
Note
Wang, X., & Wilde, M. M. (2023). Exact entanglement cost of quantum states and channels under positive-partial-transpose-preserving operations. Physical Review A, 107(1), 012429.
- Entanglement.StaticEntTheory.LogNeg(rho, varargin)¶
- \[E_{\operatorname{N}}(\rho_{AB}) = \log\|\rho^{T_{B}}_{AB}\|_1\]
- Parameters:¶
rho (
numeric) – The density matrix of the bipartite state.varargin (
numeric) – The array storing dimensions of subsystems A and B.
- Returns:¶
The logarithmic negativity of \(\rho_{AB}\).
- Return type:¶
numeric
- Raises:¶
error– If either input/output dimension does not match, an error is raised.
Note
Plenio, M. B. (2005). Logarithmic negativity: a full entanglement monotone that is not convex. Physical review letters, 95(9), 090503.
- Entanglement.StaticEntTheory.MaxRains(rho, varargin)¶
- \[E_{\operatorname{W}}(\rho_{AB}) = \log\max\operatorname{Tr}[\rho^{T_B}_{AB} R_{AB}], \ s.t.\ \|R_{AB}\|_{\infty} \leq 1, R^{T_{B}}_{AB} \geq 0.\]
- Parameters:¶
rho (
numeric) – The density matrix of the bipartite state.varargin (
numeric) – The array storing dimensions of subsystems A and B.
- Returns:¶
The improved logarithmic negativity (max-Rains entropy) of \(\rho_{AB}\).
- Return type:¶
numeric
- Raises:¶
error– If either input/output dimension does not match, an error is raised.
Note
Wang, X., & Duan, R. (2016). Improved semidefinite programming upper bound on distillable entanglement. Physical Review A, 94(5), 050301.
- Entanglement.StaticEntTheory.RainsBound(rho, varargin)¶
- \[R(\rho_{AB}) = \min\operatorname{D}(\rho_{AB}\|\sigma_{AB}), \ s.t. \ \sigma_{AB} \in \operatorname{PPT'}(A:B).\]
- Parameters:¶
rho (
numeric) – The density matrix of the bipartite state.varargin (
numeric) – The array storing dimensions of subsystems A and B.
- Returns:¶
The Rains’ bound of \(\rho_{AB}\).
- Return type:¶
numeric
- Raises:¶
error– If either input/output dimension does not match, an error is raised.
Note
Rains, E. M. (2001). A semidefinite program for distillable entanglement. IEEE Transactions on Information Theory, 47(7), 2921-2933.
- Entanglement.StaticEntTheory.TempLogNeg(rho, varargin)¶
- \[E_N^{\tau}(\rho_{AB}).\]
- Parameters:¶
rho (
numeric) – The density matrix of the bipartite state.varargin (
numeric) – The array storing dimensions of subsystems A and B.
- Returns:¶
The Temperal logarithmic negativity of \(\rho_{AB}\).
- Return type:¶
numeric
- Raises:¶
error– If either input/output dimension does not match, an error is raised.
Note
Lami, L., & Regula, B. (2023). No second law of entanglement manipulation after all. Nature Physics, 19(2), 184-189.