Quantum States and Channels

In quantum theory, the state of a system is modeled by density operators on a Hilbert space HA. 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.

  1. The trace-preserving property of Φ means that Tr[Φ(X)] = Tr(X) for all linear operators X on the Hilbert space HA. 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.
  2. The completely positive property of Φ means that 1R 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 dA be the dimension of the Hilbert space HA and let us fix an orthonormal basis {|iA}i=0dA1 of HA. Then, define the state

state_channel0xwhere HA is a Hilbert space with the same dimension d as HA. 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 |ΓAA, we define the Choi representation C(Φ) of the channel Φ to be

state_channel2xOften, 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 HA. Indeed, we can write C(Φ) as

state_channel3xIn particular, the action of Φ can be written explicitly in terms of its Choi representation as

state_channel4xNow, if Φ is trace-preserving, observe that

state_channel5xwhich means that

state_channel6xTherefore, the operator ρΦAB defined as

state_channel7xis positive semi-definite (by Choi’s theorem, since Φ is a channel) and has trace one. ρΦAB 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 ρAB and we want to associate a quantum channel Φρ to it. If we define Φρ as

state_channel8xthen Φρ will certainly be completely positive (since its Choi representation is simply ρAB, which is positive semi-definite), but it will not be trace-preserving unless ρA : = TrB[ρAB] = (1/dA)1A. 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 TrB(C(Φ)) = 1A for the trace-preservation of Φ arises from the fact that the Choi representation C(Φ) has the vector |Γin its definition and the fact that TrA(|Γ⟩⟨Γ|) = 1A. What we would thus like is, given ρAB, a vector |ψρAA that has marginal on Aequal to ρA and a channel Φρ such that

state_channel10xA slight subtlety is that the marginal on A of |ψρmust have full support or else we cannot guarantee that the above expression will uniquely specify Φρ on the entire operator space of HA.

Now, observe that if we define |ψρas

state_channel11xthen TrA[|ψρ⟩⟨ψρ|AA] = ρA. Assuming ρAB is such that ρA has full support, let’s use the above definition of |ψρand see if we can find a channel Φρ such that

state_channel12xExpanding the left-hand side of this using the definition of |ψρ, we get

state_channel13xWe require this to be equal to ρAB, and since ρA is fully supported it holds that ρA, and hence √ρA, is invertible. (Another equivalent way of stating this fact is to say that ρA has full rank, meaning that it has rank dA, the dimension of HA. Another equivalent way of stating it is that ρA does not have zero eigenvalues. Yet another equivalent statement is that ρA is positive definite.) Therefore, we can write

state_channel15xNow, we know how to write down the action of Φ given C(Φ). Therefore,

state_channel16xIt is easy to check that Φρ is trace-preserving using either the explicit form Φρ above or its Choi representation by checking that TrB(Cρ)) = 1A. 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 ρAB is such that ρA 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). Φρ will then have the action given by the expression above only on the support of ρA and will therefore be a channel on the support of ρA.

Operators of the form

state_channel17xare called conditional states. Specifically,

state_channel18xis 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

state_channel19x

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

state_channel20xwhere the operation is defined as

state_channel21xNote that it is common to suppress identity operators, so that the expression above should be interpreted as ρAB (ρA1 1B). With the definition of the conditional state, the action of Φρ as given above is

state_channel22xwhich is reminiscent of the action of stochastic maps in classical probability theory,

state_channel23xThe transpose is necessary in order to make Φρ 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.