## Operational distance and fidelity for quantum channels (2005)

Venue: | J. Math. Phys |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Belavkin05operationaldistance,

author = {Viacheslav P. Belavkin and Giacomo Mauro D’ariano and Maxim Raginsky},

title = {Operational distance and fidelity for quantum channels},

journal = {J. Math. Phys},

year = {2005},

volume = {46},

pages = {062106}

}

### OpenURL

### Abstract

ABSTRACT. We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well-defined for channels between finitedimensional algebras, but it also applies to a certain class of channels between infinitedimensional algebras (explicitly, those channels that possess an operator-valued Radon– Nikodym density with respect to the trace in the sense of Belavkin–Staszewski) and induces a metric on the set of quantum channels which is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (‘generalized transition probability’) of Uhlmann, is topologically

### Citations

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110 |
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Citation Context ...omplex separable Hilbert space associated to a quantum-mechanical system. Given a pair of density operators ρ, σ, i.e., positive trace-class operators with unit trace, one can use either the fidelity =-=[4, 5, 6, 7]-=- F (ρ, σ) := Tr � (ρ 1/2 σρ 1/2 ) 1/2� (1) or the trace-norm (half-) distance D(ρ, σ) := 1 2 �ρ − σ�⊺, (2) where �ρ�⊺ := Tr |ρ|, |ρ| := (ρ † ρ) 1/2 [8, 9]. Loosely speaking, two states ρ and σ are clo... |

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53 |
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Citation Context ...where �ρ�⊺ := Tr |ρ|, |ρ| := (ρ † ρ) 1/2 [8, 9]. Loosely speaking, two states ρ and σ are close to each other if F (ρ, σ) is large, or if D(ρ, σ) is small. In fact, as follows from the key inequality =-=[5, 10]-=- 1 − F (ρ, σ) ≤ D(ρ, σ) ≤ � 1 − F 2 (ρ, σ), (3) the fidelity and the trace-norm distance are equivalent in the sense that any two density operators that are close to one another in the sense of (1) ar... |

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18 |
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Citation Context ... trace, one can use either the fidelity [4, 5, 6, 7] F (ρ, σ) := Tr � (ρ 1/2 σρ 1/2 ) 1/2� (1) or the trace-norm (half-) distance D(ρ, σ) := 1 2 �ρ − σ�⊺, (2) where �ρ�⊺ := Tr |ρ|, |ρ| := (ρ † ρ) 1/2 =-=[8, 9]-=-. Loosely speaking, two states ρ and σ are close to each other if F (ρ, σ) is large, or if D(ρ, σ) is small. In fact, as follows from the key inequality [5, 10] 1 − F (ρ, σ) ≤ D(ρ, σ) ≤ � 1 − F 2 (ρ, ... |

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Citation Context ...characterization of dimension-free bounds (whenever they exist) on other operationally meaningful distance measures for quantum operations [3] in terms of the CB-norm distance. As pointed out in Ref. =-=[15]-=-, such bounds are crucial for a successful generalization of the usual quantum capacity of a channel [1, 2] (i.e., with respect to the identity channel) to the case of comparing quantum channels to an... |

9 |
Functional Analysis, Springer-Verlag
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Citation Context ... more generally as an integral � Φ (B) = Z F (z) † BF (z) dµ (z) , (12) with respect to a positive measure µ on a measurable space (Z, BZ), where the integration is understood in the sense of Bochner =-=[18]-=-, then the predual map Φ⊺ has the transposed integral form � Φ⊺ (ρ) = F⊺ (z) † ρF⊺ (z) dµ (z) , Z where g ∋ ξ ↦−→ 〈ξ|F⊺ (z) are Hilbert-transposed to the operators h ∋ η ↦−→ 〈η|F (z), that is F⊺ (z) =... |

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Citation Context ...lass of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon–Nikodym density with respect to the trace in the sense of Belavkin– Staszewski =-=[14]-=-) and is equivalent to the CB-norm distance, echoing the way the Uhlmann fidelity (1) for density operators is equivalent to the trace-norm distance (2). Apart from these technical features, the minim... |

7 |
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Citation Context ...) = 1 m m� i=1 k=1 m� i=1 k=1 m� Φ⊺ (|i〉〈k|) ⊗ |i〉〈k| ≡ ρ m� Ψ⊺ (|i〉〈k|) ⊗ |i〉〈k| ≡ σ. The fidelity F (ρ, σ), taken as the channel fidelity F(Φ, Ψ) := F � Φ⊺ ⊗ id(π), Ψ⊺ ⊗ id(π) � by Raginsky in Ref. =-=[11]-=-, enjoys many properties parallel to those of the fidelity (1) for quantum states. Alternatively, one can adopt the (half-) distance [1, 12, 13] D(Φ, Ψ) := 1 2 �Φ − Ψ�cb, (5) where � • �cb denotes the... |

5 |
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Citation Context ...omplex separable Hilbert space associated to a quantum-mechanical system. Given a pair of density operators ρ, σ, i.e., positive trace-class operators with unit trace, one can use either the fidelity =-=[4, 5, 6, 7]-=- F (ρ, σ) := Tr � (ρ 1/2 σρ 1/2 ) 1/2� (1) or the trace-norm (half-) distance D(ρ, σ) := 1 2 �ρ − σ�⊺, (2) where �ρ�⊺ := Tr |ρ|, |ρ| := (ρ † ρ) 1/2 [8, 9]. Loosely speaking, two states ρ and σ are clo... |

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Gaussian noise and quantum-optical communication,” Phys
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