Questions tagged [complete-positivity]
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22 questions
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Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 73
2
votes
0
answers
267
views
Example of a unital contractive map that is not completely positive on an operator system
I am aware of maps that are positive but not completely positive (for example transpose map). BUT I can not think of an example of the following type.
Does there exist an operator $T$ such that a map $...
- 231
3
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0
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179
views
Stinespring's theorem: can we choose the dilation to be an isometry?
Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
- 175
4
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0
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131
views
"Dual" of a CP map
Let $M,N$ be von Neumann algebras, and let $\phi:M\rightarrow N$ be a normal completely positive map. I am interested in conditions when there is a "dual" normal completely positive map $\...
- 18.7k
7
votes
1
answer
264
views
Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?
Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...
- 619
3
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answers
181
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Completely positive, unital maps acting on unitary operators [solved]
Call a completely positive, irreducible, unital map $E$ on the operators on a finite-dimensional Hilbert-space primitive if there is only a single eigenvalue with modulus 1 (all others have modulus &...
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4
votes
1
answer
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Complete positivity with infinite dimensional auxillary spaces
The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^*$ algebras of) the bounded operators on ...
- 143
8
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1
answer
547
views
Maps which are both completely positive and positive
Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
- 43.2k
3
votes
1
answer
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Reference on completely positive maps which are isometries
Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
- 627
3
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1
answer
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Completely Positive Maps and their dual in Separable Hilbert Space
Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the set of positive operator with trace less ...
- 1,503
1
vote
1
answer
190
views
Bounded operators on the Stinespring representation space
Let $A$ be a $C^*$-algebra and let $\phi:A\to B(H)$ be a completely positive map. The Stinespring representation theorem constructs a representation of $A$ on a Hilbert space $K$, which is constructed ...
- 99
1
vote
1
answer
441
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Extensions of completely positive maps
It is known that for a completely bounded map $\psi:A\to B(H)$ there exist completely positive maps $\phi_1,\phi_2:A\to B(H)$ such that
$$\Vert \phi_i\Vert_{cb}=\Vert \psi\Vert_{cb},$$
and the map $\...
- 99
5
votes
2
answers
288
views
Completely positive maps with commuting ranges can be extended to maximal tensor product
I'm trying to do the following exercise from Brown and Ozawa's book.
Exercise $3.5.1.$ Let $\varphi: A \to B(H)$ and $\psi: B \to B(H)$ be
c.p. maps with commuting ranges. Show that there exists ...
- 51
4
votes
0
answers
264
views
Does the Cauchy–Schwarz inequality imply 2-positivity?
Recall the following generalisation of Cauchy–Schwarz.
Theorem. Let $f\colon \mathscr{A} \to \mathscr{B}$ be a linear 2-positive map between C$^*$-algebras. Then for all $a,b \in \mathscr{A}$ we ...
- 355
2
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1
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Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries
Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$
Let $\phi: S \rightarrow B_2.$
Given that the ...
- 1,429
5
votes
1
answer
516
views
Stinespring's dilation without $C^{\ast}$-algebras
Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra?
I will now state the version of Stinespring's dilation ...
- 161
2
votes
1
answer
559
views
When are completely positive maps monic/epic?
In the category of C*-algebras and $*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see Mono- and epi-morphisms for C*-algebras). Is ...
- 3,937
2
votes
1
answer
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Arveson index and curvature
Can someone explain me what is the intuitive idea behind Arveson Index and curvature of $E_0$ semigroups. I was reading the standard paper of Arveson, but is lost and yet to get intuition about it. An ...
- 421
4
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282
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Extensions of completely positive mappings
I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...
- 619
2
votes
1
answer
213
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Positive definite functions on G from Hilbert space vectors?
Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a positive definite function on $G$ using $\xi$?
This question is rather vague and ...
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7
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What are known examples of positive but not completely positive maps?
The only example I know of a positive map which is not completely positive is the transpose map on $M_n(\mathbb{C})$. Of course, one can come up with minor perturbations of this (compose it with, or ...
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When is this map completely positive?
Consider the complex $n$-by-$n$ matrices $M_n$.
Suppose that $A_i$, for $i=1,\ldots,n^2$, satisfy $\mathrm{Tr}(A_i^*
A_j)=\delta_{ij}$, so that together they form an orthonormal basis for
$M_n$. ...
- 3,937