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This paper proposes the continuous controller design method for quantum Shannon entropy, which can continuously drive the entropy to track a desired trajectory. We also analyzed the controllability of Shannon entropy in very short time interval. Simulations are done on five dimensional quantum system, which can verify the validation of the method.

Quantum control has become an important topic in quantum information [

For pure state in closed quantum systems, our previous work [

This paper is organized as follows. Section 2 shows the definition of quantum Shannon entropy, and presents our control goal. Section 3 provides the continuous controller design methods. Section 4 shows the numerical simulation examples. Concluding remarks are given in Section 5.

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

where

where both the wave function and the coefficients should be normalized:

Defining the state of the system as

we can get the state space control mode

where both A and

Assuming a system that consists of n states, in which the probability for the i-th state to happen is

which shows the degree of randomness of the system. For example, when

where

reaches its maximum value

where k is a given integer. Here

Our control goal is to drive the entropy to track a desired trajectory. The control of

Here we provide the continuous controller design method which can drive the entropy (8) to track a desired trajectory. Such control task is called “temporal control”, which means not only the destiny should satisfy the requirement, but also the entropy at any instant of the entire process should follow the pre-specified value.

Here we only consider finite-dimensional quantum systems with dimension n. First we can get the time derivative of (8) as

where

because the sum of probabilities should always equal 1. So we have

Here we use

which leads to

Here “

Next we define

which gives

So we can get the controller

If the desired trajectory of

When

Proposition 1. When

where

and

Proof. From (18) we can get

where

So we have

Hence Proposition 1 has been proved. □

Based on Proposition 1, we can get the conditions under which in very short time the entropy can only increase or decrease, which are shown in Theorem 1. It can be seen that Theorem 1 gives the necessary and sufficient conditions.

Theorem 1. In very short time, the entropy can only increase when

and can only decrease when

Proof. Assuming the sampling period is T,

which gives

Similarly we can get

If

Here

If

If

Here

When

(a) If

(b) If

(c) If

Above all, we can get the conclusion in Theorem 1. □

In quantum mechanics, A is often chosen to be diagonal, thus all the elements in A are pure imaginary since A is skew-Hermitian. Assume

From (18) we can see

When

Theorem 2. If A is diagonal, in very short time the entropy can only increase when

and can only decrease when

Proof. From

The discussion can be divided into 3 cases:

(a)

We have

(b)

Here

(c)

Similarly we know if

Above all, we can get the conclusion in Theorem 2. □

When the entropy has reached the target, it needs to be maintained constant. If A is diagonal, from (21) we know we only need

We present simulations for continuous controller on a five-level quantum system. For the five-level case in which the discrete controller is difficult to apply, the continuous controller is adopted to achieve good performance, and the controllability result is verified by simulation.

For five-level quantum system

the discrete controller is difficult to apply, while the continuous controller can be adopted to achieve good performance. For initial state

changes as follows in seven steps: (a) increases to 1.61; (b) keeps constant; (c) increases to 1.66; (d) keeps constant; (e) decreases to 1.61; (f) increases to 1.66; (g) keeps constant. If

Combining (22) with (23) we can get the simulation results for both

From

continuous method can lead to very accurate tracking. At some instant

Next we verify the controllability result by simulation. For initial state

where

From

This paper proposed the continuous controller design method for quantum Shannon entropy. Different from our previous work on discretized controller, the new method can continuously drive the entropy to track a pre-specified target trajectory. Controllability analysis is also provided. The simulation results verified the validation of the method.

This work is supported by the Postdoctoral International Exchanging Program of China.

Yifan Xing,Wensen Huang,Jinhui Zhao, (2016) Continuous Controller Design for Quantum Shannon Entropy. Intelligent Control and Automation,07,63-72. doi: 10.4236/ica.2016.73007