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PhyslibAlpha.ProbabilisticTheory.Composite.CompletePositivity

Complete positivity of classical channels

i. Overview

A channel acting on one part of a composite system should keep every nonnegative composite observable nonnegative, whatever the other part is. This is complete positivity, and it depends on which composite is used. For a channel into or out of a classical system it holds for every choice of composites: a classical system composes uniquely, so there is only one composite to check, and positive maps preserve the minimal and the maximal cone.

Precisely, let `A` be any Archimedean ancilla. Then `φ ⊗ id` maps every composite observable in the maximal cone into every Archimedean cone that contains the minimal cone, whenever the input or the output of `φ` is classical. In particular it maps any composite of the input with `A` into any Archimedean composite of the output with `A`. For a classical input `φ` must be a channel; for a classical output any positive map works.

ii. Key results

- `CompositeCone.map_mem_of_isNuclear_left` : a channel out of a nuclear system is completely positive. - `CompositeCone.map_mem_of_isNuclear_right` : a positive map into a nuclear system is completely positive. - `CompositeCone.map_mem_of_isNuclear_left'`, `CompositeCone.map_mem_of_isNuclear_right'` : the same, stated for composites. - `CompositeCone.map_mem_of_isClassical_left`, `CompositeCone.map_mem_of_isClassical_right` : the same for classical systems.

iii. Table of contents

  • A. Tensoring with the identity
  • B. Nuclear inputs and outputs
  • C. Classical inputs and outputs

A. Tensoring with the identity

B. Nuclear inputs and outputs

C. Classical inputs and outputs

8 declarations

theorem

(ϕ⊗id)(1⊗1)=1⊗1(\phi \otimes \text{id})(1 \otimes 1) = 1 \otimes 1 for quantum channels

Let E,FE, F, and AA be ordered unit spaces. For any quantum channel ϕ:E→F\phi : E \to F, the tensor product map ϕ⊗idA:E⊗A→F⊗A\phi \otimes \text{id}_A : E \otimes A \to F \otimes A maps the unit of the composite system 1E⊗1A1_E \otimes 1_A to the unit 1F⊗1A1_F \otimes 1_A. That is, (ϕ⊗idA)(1E⊗1A)=1F⊗1A(\phi \otimes \text{id}_A)(1_E \otimes 1_A) = 1_F \otimes 1_A.

theorem

(ϕ⊗id)(\phi \otimes \text{id}) preserves the Archimedean closure of the minimal tensor cone

Let E,FE, F, and AA be ordered unit spaces. Let ϕ:E→F\phi : E \to F be a quantum channel (a unital completely positive map). If an element z∈E⊗RAz \in E \otimes_{\mathbb{R}} A belongs to the Archimedean closure of the minimal tensor cone of E⊗RAE \otimes_{\mathbb{R}} A, then its image under the tensor product map (ϕ⊗idA)(z)(\phi \otimes \text{id}_A)(z) belongs to the Archimedean closure of the minimal tensor cone of F⊗RAF \otimes_{\mathbb{R}} A. The Archimedean closure of the minimal tensor cone is defined as the set of elements xx such that x+ϵ(1⊗1)x + \epsilon (1 \otimes 1) lies in the minimal tensor cone for every ϵ>0\epsilon > 0.

theorem

A channel out of a nuclear system is completely positive (Left-sided)

Let EE and FF be ordered unit spaces and AA be an ancilla system. Suppose EE is a nuclear system. Let ϕ:E→F\phi: E \to F be a quantum channel. For any element zz in the maximal tensor cone of E⊗AE \otimes A, and for any Archimedean tensor cone C⊆F⊗AC \subseteq F \otimes A that contains the minimal tensor cone of F⊗AF \otimes A, the image of zz under the tensor product map, (ϕ⊗idA)(z)(\phi \otimes \text{id}_A)(z), belongs to CC. In summary, a quantum channel out of a nuclear system is completely positive: it maps every composite observable in the maximal cone into any Archimedean cone that contains the minimal cone.

theorem

A Positive Map into a Nuclear System is Completely Positive

Let EE, FF, and AA be ordered real vector spaces. Suppose FF is a nuclear system. Let ϕ:E→F\phi : E \to F be a positive linear map. For any element zz in the maximal tensor cone maxTensorCone(E,A)\text{maxTensorCone}(E, A), the tensor (ϕ⊗idA)(z)(\phi \otimes \text{id}_A)(z) belongs to every Archimedean pointed cone C⊆F⊗RAC \subseteq F \otimes_{\mathbb{R}} A that contains the minimal tensor cone minTensorCone(F,A)\text{minTensorCone}(F, A).

theorem

A quantum channel from a nuclear system maps any composite cone C1C_1 to any Archimedean composite cone C2C_2

Let EE, FF, and AA be ordered unit spaces. Suppose EE is a nuclear system and ϕ:E→F\phi : E \to F is a quantum channel. For any composite cone C1⊆E⊗RAC_1 \subseteq E \otimes_{\mathbb{R}} A and any Archimedean composite cone C2⊆F⊗RAC_2 \subseteq F \otimes_{\mathbb{R}} A, if an element zz belongs to C1C_1, then its image under the tensor product map, (ϕ⊗idA)(z)(\phi \otimes \text{id}_A)(z), belongs to C2C_2.

theorem

Positive Maps into Nuclear Systems Preserve Composite Cone Membership

Let E,FE, F, and AA be real ordered vector spaces. Suppose FF is a nuclear system. Let ϕ:E→F\phi : E \to F be a positive linear map and idA\text{id}_A be the identity map on AA. For any composite cone C1⊆E⊗RAC_1 \subseteq E \otimes_{\mathbb{R}} A and any Archimedean composite cone C2⊆F⊗RAC_2 \subseteq F \otimes_{\mathbb{R}} A, if a composite observable z∈E⊗RAz \in E \otimes_{\mathbb{R}} A belongs to C1C_1, then its image under the map ϕ⊗idA\phi \otimes \text{id}_A belongs to C2C_2.

theorem

Channels from classical systems are completely positive

Let EE be a classical Archimedean order unit space and FF be an order unit space. Let ϕ:E→F\phi : E \to F be a quantum channel. For any ancilla system AA, any composite cone C1⊆E⊗AC_1 \subseteq E \otimes A, and any Archimedean composite cone C2⊆F⊗AC_2 \subseteq F \otimes A, if a composite observable zz belongs to C1C_1, then its image under the map ϕ⊗idA\phi \otimes \text{id}_A belongs to C2C_2. In other words, (ϕ⊗idA)(z)∈C2(\phi \otimes \text{id}_A)(z) \in C_2.

theorem

Positive maps into classical systems are completely positive

Let EE, FF, and AA be Archimedean order unit spaces. If FF is a classical system, then any positive linear map ϕ:E→F\phi : E \to F is completely positive in the following sense: for any composite cone C1⊆E⊗AC_1 \subseteq E \otimes A and any Archimedean composite cone C2⊆F⊗AC_2 \subseteq F \otimes A, if a composite observable zz belongs to C1C_1, then its image under the map ϕ⊗idA\phi \otimes \text{id}_A belongs to C2C_2. That is, (ϕ⊗idA)(z)∈C2(\phi \otimes \text{id}_A)(z) \in C_2.