In quantum theory, the state of a system is modeled by density operators on a Hilbert space *H*_{A}. For simplicity, we will restrict ourselves to finite-dimensional Hilbert spaces. In this case, density operators can be thought of as simply matrices that are Hermitian, positive semi-definite, and have trace equal to one.

The evolution of a quantum system is described by quantum channels. These are, by definition, linear maps on operators that are trace-preserving and completely positive. In particular, they are maps that take quantum states to quantum states. Let Φ be a quantum channel.

- The trace-preserving property of Φ means that Tr[Φ(X)] = Tr(X) for all linear operators X on the Hilbert space
*H*_{A}. In particular, for any density operator ρ, we get that Tr[Φ(ρ)] = 1. Physically, one can think of this property as a statement of conservation of probability. - The completely positive property of Φ means that 1
_{R}⊗ Φ is positive for any reference quantum system R of any dimension. Complete positivity is an important requirement physically since in quantum theory it is possible for particles to be entangled with an external system that is outside the control of the experimenter. Positivity of states passing through the channel should be preserved even in the presence of such entanglement.

There are many ways of representing quantum channels. One important representation of quantum channels is the *Choi representation*. To define it, let d_{A} be the dimension of the Hilbert space *H*_{A} and let us fix an orthonormal basis {|i⟩_{A}}_{i=0}^{dA−1} of *H*_{A}. Then, define the state

where *H*_{A′} is a Hilbert space with the same dimension d as *H*_{A}. We call such a state maximally entangled. It is the higher-dimensional analogue of the two-qubit Bell state |Φ^{+}⟩ = (1/√2)(|0, 0⟩ + |1, 1⟩). Using |Γ⟩_{A′A}, we define the Choi representation C(Φ) of the channel Φ to be

Often, one is given a map Φ and has to determine whether or not it is completely positive. Choi proved that Φ is completely positive if and only if C(Φ) is positive semi-definite. This is a remarkable result since we saw in the definition of completely positive that the reference system R can be of arbitrary dimension. What Choi’s result tells us is that to determine complete positivity it is enough to simply let R be a copy of A.

The Choi representation of a channel uniquely characterizes it since it gives us the action of Φ on the orthonormal operator basis {|i⟩⟨j|} of *H*_{A}. Indeed, we can write C(Φ) as

In particular, the action of Φ can be written explicitly in terms of its Choi representation as

Now, if Φ is trace-preserving, observe that

which means that

Therefore, the operator ρ_{Φ}^{AB} defined as

is positive semi-definite (by Choi’s theorem, since Φ is a channel) and has trace one. ρ_{Φ}^{AB} is therefore a density operator. The Choi representation thus gives us a way to associate to each quantum channel a quantum state. What about the other way around?

Suppose we are given a quantum state ρ^{AB} and we want to associate a quantum channel Φ_{ρ} to it. If we define Φ_{ρ} as

then Φ_{ρ} will certainly be completely positive (since its Choi representation is simply ρ^{AB}, which is positive semi-definite), but it will not be trace-preserving unless ρ^{A} : = Tr_{B}[ρ^{AB}] = (1/d_{A})1_{A}. So the Choi representation only provides a one-to-one correspondence between channels and states with maximally-mixed marginal on A.

Observe that the condition Tr_{B}(C(Φ)) = 1_{A} for the trace-preservation of Φ arises from the fact that the Choi representation C(Φ) has the vector |Γ⟩ in its definition and the fact that Tr_{A}(|Γ⟩⟨Γ|) = 1_{A′}. What we would thus like is, given ρ^{AB}, a vector |ψ_{ρ}⟩_{A′A} that has marginal on A′ equal to ρ_{A} and a channel Φ_{ρ} such that

A slight subtlety is that the marginal on A of |ψ_{ρ}⟩ must have full support or else we cannot guarantee that the above expression will uniquely specify Φ_{ρ} on the entire operator space of *H*_{A}.

Now, observe that if we define |ψ_{ρ}⟩ as

then Tr_{A}[|ψ_{ρ}⟩⟨ψ_{ρ}|_{A′A}] = ρ_{A}. Assuming ρ_{AB} is such that ρ_{A} has full support, let’s use the above definition of |ψ_{ρ}⟩ and see if we can find a channel Φ_{ρ} such that

Expanding the left-hand side of this using the definition of |ψ_{ρ}⟩, we get

We require this to be equal to ρ_{AB}, and since ρ_{A} is fully supported it holds that ρ_{A}, and hence √ρ_{A}, is invertible. (Another equivalent way of stating this fact is to say that ρ_{A} has full rank, meaning that it has rank d_{A}, the dimension of *H*_{A}. Another equivalent way of stating it is that ρ_{A} does not have zero eigenvalues. Yet another equivalent statement is that ρ_{A} is positive definite.) Therefore, we can write

Now, we know how to write down the action of Φ given C(Φ). Therefore,

It is easy to check that Φ_{ρ} is trace-preserving using either the explicit form Φ_{ρ} above or its Choi representation by checking that Tr_{B}(C(Φ_{ρ})) = 1_{A}. It is also clearly completely positive since its Choi representation, being the product of positive semi-definite operators, is positive semi-definite.

Now, to get this result, we assumed that our given state ρ_{AB} is such that ρ_{A} has full support, and hence is invertible. If this is not the case, then we can interpret the inverse as a generalized inverse (that is, the *Moore-Penrose pseudo-inverse*, which takes the inverse of an operator on its support). Φ_{ρ} will then have the action given by the expression above only on the support of ρ_{A} and will therefore be a channel on the support of ρ_{A}.

Operators of the form

are called *conditional states*. Specifically,

is called the *state of B conditioned on A*. (Note that it is not actually a state, since it does not have unit trace.) The name comes from the fact that it looks analogous to the definition

of the conditional probability distribution *P(B|A)*, especially if we write it as

where the operation ⋆ is defined as

Note that it is common to suppress identity operators, so that the expression above should be interpreted as ρ_{AB} ⋆ (ρ_{A}^{−1} ⊗ 1_{B}). With the definition of the conditional state, the action of Φ_{ρ} as given above is

which is reminiscent of the action of stochastic maps in classical probability theory,

The transpose is necessary in order to make Φ_{ρ} completely-positive; without it, the map would only be positive.

You can find more information about conditional states in the paper by Leifer and Spekkens.