## 7.7 Completely positive trace-preserving maps

It is easy to verify that quantum channels preserve positivity and trace, but the converse is not true!
You may find it surprising, but there are linear maps that preserve positivity and the trace, but which are *not* quantum channels.
The matrix transpose operation *cannot* be written in the Stinespring (or the Kraus) form;
it is not induced by a unitary operation on some larger Hilbert space, and it cannot be implemented.
So, we then ask, *what is* the class of physically admissible maps?

Mathematically speaking, a quantum channel ^{113}

Firstly, for such a map

\mathcal{E} to be a channel it must*respect the mixing of states*. Consider an ensemble of systems, with a fractionp_1 of them in the state\rho_1 , and the remainingp_2 of them in the state\rho_2 . The overall ensemble is described by\rho=p_1\rho_1+p_2\rho_2 . If we apply\mathcal{E} to each member of the ensemble individually, then the overall ensemble will be described by the density operator\rho'=\mathcal{E}(\rho) , which is given by\rho'=p_1\mathcal{E}(\rho_1)+p_2\mathcal{E}(\rho_2) . We conclude that\mathcal{E} must be a*linear map*.Next, since

\mathcal{E} must*map density operators to density operators*it must be both*positive*(\mathcal{E}(\rho)\geqslant 0 whenever\rho\geqslant 0 ) and*trace preserving*(\operatorname{tr}\mathcal{E}(\rho)=\operatorname{tr}\rho for all\rho ).Finally comes a subtle point. It turns out that being positive is not good enough; we must further require that the map

\mathcal{E} *remains positive even when extended to act on a part of a larger system*. Suppose that Alice and Bob share a bipartite system\mathcal{AB} in an entangled state\rho_\mathcal{AB} , and, whilst Alice does nothing, Bob applies the local operation\mathcal{E} to his subsystems, and his subsystems only. Then the resulting map on the whole bipartite system is given by\mathbf{1}\otimes\mathcal{E} , and this must give a proper density operator\rho'_\mathcal{AB} of the composed system. It turns out that this is a strictly stronger property than mere positivity; we are asking for something called**complete positivity**. Needless to say, complete positivity of\mathcal{E} implies positivity, but the converse does not hold: there are maps which are positive but not completely positive. The matrix transpose operation\rho\rightarrow \rho'=\rho^T is a classic example of such a map.

In fact, we can study the matrix transpose a bit further.
Consider the transpose operation on a single qubit: *only one* of the two qubits (say, the second one), then the density matrix of the two qubits evolves under the action of the partial transpose *not* preserve positivity, and therefore *not* a completely positive map.
If you prefer to see this more explicitly, then you can use the matrix representation of

So we have seen that, at the very least, we want to be considering *completely positive trace-preserving* maps, but how do we know whether or not there are any restrictions left to impose?
Needless to say, here is where mathematics alone cannot guide us, since we are trying to characterise maps which are *physically admissible*, and mathematics knows nothing about the reality of our universe!
However, one thing that we can do is compare our abstract approach with the derivations of quantum channels defined in terms of the Stinespring (or the Kraus) representation.
As it happens, we can show that a map is completely positive and trace preserving if and only if it can be written in the Stinespring (or the Kraus) form.
In other words:

Quantum channels are exactly the completely positive trace-preserving (CPTP) maps.

One direction of this claim is much simpler than the other.
Any channel *and* the extended map **Choi matrix**^{114}, which is yet another way to characterise linear maps between operators.

**!!to-do: tim footnote: here we work only with finite dimensional spaces, and if**X is f.d. then every linear operatorf\colon X\to Y between normed vector spaces is continuous (and thus bounded)!!- talk about unbounded operators

Recall that, given a pair of Hilbert spaces

\mathcal{H} and\mathcal{H}' , we denote the set of (bounded) linear operators from\mathcal{H} to\mathcal{H}' by\mathcal{B}(\mathcal{H},\mathcal{H}') . We write\mathcal{B}(\mathcal{H}) as shorthand for\mathcal{B(H,H)} ↩︎Man-Duen Choi was brought up in Hong Kong. He received his Ph.D. degree under the guidance of Chandler Davis at Toronto. He taught at the University of California, Berkeley, from 1973 to 1976, and has worked since then at the University of Toronto. His research has been mainly in operator algebras, operator theory, and polynomial rings. He is particularly interested in examples/counterexamples and two-by-two matrix manipulations.↩︎