This paper proposes a new quantum control method which controls the Shannon entropy of quantum systems. For both discrete and continuous entropies, controller design methods are proposed based on probability density function control, which can drive the quantum state to any target state. To drive the entropy to any target at any prespecified time, another discretization method is proposed for the discrete entropy case, and the conditions under which the entropy can be increased or decreased are discussed. Simulations are done on both two- and three-dimensional quantum systems, where division and prediction are used to achieve more accurate tracking.

Quantum control has become an important topic in quantum information [

Shannon entropy in atomic calculations has further been related to various properties such as atomic ionization potential [

In the recent research about quantum sliding-mode control (SMC) [

For the biological and physiological datasets, quantifying disorder of the system has become popular as an intense area of promising recent research. In the recent study of a complexity measure for nonstationary signals [

Quantum von Neumann entropy is a good measure of entanglement, and it will reduce to Shannon entropy for the pure state case. It can provide a real-time noise observation and a systematic guideline to make reasonable choice of control strategy. The von Neumann entropy is just a measure of the purity of the given density matrix without explicit reference to information contained in individual measurements [

This paper provides two primary methods to steer the discrete and continuous quantum Shannon entropy via quantum PDF control. And for the discrete case, a method based on discretization approximation is provided which can directly control the entropy and achieve more accurate performance. This paper is organized as follows. Section

In quantum control, the state of a closed quantum system is represented by a state vector (wavefunction)

Assuming a system that consists of

Here, we consider finite dimensional quantum systems. From definition (

Denote the target of

When the state has reached its target, in order to keep it unchanged, we can do the following calculation about the derivative of the probability density:

From definition (

Define the target distribution of

When

In the above two methods, the entropy does not truly enter the control procedure and cannot be driven to the target at any prespecified time. To achieve more direct and accurate control, we can adopt discretization to clarify the relationship between the entropy and the controller.

Assuming the sampling period is

When

Assume

Theorem

Consider the following:

Define two column-vectors as follows:

We can do the following calculation:

For

For

Proposition

Let

Consider the following:

Assume

For

For

Based on Proposition

The entropy cannot be reduced in very small time

From (

So the conditions under which the entropy can only be reduced are

The relationship between

Theorem

We can show the essence of the algorithm in Figure

The relationship between entropy and probability for two-level quantum systems.

For arbitrary point

It should be noted that for the entropy’s maximum point

In order to illustrate the effectiveness of our algorithm, we present simulation examples on both two-level and three-level quantum systems.

Consider the system

For initial state

Evolutions of the entropy for system (

From Figure

Evolutions of the entropy for system (

We can see the entropy can be driven to its destination at any prespecified time, which can be accomplished very quickly in one step. When the entropy has reached its target, from (

For initial state

Change of the entropy with respect to

From Figure

Evolutions of the entropy under

From Figure

Consider the following system:

Consider the following:

Evolutions of the entropy and states for system (

In order to overcome the delays, we can divide one step into halves and use predictions, which can be shown in Figure

Improved control strategy with division and prediction.

The time interval

The simulations are shown in Figure

Evolutions of the entropy and states for system (

This paper proposes a new quantum control method which controls the Shannon entropy of quantum systems. Simulation examples evidenced the effectiveness of the method. A strength of our method is that it provides a direct control algorithm for discrete quantum entropy, rather than the indirect one via PDF control. Our method provides a universal tool for entropy control, which can also contribute to classical information theory. Some immediate extensions of the method include quantum sliding-mode control and coherent control. The extension of the methods to the mixed state case deserves our future research. The applications in correlation energy and biological control are also of keen interests and currently being pursued.

This work is supported by the National Natural Science Foundation of China (Grant no. 60736021).