Quantifying channels output similarity with applications to quantum control

In this work we aim at quantifying quantum channel output similarity. In order to achieve this, we introduce the notion of quantum channel superfidelity, which gives us an upper bound on the quantum channel fidelity. This quantity is expressed in a clear form using the Kraus representation of a quantum channel. As examples, we show potential applications of this quantity in the quantum control field.


Introduction
Recent applications of quantum mechanics are based on processing and transferring information encoded in quantum states. The full description of quantum information processing procedures is given in terms of quantum channels, i.e. completely positive, trace preserving maps on the set of quantum states.
In many areas of quantum information processing one needs to quantify the difference between ideal quantum procedure and the procedure which is performed in the laboratory. This is especially true in the situation when one deals with imperfections during the realization of experiments. These imperfections can be countered, in a quantum control setup, using various techniques, such us dynamical decoupling [1][2][3][4], sliding mode control [5] and risk sensitive quantum control [6,7]. A different approach is to model the particular setup and optimize control pulses for a specific task in a specific setup [8][9][10][11]. In particular the problem of quantifying the distance between quantum channels was studied in the context of channel distinguishability.
The main aim of this paper is to provide a succinct expression for the channel superfidelity. The main aim of this paper is to provide a succinct expression for the channel output similarity. As a measure of similarity we will consider the superfidelity function and define channel superfidelity.

Quantum states and channels
First, we introduce two basic notions: density operators and superoperatros: Definition 1 We call an operator ρ ∈ L(X ) a density operator iff ρ ≥ 0 and Trρ = 1. We denote the set of all density operators on X by D(X ).
Definition 2 A superoperator is a linear mapping acting on linear operators L(X ) on a finite dimensional Hilbert space X and transforming them into operators on a another finite dimensional Hilbert space Y. Thus Now we define the tensor product of superoperators Definition 3 Given superoperators we define the product superoperator to be the unique linear mapping that satisfies the equation for all operators A 1 ∈ L(X 1 ), A 2 ∈ L(X 2 ).
In the most general case, the evolution of a quantum system can be described using the notion of a quantum channel [12][13][14].
Definition 4 A quantum channel is a superoperator Φ that satisfies the following restrictions: 2. Φ is completely positive, that is for every finite-dimensional Hilbert space Z the product of Φ and identity mapping on L(Z) is a nonnegativity preserving operation, i.e.
Many different representations of quantum channels can be chosen, depending on the application. In this paper we will use only the Krauss representation.

Definition 5
The Kraus representation of a completely positive superoperator (Def. 4(2)) is given by a set of operators K i ∈ L(X , Y). The action of the superoperator Φ is given by: This form ensures that the sueroperator is completely positive. For it to be also trace-preserving we need to impose the following constraint on the Kraus operators

Supporting definitions
In this section we define additional operations used in our proof. We begin with the partial trace Definition 7 For all operators A ∈ L(X ) and all operators B ∈ L(Y) the partial trace is a linear mapping defined as: The extension to operators not in the tensor product form follows from linearity.
Next, we introduce the purification of quantum states: Definition 8 Given Hilbert spaces X and Y, we will call |ζ ∈ X ⊗ Y a purification of ρ ∈ D(X ) if We will also need the notion of conjugate superoperator Definition 9 Given a quantum channel Φ : Note, that the conjugate to completely positive superoperator is completely positive, but is not necessarily trace-preserving Definition 10 We define the linear mapping for dyadic operators as vec(|ψ φ|) = |ψ |φ , for |ψ ∈ Y and |φ ∈ X and uniquely extended by linearity.
We introduce the inverse of the vec(·)
Remark 1 For every choice of Hilbert spaces X 1 , X 2 , Y 1 and Y 2 and every choice of operators A ∈ L(X 1 , Y 1 ), B ∈ L(X 2 , Y 2 ) and X ∈ L(X 2 , X 1 ) it holds that:

Quantum channel fidelity
First, we introduce the fidelity and channel fidelity Definition 12 Given operators A ∈ L(X ) and B ∈ L(X ) such that A ≥ 0, B ≥ 0 we define the fidelity between A and B as: Definition 13 Quantum channel fidelity of a channel Φ : L(X ) → L(X ) for some σ ∈ D(X ) is defined as: where the infimum is over all Hilbert spaces Z and all ξ ∈ D(X ⊗ Z) such that Tr Z ξ = σ It can be shown that this infimum is independent of ξ and is given by where Φ has the Kraus form given by

Our results
Definition 14 Consider two quantum channels Φ, Ψ : L(X ) → L(X ) and a density operator σ ∈ D(X ). We define the quantum channel superfidelity to be: where the infimum is over all Hilbert spaces Z and over all purifications ξ = |ζ ζ| ∈ D(X ⊗ Z) of σ.
The channel superfidelity G ch (Φ, Ψ; σ) places a lower bound on the output superfidelity of two quantum channels in the case of the same input states. Henceforth, where unambigous, we will write the channel superfidelity as G ch .
Theorem 1 Given quantum channels Φ, Ψ : L(X ) → L(X ) with Kraus forms given by the sets {A i : A i ∈ L(X )} i and {B j : B j ∈ L(X )} j respectively the quantum channel superfidelity is given by: Proof. As we limit ourselves only to pure states ξ, in order to calculate the superfidelity, we need to compute the following quantities: As the general idea is shared between all of these quantities, we will show here the calculation for the first one. We get where the first equality follows from the definition of the conjugate superoperator. The Kraus form of the suoperoperator Φ † • Ψ is given by the set for some U ∈ L(Z, X ) such that U U † = Π im(σ) . We obtain: This quantity is independent of the particular purification of σ. Following the same path for the other two quantities shown in Eq (25), we recover the expression for the channel superfidelity from Eq (24).
The following simple corollaries are easily derived from Theorem 1.
Proof. If Φ is a unitary channel, then the second term in Eq. (24) vanishes. Let us assume that Ψ has a Kraus form The Kraus form of the channel Ψ is given by the set {U † A i : A i ∈ L(X )}. Using this in Eq. (21) we get F ch (Ψ ; σ) = i |TrσU † A i | 2 . This completes the proof.

Corollary 2
In general, the quantum channel superfidelity is an upper bound on the quantum channel fidelity, that is: Proof. Obvious.

Corollary 3
If σ ∈ D(X ) is a pure state i.e. σ = |ψ ψ| then Proof. Let us only focus on the first terms in Eq. (8) and Eq. (24). We will denote these terms T and T ch respectively. Let us assume that channels Φ and Ψ have Kraus forms {A i ∈ L(X )} i and {B j ∈ L(X )} j respectively. We get: Performing similar calculations for other terms, we recover Eq. (31)

Simple examples
In this section we provide a number of examples of the application of Theorem 1.

Erasure channel
Definition 15 Given a quantum state ξ ∈ D(X ) the erasure channel is given by: for any A in L(X ). The Kraus form of this channel is given by the set Let us consider the superfidelity between the erasure channel Φ and a unitary channel ∀σ ∈ D(X ) Ψ : σ → U σU † for some U ∈ U(X ). We note that the second term in Eq. (24) vanishes. What remains is: where µ ↓ i and λ ↓ i denote the eigenvalues of σ and ξ respectively, sorted in a descending order. The last inequality follows from von Neumann's trace inequality [18].

Sensitivity to channel error
Consider a quantum channel Φ with the Kraus form {A i : A i ∈ L(X )} and a quantum channel Ψ : H). We get: Now, we concentrate on the change of the quantum channel superfidelity under the change of . As we are interested only in small values of , we expand Eq. (35) up to the linear term in the Taylor series. For small values of we get: Note that this depends on the value of the observable H of the operator A j σA † i .

Sensitivity to Hamiltonian parameters
In this section we will show how the channel superfidelity is affected by errors in the system Hamiltonian parameters. First, we will show analytical results for a single qubit system at a finite temperature. Next, we show numerical results for a simple, three-qubit spin chain.

Single qubit at a finite temperature
A single qubit at a finite temperature is described by the master equatioṅ where σ + = |1 0|, σ − = σ † + and is the error in Ω. Our goal is to calculate the quantum channel superfidelity between the case when there is no error in Ω, i.e. = 0 and the case with error in Ω. Henceforth, we will assume γ − = γ + = 1 for clarity. For a given time T , Eq. (37) may be rewritten as where Φ T is a quantum channel in the quantum dynamical semigroup. A natural representation M Φ T for the channel Φ T may be found as [19]: where In this representation we may rewrite Eq. (38) as vec(ρ(T )) = M Φ T vec(ρ(0)).
The Choi-Jamio lkowski representation of the channel Φ T is given by Here, M R denotes the reshuffle operation on matrix M [12]. Now, it is simple to find the Kraus form of the channel Φ T . The Kraus operators are related to the eigenvalues λ i and eigenvectors |λ i of D Φ T in the following manner: Inserting these Kraus operators into Eq. (24) we get where ρ ii (0) = i|ρ(0)|i . Note that, we get G ch = 1 in two cases. First, for large T and second when T = π 2 . As we are mainly interested in small values of , we expand cos T up to the second term in the Taylor series. We get: In this setup the channel superfidelity has a quadratic dependence on the error parameter . This should be compared with the results in Sec. 4.2.

Quantum control example
In this section, we consider a three-qubit spin chain with dephasing interactions with the environment. We will consider piecewise constant control pulses. The time evolution of the system is governed by the equation: where H = H d + H c . Here H d is the drift term of the Hamiltonian given by where σ i α denotes σ α acting on site i. We set the control Hamiltonian H c to: where h x (t i ) and h y (t i ) denote the control pulses in the time interval t i . We set the target to be i.e. a NOT gate on the third qubit. We fixed the number of time intervals N = 64, the total evolution time T = 6.1 and the maximum amplitude of a single control pulse ∀k ∈ x, y max(|h k |) = 10. First, we optimize control pulses for the system, such that we achieve a high fidelity of the gate U T . Next, to each control pulse we add a noise term h witch has a normal distribution, h = N (0, s). Fig. 1 shows the change of G ch as a function of the standard deviation s. We have conducted 100 simulations for each value of s. As expected, the quantum channel superfidelity decreases slowly for low values of s. After a certain value the decrease becomes rapid. As values of s increase, the minimum and maximum achieved fidelity diverge rapidly. This is represented by the shaded area in Fig. 1. We can approximate the average value of the channel fidelity as G ch ≈ 1 − cs 2 . Fitting this function to the curve shown in Fig. 1b gives a relative error which is less then 0.5%.

Conclusions
We have studied the superfidelity of a quantum channel. This quantity allows us to provide an upper bound on the fidelity of the output of two quantum channels. We shown an example of application of this quantity to a unitary and an erasure channel. The obtained superfidelity can be easily limited from above by the product w eigenvalues of the input state σ and the result of the erasure channel ξ.
Furthermore, as shown in our examples, the quantum channel superfidelity may have potential applications in quantum control theory as an easy to compute figure of merit of quantum operations. In a simple setup, where the desired quantum channel is changed by a unitary transformation U = exp(−i H) we get a linear of the decrease of channel superfidelity on the noise parameter . On the other hand, when we introduce the noise as a control error in a single qubit quantum control setup, we get a quadratic dependence on the noise parameter.
Finally, we shown numerical results for a more complicated system. We calculated the quantum channel superfidelity for a three-qubit quantum control setup. First we found control pulses which achieve a high fidelity of the  Figure 1: Quantum channel superfidelity as a function of noise in the system's control pulses. The shaded area represents the range of the achieved channel superfidelity, the black line is the average channel superfidelity. desired quantum operation, next we introduced Gaussian noise in the control pulses. Our results show, that the quantum channel superfidelity stayed high for a wide range of the noise strength.