# Local Channels under Decomposition of Quantum Gates in Bipartite Systems

Authors: Yihui Lao; Pei Wang; Shanli Liao
Journal: Journal of Mathematics
 (2023)

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## Abstract

For a bipartite quantum system with Hilbert spaceHm⊗Hn, this paper considers a quantum channel ϕ from the system to itself which preserves the set of separable states. Within this class of quantum channels, the paper characterizes those channels which preserve the Tsallis entropy of combinations of separable states through decomposing quantum gates, where the characterization is given in terms of the action of the channel on separable states.


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## Body

## 1. Introduction

In the mathematical framework of quantum information theory, quantum states (density matrices) are positive operators with trace 1 on complex Hilbert spaceHm with dimension m. The convex set of all states on Hm are denoted by SHm. A pure state is a rank one state, i.e., a rank one projection, and the set of all pure states are denoted by P1Hm. Furthermore, ρ∈P1Hm if ρ2=ρ; ρ is a mixed state if ρ2≠ρ. In general, people deal with multipartite systems. The underlying space Hm of a multipartite composite quantum system is a tensor product of underlying spaces Hmi of its subsystems, that is, Hm=Hm1⊗Hm2⊗⋯⊗Hmk. The system is called a a bipartite one in case k=2. ρ∈SHm⊗Hn is said to be separable if ρ can be written as ρ=∑i=1kpiρi⊗σi, where ρi and σi are states on Hm and Hn, respectively, and pi are positive numbers with ∑i=1kpi=1. Otherwise, ρ is said to be inseparable or entangled. The full separability of multipartite states can be defined similarly. SsepHm⊗Hn stands for the convex set of all separable states on Hm⊗Hn and PursepHm⊗Hn for the set of all separable pure states on Hm⊗Hn. It is obvious that a tensor product ω⊗τ is a pure state if and only if ω and τ are both pure states. The set of all m−th square matrices are denoted by MHm. Symbol A∗ stands for the conjugate transpose of a matrix A.Entanglement is a basic physical resource to realize various quantum information and quantum communication tasks. It is important to determine whether or not a state in a composite system is separable, which is also a very laborious work in this field. It is sensible and helpful to find the characterization of linear maps on states leaving some invariant quantum properties, which will simplify a given state so that it is easier to detect the entanglement in it. Many excellent works of these problems corresponding to multipartite quantum systems [1–4] are worthy to be learnt. Especially, lots of scholars focused on characterizing maps on quantum states preserving entropy of convex combinations. In reference [5], He et al. have discussed von Neumann entropy-preserving maps on quantum states in finite dimensional Hilbert space. In reference [6], Karder and Petek have established the form of maps preserving generalized entropy of convex combinations of states in the left quaternionic Hilbert space H with dimension 3≤dimH≤n. In their investigation [7] on von Neumann entropy, Yuan and Wang proved that Sϕρ=Sρ∀ρ∈SH if and only if there exists a unitary operator U such that ϕρ=UρU†, where ϕρ is a bistochastic quantum operation, i.e., ϕρ=∑iEiρEi†∀ρ∈SH, and Ei is a set of matrices known as operation elements and ∑iEi†Ei=idH. Moreover, in reference [8], Yuan and Paul extended the result to infinite dimensional separable Hilbert space case. Inspired by the results in references [5, 7]), we now investigate Tsallis entropy-preserving quantum channels on the tensor product (bipartite systems) states in complex Hilbert space with finite dimensions. Next, we will review the definition of the quantum channels and Tsallis entropy, respectively. A quantum channel is described by a trace-preserving completely positive map Φ:MHm⟶MHn if and only if it admits an expression ΦX=∑iAiXAi∗, where Ai are m×n matrices and ∑iAi∗Ai=idn. The Tsallis entropy [9] is defined as follows:(1)Srρ=1−r−1trρr−1,r≠1.In this paper, we always assume thatr>0 and r≠1, i.e., r∈ℝ+\1. It is well-known that Srρ has concavity and subadditivity [10]. Using the spectral decomposition theorem, it is easy to see that limr⟶1Srρ=Sρ, where Sρ=−trρlnρ is the von Neumann entropy. In fact, assuming that quantum state ρ=diagλ1,λ2,…,λmλi≥0,i=1,2,…,m acting on a complex finite Hilbert space Hm with dimension m, noting ∑i=1mλi=1 and the convention 0ln0≡0 which comes from limx⟶0xlnx=0, one can get this relationship through L′ Hospital rule.In quantum computation, a quantum gate usually represented by a unitary matrix is a basic quantum circuit that operates on a small number of qubits. Such a gate has the same number of inputs as outputs. Efforts in this paper to decompose a general quantum gate into the product of two-state unitary matrices are our hope that may simplify sorts of logic operations in quantum computation involving the encoding of computational tasks into the temporal evolution of a quantum system.Theorem 1.
LetHm,Hn be the complex Hilbert spaces with dimHm=m,dimHn=n, respectively. Let ϕ:SHm⊗Hn⟶SHm⊗Hn be a quantum channel and ϕSsepHm⊗Hn=SsepHm⊗Hn. Then, the following statements are equivalent:(1)
For anyρ,σ∈SsepHm⊗Hn, t∈0,1, and r∈ℝ+\1, quantum channel ϕ preserves Tsallis entropy Srtρ+1−tσ=Srtϕρ+1−tϕσ.(2)
There exist unitary operatorsUm and Vn acting on Hm,Hn, respectively, such that ϕρ=Um⊗VnρUm⊗Vn∗.Remark 1.
Actually in reference [11], we have discussed the similar topic both in single finite and infinite dimensional complex separable Hilbert space H and obtained the conclusion that ϕρ=UρU∗ρ∈H. Even if the state ρ in the formula ϕρ=UρU∗ is separable, the unitary operator U (quantum gate) may not be separable when this issue is discussed under the bipartite systems context. Fortunately, we succeeded in splitting U into two parts via making full use of the properties of Tsallis entropy.
The remaining part of this paper is organized as follows: In Section2, we give some lemmas that will offer services and be helpful to prove the main theorem in Section 3.


## 2. Lemmas

Before the proof of the main theorem, we need the following lemmas.Lemma 1 (see Lemma3, [10]).
LetHm be a complex Hilbert space, for any r∈ℝ+\1, ρ,σ∈SHm, Srρ=0 if and only if ρ∈PurHm. Srρ⊗σ=Srρ+Srσ+1−rSrρSrσ.Lemma 2 (see Lemma2.2 [12]).
LetA, C, E∈MHm and B,D, F∈MHn. If A⊗B=C⊗D+E⊗F, then either C and E are linearly dependent or D and F are linearly dependent.Lemma 3 (see Lemma 13, [11]).
LetHm be a complex Hilbert space, Tsallis entropy Srρ is strictly concave, i.e., Srtρ+1−tσ≥tSrρ+1−tSrσ for any ρ,σ∈SHm, t∈0,1, r∈ℝ+\1. Moreover, equality holds if and only if ρ=σ.Lemma 4 (see Theorem 1, [11]).
LetHm be a complex Hilbert space. Then, for any r∈ℝ+\1 and ρ,σ∈SHm, Srρ=Srσ if and only if there exists a unitary Wσ acting on Hm such that ρ=WσσWσ∗, where Wσ is dependent on σ.Lemma 5.
LetHn be a complex Hilbert space and ϕ be a quantum channel. Then, for any r∈ℝ+\1 and ρ∈SHn, Srϕρ=Srρ if and only if there exists a unitary W acting on Hn such that ϕρ=WρW∗, where W is linearly independent.Proof of Lemma 1.
Sufficiency is straightforward.
Necessity. By Lemma4, for r∈ℝ+\1, Srϕρ=Srρ implies ϕρ=WρρWρ∗, where Wρ is a unitary on Hm and is dependent on ρ. Then, Wρ∗∑iAiρAi∗Wρ=ρ. Using Theorem 3.5 [13], we have ∑iAi∗WρρWρ∗Ai=ρ, which means ∑i,jAi∗WρWρ∗AjρAj∗WρWρ∗Ai=ρ. Then, ∑i,jAi∗AjρAj∗Ai=ρ. By Corollary 3.4 in reference [14], it follows Ai∗Ajρ=ρAi∗Aj, for 1≤i,j≤n and any quantum state ρ. Since ∑iAi∗Ai=I, we obtain Ai∗Aj=λijI and Ai∗Ai=λiiI, where λii>0, ∑inλii=1 and 1≤i,j≤n. It is obvious that(2)λij=1ntrAi∗Aj=1ntrAj∗Ai¯=1nnλji¯=λji¯,where λ¯ is the complex conjugate of λ. Thus,(3)A1∗A1A1∗A2⋯A1∗AnA2∗A1A2∗A2⋯A2∗An⋮⋮⋮⋮An∗A1An∗A2⋯An∗An=λ11Iλ12I⋯λ1nIλ21Iλ22I⋯λ2nI⋮⋮⋮⋮λn1Iλn2I⋯λnnI=M⊗I,where M=λijn×n. Since M≥0, then for 1≤i,j≤n, we have λij2≤λiiλjj. Taking tr on the both sides of Equality ∑i,jAi∗WρWρ∗AjρAj∗WρWρ∗Ai=ρ, we have ∑i,jλij2=1=∑inλii2. By direct calculation, we obtain ∑i≠jλij2=∑i≠jλiiλjj. Noticing λij2≤λiiλjj, we have λij2=λiiλjj,1≤i,j≤n. Thus, for 1≤j≤n, we obtain Aj∗A1=λj1I=λ11λjje−1θjI, which means(4)A1λ11=Ajλjje−1θj,A1∗λ11=Aj∗λjje−−1θj.
Finally, we know(5)ϕρ=A1∗ρA1+A2∗ρA2+⋯+An∗ρAn=λ11+λ22+⋯λnnλ11A1∗ρA1=A1λ11ρA1∗λ11.
LetU=A1∗/λ11, then U is a unitary matrix such that ϕρ=UρU∗.Remark 2.
The proof of Lemma5 can be seen in [7] where von Neumann entropy was discussed.Lemma 6.
LetHm,Hn be complex Hilbert spaces with dimHm=m,dimHn=n, respectively. ϕ:SHm⊗Hn⟶SHm⊗Hn be a quantum channel and ϕSsepHm⊗Hn=SsepHm⊗Hn. For any ρ,σ∈SsepHm⊗Hn, t∈0,1, and r∈ℝ+\1, ϕ preserves Tsallis entropy Srtρ+1−tσ=Srtϕρ+1−tϕσ. Then,(1)
Srρ=Srϕρ(2)
ϕPursepHm⊗Hn=PursepHm⊗HnProof of Lemma 6.
(1)
Takingt=1 in equality Srtρ+1−tσ=Srtϕρ+1−tϕσ, we have Srρ=Srϕρ for any ρ∈SsepHm⊗Hn.(2)
Ifρ is a pure state, i.e., ρ∈PursepHm⊗Hn, then by Lemma 1, we obtain Srρ=0, which follows that Srϕρ=0. By Lemma 1 again, we have ϕρ∈PursepHm⊗Hn. One can deal with the converse case similarly. The proof of Lemma 6 is now completed.
For pure stateP∈PurHm and Q∈PurHn, let(6)LP=P⊗Q1∈SHm⊗Hn:∀Q1∈PurHn;RQ=P1⊗Q∈SHm⊗Hn:∀P1∈PurHm.Lemma 7.
LetHm,Hn be complex Hilbert spaces with dimHm=m,dimHn=n, respectively. ϕ:SHm⊗Hn⟶SHm⊗Hn be a quantum channel and ϕSsepHm⊗Hn=SsepHm⊗Hn. For any ρ,σ∈SsepHm⊗Hn, t∈0,1, and r∈ℝ+\1, ϕ preserves Tsallis entropy Srtρ+1−tσ=Srtϕρ+1−tϕσ. Then, one of the following statements holds.(1)
For any pure stateP∈PurHm, there exists some pure state P¯∈PurHm such that ϕLP⊆LP¯(2)
For any pure stateP∈PurHm, there exists some pure state Q¯∈PurHn such that ϕLP⊆RQ¯Proof of Lemma 3.
By Lemma6, we have ϕPursepHm⊗Hn=PursepHm⊗Hn. Then, for any separable pure state P⊗Q∈PursepHm⊗Hn, there exists some separable pure state P¯⊗Q¯∈PursepHm⊗Hn such that ϕP⊗Q=P¯⊗Q¯.
First, we fix a pure stateP∈PurHm, for any Q1, Q2∈PurHn with Q1≠Q2; by Lemma 6, there exist pure states P¯i∈PurHm and Q¯i∈PurHn,i=1,2 such that(7)ϕP⊗Q1=P¯1⊗Q¯1,ϕP⊗Q2=P¯2⊗Q¯2.
Since a quantum channel must be affine on quantum states, we see thatϕ is bijective and ϕtρ+1−tσ=tϕρ+1−tϕσ for quantum states ρ,σ. Thus, if P⊗Q1≠P⊗Q2, we have P¯1⊗Q¯1≠P¯2⊗Q¯2, and(8)P¯t,1⊗Q¯t,1=ϕP⊗tQ1+1−tQ2=ϕtP⊗Q1+1−tP⊗Q2=tϕP⊗Q1+1−tϕP⊗Q2=tP¯1⊗Q¯1+1−tP¯2⊗Q¯2.
By Lemma2, we know that either (i) Q¯1 and Q¯2 are linearly dependent or (ii) P¯1 and P¯2 are linearly dependent.(i)
IfQ¯1 and Q¯2 are linearly dependent, since P¯i and Q¯i,i=1,2 are all pure states and P¯1⊗Q¯1≠P¯2⊗Q¯2, we have Q¯1=Q¯2 and P¯1, P¯2 are linearly independent. Let Q¯1=Q¯2=Q¯, then(9)ϕP⊗Q1=P¯1⊗Q¯,ϕP⊗Q2=P¯2⊗Q¯.Now, for anyQ∈PurHn, let ϕP⊗Q=P¯⊗Q¯¯. Since P¯ is linearly independent of at least one of P¯1 and P¯2, otherwise, we will have P¯1=P¯2. Let P¯ and P¯1 are linearly independent. Noticing the following equality and recalling Lemma 2(10)P¯t,2⊗Q¯t,2=ϕP⊗tQ1+1−tQ=ϕtP⊗Q1+1−tP⊗Q=tϕP⊗Q1+1−tϕP⊗Q=tP¯1⊗Q¯+1−tP¯⊗Q¯¯.We getQ¯=Q¯¯. Thus, for a fixed P∈PurHm, there exists some pure state Q¯∈PurHn such that ϕLP⊆RQ¯.(ii)
IfP¯1 and P¯2 are linearly dependent. Similarly, one can get that, for a fixed P∈PurHm, there exists some pure state P¯∈PurHm such that ϕLP⊆LP¯.
Now, in the claim for a fixedP0∈PurHm, there exists some pure state P0¯∈PurHm such that ϕLP0∈LP0¯. Next, we will show that for any P∈PurHm, there exists some pure state P¯∈PurHm such that ϕLP⊆LP¯.
Let(11)A=P∈PurHm:ϕLP⊆LP¯,P¯∈PurHm,B=P∈PurHm:ϕLP⊆RQ¯,Q¯∈PurHn.
Then,A∪B=PurHm. Assume that B≠∅, then there exist a P1∈B and a Q¯∈PurHn such that ϕP1⊗Q=PQ¯⊗Q¯ for all Q∈PurHn. Noticing that ϕLP0⊆LP0¯, we have(12)P¯t,3⊗Q¯t,3=ϕtP1+1−tP0⊗Q,=ϕtP1⊗Q+1−tP0⊗Q,=tϕP1⊗Q+1−tϕP0⊗Q,=tPQ¯⊗Q¯+1−tP0¯⊗QP0¯.
Due to the arbitrariness ofQ, we have P0¯=PQ¯ and Q¯=QP0¯. Thus,(13)ϕP1⊗Q=P¯0⊗Q¯,=ϕP0⊗Q.
This contradicts to the injectivity ofϕ, then B=∅, which follows that for any P∈PurHm, there exists some pure state P¯∈PurHm such that ϕLP⊆LP¯. Similarly, if for a fixed P0∈PurHm, there exists some pure state Q0¯∈PurHn such that ϕLP0⊆RQ0¯. Then, for any P∈PurHm, there exists some pure state Q¯∈PurHn such that ϕLP⊆RQ¯. We complete the proof of Lemma 3.
A similar argument to Lemma7 will result in Lemma 8.Lemma 8.
LetHm,Hn be the complex Hilbert spaces with dimHm=m,dimHn=n, respectively. ϕ:SHm⊗Hn⟶SHm⊗Hn be a quantum channel and ϕSsepHm⊗Hn=SsepHm⊗Hn. For any ρ,σ∈SsepHm⊗Hn, t∈0,1, and r∈ℝ+\1, ϕ preserves Tsallis entropy Srtρ+1−tσ=Srtϕρ+1−tϕσ. Then, one of the following statements holds:(1)
For any pure stateQ∈PurHn, there exists some pure state Q¯∈PurHn such that ϕRQ⊆RQ¯(2)
For any pure stateQ∈PurHn, there exists some pure state P¯∈PurHm such that ϕRQ⊆LP¯
Combining Lemmas7 and 8 will give.Lemma 9.
Ifϕ:SsepHm⊗Hn⟶SsepHm⊗Hn satisfies the same hypotheses as that in Lemma 7, then one of the following statement holds:(1)
Case (1) in Lemma7 and Case (3) in Lemma 8 hold(2)
Case (2) in Lemma7 and Case (4) in Lemma 8 holdProof of Lemma 4.
If Case (1) in Lemma7 and Case (4) in Lemma 8 hold simultaneously, then for any P∈PurHm and Q∈PurHn, there exist P¯, P¯¯∈PurHm such that ϕLP⊆LP¯ and ϕRQ⊆LP¯¯; then,(14)P¯t,4⊗Q¯t,4=ϕtP+1−tP⊗Q=ϕtP⊗Q+1−tP⊗Q=tϕP⊗Q+1−tϕP⊗Q=tP¯⊗Q¯+1−tP¯¯⊗Q¯.
Due to the arbitrariness ofQ¯, we have(15)tP¯+1−tP¯¯=P¯t,4.
Recalling thatP¯t,4 is a pure state and P¯≠P¯¯. It forces tP¯+1−tP¯¯=P¯t,4. A contradiction appears. Similarly, Case (2) in Lemma 7 and Case (3) in Lemma 8 do not occur.Lemma 10 (see [10]).
Forr∈ℝ+\1, let ρi∈SHm,i=1,…,m, 0<λi<1 and ∑i=1mλi=1, then(16)Sr∑i=1mλiρi≤∑i=1mλirSrρi+1−∑i=1mλirr−1,where the equality holds if and only if ρiρj=0 for all 1≤i,j≤m.We denoteρσ=0 if ρ and σ are orthogonal. For ρ,σ∈SsepHm⊗Hn, let ρ=ρ1⊗ρ2,σ=σ1⊗σ2, where ρ1,ρ2∈SHm,σ1,σ2∈SHn, then ρσ=ρ1⊗ρ2σ1⊗σ2=ρ1σ1⊗ρ2σ2. Thus, ρσ=0 if and only if either ρ1σ1=0 holds or ρ2σ2=0 holds.Lemma 11 (see [11]).
For anyr∈ℝ+\1 and ρ∈SHm, Srρ=1−m1−r/r−1 if and only if ρ=Im/m, where Im is the identity on Hm.


## 3. Proof of the Main Theorem

Now, we come to the proof of the main theorem.Proof of the Main Theorem 1.
Assume that Case (1) in Lemma9 holds, that is, for any P∈PurHm and Q∈PurHn, we have ϕLP⊆LP¯ and ϕRQ⊆RQ¯. Then, for any P∈PurHm, there exist τ1:PurHm⟶PurHm, τ2:PurHm⊗PurHn⟶PurHn (τ2 is dependent on P) such that(17)ϕP⊗Q=τ1P⊗τ2P,Q,for any Q∈PurHn. In order to prove τ2P,Q is independent on P, now assume that P1,P2∈PurHm and P1≠P2, then(18)ϕP1⊗Q=τ1P1⊗τ2P1,Q,ϕP2⊗Q=τ1P2⊗τ2P2,Q,which implies(19)P¯⊗Q¯=ϕtP1+1−tP2⊗Q=ϕtP1⊗Q+1−tP2⊗Q=tϕP1⊗Q+1−tϕP2⊗Q=tτ1P1⊗τ2P1,Q+1−tτ1P2⊗τ2P2,Q,for some P¯⊗Q¯∈SHm⊗Hn and for any t∈0,1. This means, by Lemma 2, τ2P1,Q and τ2P2,Q are linearly dependent or τ1P1 and τ1P2 are linearly dependent. If τ2P1,Q and τ2P2,Q are linearly dependent, immediately we have τ2P1,Q=τ2P2,Q since τ2P1,Q,τ2P2,Q are pure states, which leads to that τ2 is independent on P. If τ1P1 and τ1P2 are linearly dependent; similarly, we have τ1P1=τ1P2, thus(20)ϕtP1+1−tP2⊗Q=τ1P1⊗tτ2P1,Q+1−tτ2P2,Q.
Noting that there some pure stateQ¯ such that ϕRQ⊆RQ¯. Then,(21)ϕtP1+1−tP2⊗Q=tτ1P1,Q+1−tτ1P2,Q⊗τ2Q,which means(22)tτ2P1,Q+1−tτ2P2,Q=τ2Q.
Recalling thatτ2P1,Q, τ2P2,Q, and τ2Q are all pure states, thus τ2P1,Q=τ2P2,Q=τ2Q. As a result, we have proved that for any separable pure state P⊗Q, there are two bijectives (ϕ is affine) τ1:PurHm⟶PurHm and τ2:PurHn⟶PurHn such that(23)ϕP⊗Q=τ1P⊗τ2Q.
Similarly, if the Case (2) in Lemma9 holds, we can obtain that for any separable pure state P⊗Q, there are two bijectives τ¯1:PurHn⟶PurHm and τ¯2:PurHm⟶PurHn such that(24)ϕP⊗Q=τ¯1Q⊗τ¯2P.
In the following part, we will claim thatϕ satisfying main theorem’s hypotheses preserves orthogonality. That is, for any ρ,σ∈SsepHm⊗Hn, if ρσ=0, then ϕρϕσ=0. Actually, by Lemma 6, we know Srρ=Srϕρ for any ρ∈SsepHm⊗Hn, by Lemma 10, we have(25)trSrρ+1−trSrσ+1−tr−1−trr−1=Srtρ+1−tσ=Srtϕρ+1−tϕσ=trSrϕρ+1−trSrϕσ+1−tr−1−trr−1.
By Lemma10 again, we obtain ϕρϕσ=0, i.e., ϕ preserves orthogonality.
SincedimHm⊗Hn<∞, for any ρ∈SHm, it is clear that ρ can be written as ρ=∑i=1mλiPi, where 0≤λi≤1i=1,…,m, ∑i=1mλi=1 and Pi∈PurHmi=1,…,m. For any Q∈PurHn, noticing that Pi⊗QPj⊗Q=0,i≠j, SrPi⊗Q=SrϕPi⊗Q and equality (23), by Lemma 10, we get(26)Srϕρ⊗Q=Srρ⊗Q=Sr∑i=1mλiPi⊗Q=∑i=1mλirSrPi⊗Q+1−∑i=1mλirr−1=∑i=1mλirSrϕPi⊗Q+1−∑i=1mλirr−1=Sr∑i=1mλiϕPi⊗Q=Sr∑i=1mλiτ1Pi⊗τ2Q.
Then, there is a unitary matrixW such that(27)ϕρ⊗Q=W∑i=1mλiτ1Pi⊗τ2QW∗.
LetW=Imn, where Imn is the identity element on Hm⊗Hn. Then,(28)ϕρ⊗Q=∑i=1mλiτ1Pi⊗τ2Q.
We define a mapτ1Qρ:SHm⟶SHm by τ1Qρ=∑i=1mλiτ1Pi. Thus,(29)ϕρ⊗Q=τ1Qρ⊗τ2Q.
In equality (29), if ρ=Im/m (Im is the identity element on Hm), then τ1Qρ=τ1QIm/m=Im/m. Actually, by Lemmas 1 and 11 and noticing that Srϕρ⊗σ=Srρ⊗σ and τ2Q is a pure state, we have SrϕIm/m⊗Q=SrIm/m⊗Q=SrIm/m=Srτ1QFIm/m⊗τ2Q=Srτ1QIm/m=1−m1−r/r−1.
In equality (29), map τ1Q:SHm⟶SHm satisfies(30)Srtτ1Qρ+1−tτ1QImm=Srtρ+1−tImm.
In fact,(31)Srtρ+1−tImm=Srtρ+1−tImm⊗Q=Srtρ⊗Q+1−tImm⊗Q=Srtϕρ⊗Q+1−tϕImm⊗Q=Srtτ1Qρ⊗τ2Q+1−tτ1QImm⊗τ2Q=Srtτ1Qρ+1−tτ1QImm⊗τ2Q=Srtτ1Qρ+1−tImm.
By equality (30) and Theorem 2 in [11], there exists a unitary matrix UQ on MHm such that τ1Qρ=UQρUQ∗, then equality (29) can be written as(32)ϕρ⊗Q=UQρUQ∗⊗τ2Q.
TheUQ in equality (32) is independent on Q. Actually, for two different pure states Q1,Q2∈PurHn, we get τ1Qiρ=UQiρUQi∗i=1,2 for a state ρ∈SHm. If P∈PurHm, by equalities (23), (29), and (32), we have(33)τ1P⊗τ2Q1=ϕP⊗Q1=τ1Q1P⊗τ2Q1=UQ1PUQ1∗⊗τ2Q1,τ1P⊗τ2Q2=ϕP⊗Q2=τ1Q2P⊗τ2Q2=UQ2PUQ2∗⊗τ2Q2.
For all pure statesP∈PurHm, since ϕLP⊆LP¯, then τ1P=UQ1PUQ1∗=UQ2PUQ2∗. By Lemma 4.1 in reference [15], we know UQ1=UQ2. Let Um=UQ1=UQ2, thus, for any ρ∈SHm and Q∈PurHn, we get(34)ϕρ⊗Q=UmρUm∗⊗τ2Q.
Similarly, for any statesσ∈SHn and P∈PurHm, we get(35)ϕP⊗σ=τ1P⊗VnρVn∗.
Now, we claimτ2Q=VnQVn∗ in equality (34), where Vn is a unitary element on Hn. Actually, for any state ρ∈SHm, Q1,Q2∈PurHn,t∈0,1, we have(36)Srρ+SrtQ1+1−tQ2+1−rSrρSrtQ1+1−tQ2=Srρ⊗tQ1+1−tQ2=Srtρ⊗Q1+1−tρ⊗Q2=Srtϕρ⊗Q1+1−tϕρ⊗Q2=SrtUmρUm∗⊗τ2Q1+1−tUmρUm∗⊗τ2Q2=SrUmρUm∗⊗tτ2Q1+1−tτ2Q2=SrUmρUm∗+Srtτ2Q1+1−tτ2Q2+1−rSrUmρUm∗Srtτ2Q1+1−tτ2Q2=Srρ+Srtτ2Q1+1−tτ2Q2+1−rSrρSrtτ2Q1+1−tτ2Q2,which implies(37)SrtQ1+1−tQ2=Srtτ2Q1+1−tτ2Q2.
Recalling thatτ2 is surjective, by [10], we obtain τ2Q=VnQVn∗, Thus, equality (34) becomes(38)ϕρ⊗Q=UmρUm∗⊗VnQVn∗.
Similarly, equality (35) turns into(39)ϕP⊗σ=UmPUm∗⊗VnσVn∗.
Finally, for anyρ∈SHm,σ∈SHn, let σ=∑j=1lμjQj,1≤l≤n,(40)Srϕρ⊗σ=Srρ⊗σ=Srρ⊗∑j=1lμjQj=∑j=1lμjrSrρ⊗Qj+1−∑j=1lμjrr−1=∑j=1lμjrSrϕρ⊗Qj+1−∑j=1lμjrr−1=Sr∑j=1lμjϕρ⊗Qj=Sr∑j=1lμjUmρUm∗⊗VnQjVn∗=SrUmρUm∗⊗VnσVn∗.
Then, there exist unitary matricesWm, Wn on Hm, Hn, respectively, such that(41)ϕρ⊗σ=Wm⊗WnUmρUm∗⊗VnσVn∗Wm⊗Wn∗=WmUm⊗WnVnρ⊗σWmUm⊗WnVn∗.
LetU¯m=WmUm,V¯n=WnVn, then for any states ρ∈SHm,σ∈SHn, we have(42)ϕρ⊗σ=U¯m⊗V¯nρ⊗σU¯m⊗V¯n∗.
We complete the proof.
Ifϕ takes the form of equality (24), we obtain(43)ϕρ⊗σ=V¯n⊗U¯mσ⊗ρV¯n⊗U¯m∗.


## 4. Conclusions

In this paper, a quantum channel preserving Tsallis entropy in bipartite systems is characterized by decomposing quantum gates. It turns out that such a quantum channel is a local quantum channel. The motivation for the paper stems from the fact that the Tsallis entropy, just like the Renyi entropy and the von Neumann entropy, could be helpful to understand and quantify entanglement in the quantum systems.



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*Source: 2130374-2023-03-14.xml*