Local Channels under Decomposition of Quantum Gates in Bipartite Systems

Abstract

For a bipartite quantum system with Hilbert space , this paper considers a quantum channel 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.

1. Introduction

In the mathematical framework of quantum information theory, quantum states (density matrices) are positive operators with trace 1 on complex Hilbert space with dimension . The convex set of all states on are denoted by . A pure state is a rank one state, i.e., a rank one projection, and the set of all pure states are denoted by . Furthermore, if ; is a mixed state if . In general, people deal with multipartite systems. The underlying space of a multipartite composite quantum system is a tensor product of underlying spaces of its subsystems, that is, . The system is called a a bipartite one in case . is said to be separable if can be written as , where and are states on and , respectively, and are positive numbers with . Otherwise, is said to be inseparable or entangled. The full separability of multipartite states can be defined similarly. stands for the convex set of all separable states on and for the set of all separable pure states on . It is obvious that a tensor product is a pure state if and only if and are both pure states. The set of all square matrices are denoted by . Symbol stands for the conjugate transpose of a matrix .

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 with dimension . In their investigation [7] on von Neumann entropy, Yuan and Wang proved that if and only if there exists a unitary operator such that , where is a bistochastic quantum operation, i.e., , and is a set of matrices known as operation elements and . 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 if and only if it admits an expression , where are matrices and . The Tsallis entropy [9] is defined as follows:

In this paper, we always assume that and , i.e., . It is well-known that has concavity and subadditivity [10]. Using the spectral decomposition theorem, it is easy to see that , where is the von Neumann entropy. In fact, assuming that quantum state acting on a complex finite Hilbert space with dimension , noting and the convention which comes from one can get this relationship through 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. Let be the complex Hilbert spaces with , respectively. Let be a quantum channel and . Then, the following statements are equivalent:(1)For any , , and , quantum channel preserves Tsallis entropy .(2)There exist unitary operators and acting on , respectively, such that .

Remark 1. Actually in reference [11], we have discussed the similar topic both in single finite and infinite dimensional complex separable Hilbert space and obtained the conclusion that . Even if the state in the formula is separable, the unitary operator (quantum gate) may not be separable when this issue is discussed under the bipartite systems context. Fortunately, we succeeded in splitting into two parts via making full use of the properties of Tsallis entropy.
The remaining part of this paper is organized as follows: In Section 2, 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 Lemma 3, [10]). Let be a complex Hilbert space, for any , , if and only if . .

Lemma 2 (see Lemma 2.2 [12]). Let , , and , . If , then either and are linearly dependent or and are linearly dependent.

Lemma 3 (see Lemma 13, [11]). Let be a complex Hilbert space, Tsallis entropy is strictly concave, i.e., for any , , . Moreover, equality holds if and only if .

Lemma 4 (see Theorem 1, [11]). Let be a complex Hilbert space. Then, for any and , if and only if there exists a unitary acting on such that , where is dependent on .

Lemma 5. Let be a complex Hilbert space and be a quantum channel. Then, for any and , if and only if there exists a unitary acting on such that , where is linearly independent.

Proof of Lemma 1. Sufficiency is straightforward.
Necessity. By Lemma 4, for , implies , where is a unitary on and is dependent on . Then, . Using Theorem 3.5 [13], we have , which means . Then, . By Corollary 3.4 in reference [14], it follows , for and any quantum state . Since , we obtain and , where , and . It is obvious thatwhere is the complex conjugate of . Thus,where . Since , then for , we have . Taking on the both sides of Equality , we have . By direct calculation, we obtain . Noticing , we have . Thus, for , we obtain , which meansFinally, we knowLet , then is a unitary matrix such that .

Remark 2. The proof of Lemma 5 can be seen in [7] where von Neumann entropy was discussed.

Lemma 6. Let be complex Hilbert spaces with , respectively. be a quantum channel and . For any , , and , preserves Tsallis entropy . Then,(1)(2)

Proof of Lemma 6. (1)Taking in equality , we have for any .(2)If is a pure state, i.e., , then by Lemma 1, we obtain , which follows that . By Lemma 1 again, we have . One can deal with the converse case similarly. The proof of Lemma 6 is now completed.For pure state and , let

Lemma 7. Let be complex Hilbert spaces with , respectively. be a quantum channel and . For any , , and , preserves Tsallis entropy . Then, one of the following statements holds.(1)For any pure state , there exists some pure state such that (2)For any pure state , there exists some pure state such that

Proof of Lemma 3. By Lemma 6, we have . Then, for any separable pure state , there exists some separable pure state such that .
First, we fix a pure state , for any , with ; by Lemma 6, there exist pure states and such thatSince a quantum channel must be affine on quantum states, we see that is bijective and for quantum states . Thus, if , we have , andBy Lemma 2, we know that either (i) and are linearly dependent or (ii) and are linearly dependent.(i)If and are linearly dependent, since and are all pure states and we have and , are linearly independent. Let , then Now, for any , let . Since is linearly independent of at least one of and , otherwise, we will have . Let and are linearly independent. Noticing the following equality and recalling Lemma 2 We get . Thus, for a fixed , there exists some pure state such that .(ii)If and are linearly dependent. Similarly, one can get that, for a fixed , there exists some pure state such that .Now, in the claim for a fixed , there exists some pure state such that . Next, we will show that for any , there exists some pure state such that .
LetThen, . Assume that , then there exist a and a such that for all . Noticing that , we haveDue to the arbitrariness of , we have and . Thus,This contradicts to the injectivity of , then , which follows that for any , there exists some pure state such that . Similarly, if for a fixed , there exists some pure state such that . Then, for any , there exists some pure state such that . We complete the proof of Lemma 3.
A similar argument to Lemma 7 will result in Lemma 8.

Lemma 8. Let be the complex Hilbert spaces with , respectively. be a quantum channel and . For any , , and , preserves Tsallis entropy . Then, one of the following statements holds:(1)For any pure state , there exists some pure state such that (2)For any pure state , there exists some pure state such that Combining Lemmas 7 and 8 will give.

Lemma 9. If satisfies the same hypotheses as that in Lemma 7, then one of the following statement holds:(1)Case (1) in Lemma 7 and Case (3) in Lemma 8 hold(2)Case (2) in Lemma 7 and Case (4) in Lemma 8 hold

Proof of Lemma 4. If Case (1) in Lemma 7 and Case (4) in Lemma 8 hold simultaneously, then for any and , there exist , such that and ; then,Due to the arbitrariness of , we haveRecalling that is a pure state and . It forces . A contradiction appears. Similarly, Case (2) in Lemma 7 and Case (3) in Lemma 8 do not occur.

Lemma 10 (see [10]). For , let , and , thenwhere the equality holds if and only if for all .

We denote if and are orthogonal. For , let , where , then . Thus, if and only if either holds or holds.

Lemma 11 (see [11]). For any and , if and only if , where is the identity on .

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 Lemma 9 holds, that is, for any and , we have and . Then, for any , there exist , ( is dependent on P) such thatfor any . In order to prove is independent on , now assume that and , thenwhich impliesfor some and for any . This means, by Lemma 2, and are linearly dependent or and are linearly dependent. If and are linearly dependent, immediately we have since are pure states, which leads to that is independent on . If and are linearly dependent; similarly, we have , thusNoting that there some pure state such that Then,which meansRecalling that , , and are all pure states, thus . As a result, we have proved that for any separable pure state , there are two bijectives ( is affine) and such thatSimilarly, if the Case (2) in Lemma 9 holds, we can obtain that for any separable pure state , there are two bijectives and such thatIn the following part, we will claim that satisfying main theorem’s hypotheses preserves orthogonality. That is, for any , if , then . Actually, by Lemma 6, we know for any , by Lemma 10, we haveBy Lemma 10 again, we obtain , i.e., preserves orthogonality.
Since , for any , it is clear that can be written as , where , and . For any , noticing that , and equality (23), by Lemma 10, we getThen, there is a unitary matrix such thatLet , where is the identity element on . Then,We define a map by . Thus,In equality (29), if ( is the identity element on ), then . Actually, by Lemmas 1 and 11 and noticing that and is a pure state, we have .
In equality (29), map satisfiesIn fact,By equality (30) and Theorem 2 in [11], there exists a unitary matrix on such that , then equality (29) can be written asThe in equality (32) is independent on . Actually, for two different pure states , we get for a state . If , by equalities (23), (29), and (32), we haveFor all pure states , since , then . By Lemma 4.1 in reference [15], we know . Let , thus, for any and , we getSimilarly, for any states and , we getNow, we claim in equality (34), where is a unitary element on . Actually, for any state , , we havewhich impliesRecalling that is surjective, by [10], we obtain , Thus, equality (34) becomesSimilarly, equality (35) turns intoFinally, for any , let ,Then, there exist unitary matrices , on , , respectively, such thatLet , then for any states , we haveWe complete the proof.
If takes the form of equality (24), we obtain

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.

Data Availability

The data used to support the findings of this study are available within the article and are also available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of China (No. 12261096) and Research Fund for High-level Talents of Yulin Normal University (No. G2021ZK02).