### Citations

271 |
Completely positive linear maps on complex matrices
- Choi
- 1975
(Show Context)
Citation Context ...tion 4. A quantum operation is a completely positive map Φ : C n×n → C m×m between density matrices that does not increase the trace. It is well known that Φ must take the form Φ(ρ) = ∑nm i=1 EiρE∗ i =-=[3]-=-. By mocking up [9] we mean, similar to the POVM case, coupling the system to an ancilla and find an unitary evolution on the combined system such that the reduced state, obtained via the partial trac... |

210 | Separability of mixed states: necessary and sufficient conditions
- Horodecki, Horodecki, et al.
- 1996
(Show Context)
Citation Context ...pe were obtained in [4] and the parametrization of positive matrices allows one to explore their extensions in a non-ad hoc way. By the correspondence between positive maps and entanglement witnesses =-=[7]-=-, we thus show that certain families of bipartite mixed states are separable. The last two applications concerns the unitary dilation of completely positive maps on matrix algebras. While a celebrated... |

201 |
Completely Bounded Maps and Operator Algebras
- Paulsen
- 2002
(Show Context)
Citation Context ...d to be n-positive if the induced map Id ⊗ Φ : C n×n ⊗ L(H) → C n×n ⊗ L(K) is positive, and Φ is completely positive, or CP, if it is n-positive for all n. We state the following result without proof =-=[10]-=-. Theorem 6. (Russo-Dye) Let Φ be a positive map between unital C*-algebras, then ‖Φ‖ ≤ ‖Φ(I)‖. In particular, if Φ is unital and Γ a contraction, then Φ(Γ ∗ )Φ(Γ) = ‖Φ(Γ)‖ 2 ≤ Γ ∗ Γ ≤ I. Similarly, Φ... |

197 |
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys
- Werner
- 1989
(Show Context)
Citation Context ...is replaced by ⎡ ⎣ a c a c b c a c a ⎤ ⎦ or 17 ⎡ ⎣ a c c c b b c b b ⎤ ⎦.Separable Quantum States In physical language, trace-class positive matrices are unnormalized mixed states [1]. Definition 2. =-=[13]-=- Let the state space of a bipartite quantum system be the tensor product H = H1 ⊗ H2, where Hi are Hilbert spaces. A state σ ∈ L(H) is separable if it lies in the closure, in the trace norm, of states... |

30 |
A generalized Schwarz inequality and algebraic invariants for operator algebras
- Kadison
- 1952
(Show Context)
Citation Context ...rmal A ∈ L(H), Φ(A ∗ A) ≥ Φ(A ∗ )Φ(A) and Φ(A ∗ A) ≥ Φ(A)Φ(A ∗ ). What is known as Kadison’s inequality will also be needed: for every unital positive map Φ and every self-adjoint S, Φ(S 2 ) ≥ Φ(S) 2 =-=[8]-=-. What we will show that is essentially every positive map is 3−positive in a certain limited sense. We first notice that subnormal contractions enjoy a property stronger than that prescribed by Russo... |

19 |
Some assorted inequalities for positive linear maps on
- Choi
- 1980
(Show Context)
Citation Context ...sitive maps and contractions is applied to show that general positive maps are more than merely positive when restricted to certain subsets of positive matrices. Results of this type were obtained in =-=[4]-=- and the parametrization of positive matrices allows one to explore their extensions in a non-ad hoc way. By the correspondence between positive maps and entanglement witnesses [7], we thus show that ... |

16 |
Completing matrix contractions
- Arsene, Gheondea
- 1982
(Show Context)
Citation Context ...y already possess an elegant combinatorial structure and play a central role in our parametrizations. Next we consider matrix contractions. The 2 × 2 matrix contractions were already characterized in =-=[2]-=-. Here we extend the description to matrices of arbitrary size and point out the combinatorial aspect of this parametrization. Then the special case of unitary matrices is examined. We also review the... |

9 |
Eds.), Quantum Information
- Alber, Beth, et al.
- 2001
(Show Context)
Citation Context ...ng placed in physical context. The last two applications are phrased more directly in the language of quantum information theory. For general background in quantum information, we refer the reader to =-=[1]-=- and [9]. Although we only consider matrices of finite size, all parametrizations described in this paper can be extended to (semi-)infinite matrices, where convergence is given by an appropriate oper... |

6 | Parametrizing quantum states and channels
- Constantinescu, Ramakrishna
- 2003
(Show Context)
Citation Context ... ✲ ✲ ✲ ✲ ✲ ✲ ✲ ��✒ � ��✒ ❅ ❅ ❅ ❅ ❅❘ ❅❘ � � �✒ � � �✒ � ��✒ L33 L22 L11 L ❄ ❄ ❄ ❅ ❅ ❅❅❘ ❅❅❘ ❅ ❅❅❘ ∗ 33 L∗ ✻ ✻ 22 L∗ 11 ✲ Figure 8. Lattice structure for 4 × 4 positive matrices can be described easily =-=[6]-=-. Namely, let Γi = γiIH and ̷Li = √ miiB; it is clear that they parametrize M ⊗ A in the sense of Schur-Constantinescu. Matrices given by a strict inequality. The natural square roots given by the SCp... |

3 |
Positive maps on C*-algebras
- Stinespring
- 1955
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Citation Context ...tain families of bipartite mixed states are separable. The last two applications concerns the unitary dilation of completely positive maps on matrix algebras. While a celebrated result by Stinespring =-=[11]-=- shows that such dilations always exist, the paramatrization of contractions allows one to give a concrete constructive procedure for such dilations. 1The results on positive maps are first stated in... |

1 | Dilation theoretic parametrizations of positive matrices with applications to quantum information, preprint
- Tseng, Ramakrishna
(Show Context)
Citation Context ...r-like condition, called 12the Bloch cylinder, was obtained for positive matrices of trace 1 (quantum states) of any finite dimension. This provides an alternative to the well-known Bloch sphere. In =-=[12]-=- it was applied to show that every positive map is completely positive to a certain extent, thus establishing the separability of certain families of quantum states in arbitrary finite dimensions. In ... |