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Quantum Operations

Definition	Unitary Representation
Kraus Representation	Measurements as Quantum Operations
Choi Isomorphism	Examples

Under a certain physical process, the state of a given system evolves into another state. The time evolution of a closed system is described by unitary operators. For an open quantum system, which interacts with its environment, the evolution is not unitary any longer.
Dynamical processes of open quantum systems are described by a special kind of supermaps called quantum operations. A quantum operation transforms density operators to other density operators while preserving the elementary properties of density operators. In particular, density operators are positive, so a quantum operation needs to preserve positivity. However, it turns out that merely preserving positivity is not sufficient and a much stronger condition is required. Imagine that a system has interacted with its surroundings and established an entanglement with them. Physically, one expects the operation to preserve the properties, positivity in particular, of the density operator of the whole containing the system and surroundings. Essentially, a quantum operation needs to preserve not only the positivity of density operators of a given system but also all density operators of any extended system including the system itself and its surrounding systems. Mathematically, such a condition is satisfied by completely positive supermaps.

See also Section 5.2 and Appendix B of the

Quantum Workbook (2022)

Supermap	Describes the quantum operations
ChoiMatrix	The Choi matrix of a supermap

Functions useful to handle quantum operations.

Make sure that the

Q3: Symbolic Quantum Simulation

package is loaded to use the demonstrations in this documentation.

In[1072]:=

Needs["QuantumMob`Q3`"]

Definition

A supermap is a linear mapping of linear operators. Recall that linear operators on a vector space  themselves form a vector space

ℒ()

. A considerable amount of interest in supermaps came with the booming of quantum information theory in the 1990s when it became clear that supermaps are important in the study of entanglement. Since then, mathematical theories on supermaps have been developed at a notably fast pace and applied to a wide range of subjects in quantum computation and quantum information.

A quantum operation is a special kind of supermap. Let  and  be vector spaces. Suppose that

ℱ

is a supermap from

ℒ()

onto

ℒ()

ℱ

is called a quantum operation if it satisfies the following three axioms.

(a)

ℱ

never increases the trace. That is,

0≤Tr[ℱ(ρ)]≤1

for any density operator

(b)

ℱ

is convex linear. That is, for any probabilities

and density operators

on ,

ℱ

∑

=

∑

ℱ(

)

(c)

ℱ

is a completely positive supermap. That is, not only

ℱ(ρ)

itself is positive for any positive operator



, but

(ℱ⊗ℐ)(ρ)

is also positive for any positive operator on

⊗ℰ

with an arbitrary vector space

ℰ

Physically, the vector space

ℰ

in (c) above is associated with an environment.

ℱ⊗ℐ

acts non-trivially only on



associated with the system but trivially on

ℰ

. To be physically meaningful,

ℱ⊗ℐ

is expected to preserve the properties, especially positivity, of density operators

⊗ℰ

. Note that

may contain a considerable amount of entanglement due to prior interactions between the system and environment.

Most quantum operations preserve the trace. That is,

Tr[ℱ(ρ)]=1

for all density operators

. An important exception is the process associated with a (generalized selective) measurement. When the trace is not preserved,

Tr[ℱ(ρ)]=1

gives the probability for the dynamical process

ℱ

to occur.

In quantum information theory, quantum operations preserving trace, i.e., completely positive and trace-preserving supermaps, are called quantum channels. Physically, these describe communication channels that can transmit quantum information, as well as classical information.

Another important class of physical phenomena described by quantum operations is quantum decoherence or just decoherence for short, referring to the loss of quantum coherence. We have briefly introduced this effect in the previous section through toy models. Quantum operations offer a complete and general description of decoherence.

Kraus Representation

A quantum operation is a restricted form of completely positive supermap. Accordingly, for any quantum operation

ℱ

from

ℒ()

onto

ℒ()

, there exist linear maps from



onto



such that

ℱ(ρ)=

†

for all linear operators (not necessarily density operators)



. The linear maps

are called the Kraus elements or the Kraus maps of

ℱ

. Then, the trace-decreasing condition in

axiom (a

) imposes the inequalities

0≤

†

≤1

One can always choose Kraus elements that are mutually orthogonal with respect to the Hilbert-Schmidt inner product, that is,

Tr

†

=0

whenever

μ≠ν

. Through the procedure of choosing orthogonal Kraus elements, one can drastically optimize Kraus elements (see below).

The simplest example of the Kraus representation is of the form

ℱ



=Uρ

†

involving a single unitary operator. Naturally it describes the unitary dynamics of a closed system. Note that if there are more than one Kraus elements involved in the Kraus representation, the associated dynamics is not unitary even if the Kraus elements are all unitary. For example, consider a supermap

ℱ(ρ)=(1-p)ρ+

(XρX+YρY+ZρZ)

Here, all Kraus elements (

I,X,Y,Z

up to normalization factors) are unitary. However,

ℱ

causes depolarization in

for any non-zero value of

The Kraus representation of a quantum operation as a sum of operators provides powerful tools to analyze the quantum operation. However, at this stage, the Kraus representation follows from a mathematical theorem for completely positive supermap. How does the Kraus representation arise physically?

To see how the Kraus representation arises physically, let us consider a system interacting with its environment. We denote with



and

ℰ

the Hilbert spaces associated with the system and environment, respectively. For simplicity, we assume that the total system is initially in the product state

ρ⊗σ

. The total system is a closed system, and the dynamical process afterwards due to the system-environment interaction is described with an overall unitary operator

acting on the total system,

ρ⊗σ↦U(ρ⊗σ)

†

. Without access to the environment, one has to take a partial trace of the final state over the environment to obtain the state of the system. Putting it all together, the quantum operation

ℱ

describing the process is written as

ℱ(ρ)=

ℰ

[U(ρ⊗σ)

†

〈

|U(ρ⊗σ)

†

〉

where

〉|i=1,2,…,

ℰ

}

is an orthonormal basis of

ℰ

of dimension

ℰ

. On the right-hand side of the above equation, the Hermitian product is applied partially and only on

ℰ

, and the expression still remains as an operator on . To construct the Kraus elements, we first take the spectral decomposition of

σ=

j=1







where

is the rank of

, and the eigenvectors







have been normalized by their own eigenvalues

=





since

is a positive semidefinite operator. We then define linear maps

:↦

for each i and j by

ℰ



†

I⊗





U=〈

|U



In the quantum circuit model,

Figure 1

depicts the above definition. Physically,

describes the dynamics of the system under the condition that the environment has made a transition from the initial state





to state

〉

under the unitary interaction between the system and environment. We finally regard

μ:=(i,j)

as a collective index and identify

≡

to be the desired Kraus elements. Clearly, they satisfy the closure relation

†

Putting this equation to the expression for

ℱ(ρ)

, we arrive at the representation

ℱ(ρ)=

†

Figure 1. Quantum circuit describing the Kraus element

corresponding to the dynamics of the system on condition that the environment has made a transition from the initial state





to state

〉

under the unitary interaction

between the system and environment.

In the above arguments, we have derived a Kraus representation for a quantum operation based on a system-plus-environment model. While it provides a useful physical picture of quantum operations, the resulting representation does not look particularly useful at first glance. The Kraus elements

are not orthogonal to each other with respect to the Hilbert-Schmidt inner product. Even worse, the number of Kraus elements is equal to the dimension of the environmental Hilbert space

ℰ

and may be huge given that the dimension of

ℰ

is infinite for any realistic environment. However, neither raises a significant problem. A formal representation in the form of the Kraus representation already facilitates analysis of the quantum operation. Moreover, one can optimize the Kraus elements by reconstructing orthogonal Kraus elements as we see in the next section.

Optimization of Kraus Representation

Given a set of Kraus elements, how can one actually choose new Kraus elements that are mutually orthogonal. Let

{

}

be a basis of the space

ℒ(,)

of all linear maps from





. We expand

in this basis

μν

where

is the matrix of the expansion coefficients. Putting it back to the Kraus representation, we have

ℱ(ρ)=

μν

†

)

μν

The square matrix

†

of size

(dim)×(dim)

is positive semidefinite, and can be decomposed into

†

=VΛ

†

, where

is a unitary matrix and Λ is a diagonal matrix with all elements non-negative. We now define new Kraus elements

′

)

μν

Then, it is clear that they are mutually orthogonal. Furthermore,

ℱ(ρ)=

′

′†

where

N≤(dim)×(dim)

. Therefore, it is noted that a set of mutually orthogonal Kraus elements is optimal in the sense that it has no more elements than

(dim)×(dim)

Unitary Freedom of Kraus Elements

(to be completed)

Choi Isomorphism

The Choi isomorphism is a one-to-one correspondence between supermaps and (usual) operators. It allows us to inspect a supermap in terms of the corresponding operator. It can reveal additional properties of the supermap that is not immediately clear from the supermap itself.

Unitary Representation

A quantum operation can always be regarded as a unitary operator on an extended system, which involves an “environment” in addition to the original “system”. Although it is not particularly useful for practical applications, the unitary representation provides a clear physical insight into the underlying physical processes described by the quantum operation.

Measurements as Quantum Operations

Examples

So far, we discussed a general description of quantum noisy processes in terms of the corresponding quantum operations. Let us now present some examples for a single qubit. We consider some limiting cases that allow us to easily grasp the physical meaning of the Kraus elements.

Phase Damping

The phase damping process is a decoherence process without involving any relaxation of energy or change in the population over the states. In this sense, it may be regarded as a pure decoherence process without involving any energy relaxation. For this reason, it is also referred to as a dephasing process.

The unitary representation of a phase damping process in a single-qubit system is given by

or, more explicitly,

Due to the entanglement, the final state of the system alone cannot be a pure state and coherence in the initial state has been lost through the process. With the prescription above, the Kraus elements are given by

and the corresponding quantum operation is given by

Note that they are not orthogonal,

It is therefore more convenient and efficient to choose mutually orthogonal Kraus elements

In short, the quantum operation for the phase damping process is written as

It is also convenient to expand the density operator into

The coherence has disappeared completely.

The phase damping process is specified by two Kraus elements.

Here, p is the probability for the phase to be flipped.

The supermap transforms the above density operator as follows.

Amplitude Damping

In the unitary representation, the process is described by the overall unitary operator such that

They are already orthogonal to each other. The quantum operation describing the amplitude damping process reads as

With the expansion of the density operator ρ,

the above operation reads as

The amplitude damping process is specified by two Kraus elements.

Here, p is the probability for the phase to be flipped.

The supermap transforms the above density operator as follows.

Depolarizing

From the above unitary representation, we can get the Kraus elements

The Kraus elements are already orthogonal to each other. One can also check that they satisfy the completeness relation

as they should. In the Kraus representation with the above Kraus elements, a density operator ρˆ is transformed under the decoherence process as

In terms of the components in the expansion

This implies that under the process, the “spin” polarization (i.e., the Bloch vector corresponding to the resulting density operator)

The depolarizing process is specified by three Kraus elements.

Here, p is the probability for the phase to be flipped.

The supermap transforms the above density operator as follows.