Quantum operation

In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan.^[1] The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel.

Note that some authors use the term "quantum operation" to refer specifically to completely positive (CP) and non-trace-increasing maps on the space of density matrices, and the term "quantum channel" to refer to the subset of those that are strictly trace-preserving.^[2]

Quantum operations are formulated in terms of the density operator description of a quantum mechanical system. Rigorously, a quantum operation is a linear, completely positive map from the set of density operators into itself. In the context of quantum information, one often imposes the further restriction that a quantum operation ${\displaystyle {\mathcal {E}}}$ must be physical,^[3] that is, satisfy ${\displaystyle 0\leq \operatorname {Tr} [{\mathcal {E}}(\rho )]\leq 1}$ for any state ${\displaystyle \rho }$.

Some quantum processes cannot be captured within the quantum operation formalism;^[4] in principle, the density matrix of a quantum system can undergo completely arbitrary time evolution. Quantum operations are generalized by quantum instruments, which capture the classical information obtained during measurements, in addition to the quantum information.

Background[]

The Schrödinger picture provides a satisfactory account of time evolution of state for a quantum mechanical system under certain assumptions. These assumptions include

• The system is non-relativistic
• The system is isolated.

The Schrödinger picture for time evolution has several mathematically equivalent formulations. One such formulation expresses the time rate of change of the state via the Schrödinger equation. A more suitable formulation for this exposition is expressed as follows:

The effect of the passage of t units of time on the state of an isolated system S is given by a unitary operator U_t on the Hilbert space H associated to S.

This means that if the system is in a state corresponding to v ∈ H at an instant of time s, then the state after t units of time will be U_t v. For relativistic systems, there is no universal time parameter, but we can still formulate the effect of certain reversible transformations on the quantum mechanical system. For instance, state transformations relating observers in different frames of reference are given by unitary transformations. In any case, these state transformations carry pure states into pure states; this is often formulated by saying that in this idealized framework, there is no decoherence.

For interacting (or open) systems, such as those undergoing measurement, the situation is entirely different. To begin with, the state changes experienced by such systems cannot be accounted for exclusively by a transformation on the set of pure states (that is, those associated to vectors of norm 1 in H). After such an interaction, a system in a pure state φ may no longer be in the pure state φ. In general it will be in a statistical mix of a sequence of pure states φ₁,..., φ_k with respective probabilities λ₁,..., λ_k. The transition from a pure state to a mixed state is known as decoherence.

Numerous mathematical formalisms have been established to handle the case of an interacting system. The quantum operation formalism emerged around 1983 from work of Karl Kraus, who relied on the earlier mathematical work of . It has the advantage that it expresses operations such as measurement as a mapping from density states to density states. In particular, the effect of quantum operations stays within the set of density states.

Definition[]

Recall that a density operator is a non-negative operator on a Hilbert space with unit trace.

Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that

• If S is a density operator, Tr(Φ(S)) ≤ 1.
• Φ is completely positive, that is for any natural number n, and any square matrix of size n whose entries are trace-class operators

${\displaystyle {\begin{bmatrix}S_{11}&\cdots &S_{1n}\\\vdots &\ddots &\vdots \\S_{n1}&\cdots &S_{nn}\end{bmatrix}}}$

and which is non-negative, then

${\displaystyle {\begin{bmatrix}\Phi (S_{11})&\cdots &\Phi (S_{1n})\\\vdots &\ddots &\vdots \\\Phi (S_{n1})&\cdots &\Phi (S_{nn})\end{bmatrix}}}$

is also non-negative. In other words, Φ is completely positive if ${\displaystyle \Phi \otimes I_{n}}$ is positive for all n, where ${\displaystyle I_{n}}$ denotes the identity map on the C*-algebra of ${\displaystyle n\times n}$ matrices.

Note that, by the first condition, quantum operations may not preserve the normalization property of statistical ensembles. In probabilistic terms, quantum operations may be . In order that a quantum operation preserve the set of density matrices, we need the additional assumption that it is trace-preserving.

In the context of quantum information, the quantum operations defined here, i.e. completely positive maps that do not increase the trace, are also called quantum channels or stochastic maps. The formulation here is confined to channels between quantum states; however, it can be extended to include classical states as well, therefore allowing quantum and classical information to be handled simultaneously.

Kraus operators[]

Kraus' theorem (named after Karl Kraus) characterizes completely positive maps, that model quantum operations between quantum states. Informally, the theorem ensures that the action of any such quantum operation ${\displaystyle \Phi }$ on a state ${\displaystyle \rho }$ can always be written as ${\displaystyle \Phi (\rho )=\sum _{k}B_{k}\rho B_{k}^{*}}$, for some set of operators ${\displaystyle \{B_{k}\}_{k}}$ satisfying WIKI