Relating Entropies of Quantum Channels

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi–Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.


Introduction
One of the important areas of quantum information theory refers to an entropic picture of quantum states and operations.It is well known that the entropic uncertainty principle can be applied in quantum key distribution protocols [1,2] in order to quantify the performance of these protocols.Another possible area where such an approach prevails is resource theory [3].The entropic approach in the description of quantum states can also be useful in studies of quantum phenomena such as correlations or non-locality [4][5][6][7].Another essential aspect of quantum information theory is studying the time evolution of quantum systems interacting with the environment.Entropic characterization of quantum operations can be helpful in the investigation of decoherence induced by quantum channel [8,9].There exists also numerous approaches to the formulation of entropic uncertainty principles [10][11][12][13][14], which can be useful in the analysis of quantum key distribution, quantum communication or characterization of generalized measurements.In [8,9] entropy of quantum channel is defined as the entropy of the state corresponding to the channel by the Jamiołkowski isomorphism.
In quantum information theory, the relative entropy D(ρ σ) = Tr(ρ log ρ − ρ log σ) plays an important role [15] and can be useful in quantifying the difference between two quantum states.In terms of quantum distinguishability, relative entropy can be interpreted as a distance between two quantum states.Nevertheless, it is crucial to remember that it is not a metric as it does not fulfill the triangle inequality.It can be noticed that quantum transformation Φ cannot increase of a distinguishability between quantum states ρ, σ what can be written as D(ρ σ) ≥ D Φ(ρ) Φ(σ) .This fact, sometimes called the data processing inequality plays an important role in the context of hypothesis testing [16].The quantum relative entropy can also be useful in the quantification of quantum entanglement.In this context, the amount of the entanglement of a quantum state ρ can be defined as an optimal distinguishability of the state ρ from separable states e.g.min σ∈SEP D(ρ||σ), where minimization is performed over separable states.It can be noticed that the relative entropy can also be used to define of von Neumann entropy of ρ as S(ρ) = log d − D(ρ 1l/d).This definition shows that the entropy of a quantum state is related to its distance from the maximally mixed state.This approach to von Neumann entropy is useful to define the entropy of quantum channels.Formally it was studied by Gour and Wilde [17], where relative entropy of quantum channels was introduced.According to their approach, the von Neumann entropy of the quantum channel is given by an optimized relative entropy of the output from an extended channel relative to the output of the extended depolarizing channel.The optimization is performed over all possible input states.Moreover, there is also a possibility to define other information measures, e.g., conditional entropy or manual information, in terms of relative entropy.Recently the relative entropy of quantum channel and its generalizations are used in resource theory [18], studies of quantum channels e.g.distinguishability [19], quantum channel discrimination [20] or channel capacity [21,22].

Preliminaries
Let X , Y denote complex Euclidean spaces, let dim(X ) denote the dimension of the space X and L(X , Y) denotes a set of linear operators from X to Y.For simplicity we will write L(X ) ≡ L(X , X ).If ρ ∈ L(X ) is Hermitian (ρ = ρ † ), positive semi-definite (ρ ≥ 0) and a trace-one (Trρ = 1) linear operator then ρ is called as density operator.To keep our expressions simple, we will write the density operator corresponding to a pure states as lowercase Greek letters φ ≡ |φ φ|.In order to keep track of subspaces of composite systems we will write ρ AB ∈ L(X A ⊗ X B ).
The set of all such mapping Φ : L(X ) → L(Y) will be denoted by T(X , Y) and for brevity, we will write T(X ) ≡ T(X , Y).A mapping that is completely positive and trace-preserving is called a quantum channel.The set of all quantum channels will be denoted C(X , Y).There exists a well-known bijection between the sets C(X , Y) and L(Y ⊗ X ), the Choi-Jamiołkowski isomorphism.
It is given by the relation where |φ + = d i |ii and D Φ is called the dynamical matrix or Choi matrix.Normalized D Φ known as Choi-Jamiołkowski state and will be denoted as The von Neumann entropy of ρ ∈ L(X ) is defined by the following formula similarly to the classical Shannon entropy.This equation can be rewritten using the notion of relative entropy, which is defined for states ρ and σ analogously to its classical counterpart [23] Here we use the convention that D(ρ σ) is finite when supp(ρ) ⊆ supp(σ).
The definition of quantum relative entropy can be extended to the case of quantum channels in the following manner [17] The state ρ AR can be chosen as a pure state and the space X R can be isomorphic to X A .Utilizing Eq. ( 5) we get the following definition of the entropy of a quantum channel Then its entropy H(Φ)is defined as where The quantum entropy was also defined in the same matter in [16].However, there exists an earlier definition of entropy of a quantum channel.In [8,9], the quantum channels was characterized by the map entropy, which was defined as the entropy of corresponding Jamiołkowski state.It reads Its entropy H(Φ) is given by the entropy of the corresponding Choi-Jamiołkowski state The above entropy achieves its minimal value of zero for any unitary channel and the maximal value of 2 log dim X B for the completely depolarizing channel.
Based on these two definition we arrive at the following observation.
The two possible definitions of quantum channel entropy H(Φ) and H K (Φ) fulfill the following relation Proof.The proof follows from a direct inspection Let us denote Tr A ψ and we use the well known identity log(1l ⊗ ρ) = 1l ⊗ log ρ and we have Finally we note that Trσ BR 1l⊗log(Tr A ψ) = TrTr B σ BR log Tr A ψ and Tr B σ BR = Tr A ψ. Putting this into Eq.( 10) along with the fact that J Φ is normalized we get the desired result.
The main focus of this work is to find instances that saturate the inequality in Lemma 1.We will mainly focus on the study of unital qubit channels.

Quantum unital qubit channels
In this section we will focus our attention on unital qubit channels, that is Φ ∈ C(C 2 ) such that Φ(1l) = 1l.Our goal here is to show that the supremum present in Eq. ( 5) is achieved for the maximally entangled state |φ + .This can be formally written as the following theorem The remainder of this section contains technical lemmas which combined give the proof of Theorem 2.
A generic two-qubit state can be written as for some qubit unitary matrices U, V .Let us note that the quantum relative entropy is unitarily invariant D(ρ σ) = D(U ρU † U σU † ).Moreover, we use the same the fact that the Jamiołkowski matrix of channel Φ A (ρ) = Φ(W ρW † ) has the same spectrum as the Jamiołkowski matrix of channel Φ, where A is a unitary matrix.Thus, we can skip the unitary operations in our further investigations.We may perform the optimization taking into account only the parameter p which quantifies the amount of entanglement between the input qubits.In order to further simplify notation we will write where |X denotes the vectorization of the matrix X and and we define In the next step we will check the symmetry of D(Φ R) with respect to the parameter p.Hence, we formulate the first lemma.
Lemma 3 Let Φ ∈ C(C 2 ) and let φ P be a two-qubit state as in Eq. (14).Then D(Φ R) is symmetric in the parameter p.

Proof. Let us denote
This observation combined with the fact gives the symmetry of the entropy As for the term Tr (Φ ⊗ 1l)(φ P ) log(R ⊗ 1l)(φ P ) observe that Finally Combining all of these observations yields the lemma.
Subsequently, we prove in next lemma the concavity D(Φ R) with respect to the parameter p. Lemma 4 Given a unital channel Φ ∈ C(C 2 ) and let φ P be a two-qubit state as in Eq. (14).Then the function Proof.For the purpose of this proof let us denote ρ(p) = (Φ ⊗ 1l)(φ P ).Let us also denote A direct calculation shows that l(p) = h(p) + log 2, where h(p) is the point entropy.From this it follows that For g(p) we calculate Observing that ρ (p) = √ J Φ (1l ⊗ ρ(p)) √ J Φ we see that the second term in Eq. ( 22) is equal to zero.Hence, we have From Taylor expansion of derivative formulae for matrix logarithms [24] we get Thus, ( Now we will focus on the last integral above, where λ(ρ) is a diagonal matrix with eigenvalues of ρ on a diagonal and U is a unitary matrix.According to the above considerations Combining this with Eq (21) we see that f (p) is concave.
Based on the above lemmas, it can be concluded that supremum in D(Φ R), where Φ(1l) = 1l, is obtained for p = 1 2 , which indicates it is achieved for the maximally entangled state |φ + .Thus, combination of the lemmas proves Theorem 2.

Asymptotic case
In this section, we will show that Eq. 8 is saturated in the case of large system size.Firstly, let us denote dim(X A ) = dim(X B ) = d.Numerical investigations lead us to formulate the following conjecture.
where Φ chosen randomly according to measures introduced in [25].
To provide some intuition behind this conjecture, we first state a theorem which tells us about the distribution of eigenvalues of the output of an extended random quantum channel, when the input is also chosen randomly.This is summarized as Theorem 5.
Theorem 5 Let Φ be a random channel with Jamiołkowski matrix D Φ , we assume that the limiting distribution of eigenvalues of D Φ is given by µ.Let φ be a random pure state with the limiting distribution of Schmidt values given by ν.We define then the limiting distribution of eigenvalues of σ is given by µ ν.
where W is a random unitary matrix and {|i } is the computational basis.
where Φ W (ρ) = Φ(W ρW † ), note that Jamiołkowski matrix of channel Φ W has the same spectrum that the Jamiołkowski matrix of channel Φ.Next we write Now note, that the eigenvalues of σ are the same as eigenvalues of D Φ W (1l ⊗ diag(λ)), which gives the result.Now, we have the following intuition behind our conjecture.Combining the results from [26,27] with [25,28] we have for large d and uniform distribution of channels Next, we have the following result.Let |φ be a random pure state with the Schmidt numbers chosen according to some measure ν and let |φ be free from Φ. Then the output state has its spectrum given by the free multiplicative convolution µ ν, where µ is the distribution of eigenvalues of D Φ .
Let us consider following optimization target where |φ = (U ⊗ V ) i √ λ i |i, i for some unitary matrices U , V .Note that optimization result is invariant under local operations U , V on |φ , but it depends on λ i .It can be checked that The above expression reaches minimum for uniform distributed λ and them is equal to Trσ log γ = −2 log d.Since σ has spectrum given by µ ν, then where denotes the multiplicative free convolution of measures µ and ν [29].Assuming maximal entropy H(λ) = log(d) implies ν = δ(1/d), which behaves like in identity in the operation .Hence, we have which gives us Now, going back to the entropy of the channel Φ we have The intuition behind our assumption that µ = δ(1/d) is presented in Fig. 1.In it, we present the quantity D(σ γ) where σ and γ are as in Eqs. ( 35) and (36) respectively.The plots are presented for various distributions ν of the Schmidt numbers of the input state |ψ .The red line shows the case ν = Dir(d, 1), the blue line shows the case when ν = Dir(2, 1), the yellow line is the case ν = Dir(d, 2) and finally, the green line shows the case ν = δ(1/d).The dashed line is the quantity log(d) − 1 2 .As can be Subsequently the more non-zero Schmidt numbers and the more they are concentrated in the center of the simplex ∆ d−1 , the closer we get to the quantity we conjecture.Finally, when we choose a deterministic distribution in the center of simplex, we achieve the optimal value.

Conclusion
In this paper, we discuss two approaches to entropic quantification of quantum channels.We begin our studies with a formulation of a lemma, which describes a relation between the entropy of quantum channels proposed by Gour and Wilde [17] and entropy of Jamiołkowski matrix of quantum channels [8,9].We show that both definitions give the same value up to an additive constant in the case of the quantum unital qubit channels.This part of our considerations uses the mathematical language of distinguishability of quantum states and channels.Therefore we assume that obtained results can be used to study resource theories and hypothesis testing.We also provide a conjecture backed by numerical experiments that both formulas provide the same results up to an additive constant in the case of large system size.