Alternative Decomposition of Two-Qutrit Pure States and Its Relation with Entanglement Invariants1footnote 11footnote 1International Journal of Quantum Information Vol. 9, No. 6 (2011) 1499-1509http://dx.doi.org/10.1142/S0219749911008040

Alternative Decomposition of Two-Qutrit Pure States and Its Relation with Entanglement Invariants111International Journal of Quantum Information Vol. 9, No. 6 (2011) 1499-1509
http://dx.doi.org/10.1142/S0219749911008040

Rui-Juan Gu School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China    Fu-Lin Zhang flzhang@tju.edu.cn Physics Department, School of Science, Tianjin University, Tianjin 300072, China    Shao-Ming Fei School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China    Jing-Ling Chen Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China
July 18, 2019
Abstract

Based on maximally entangled states in the full- and sub-spaces of two qutrits, we present an alternative decomposition of two-qutrit pure states in a form . Similar to the Schmidt decomposition, all two-qutrit pure states can be transformed into the alternative decomposition under local unitary transformations, and the parameter is shown to be an entanglement invariant.

Decomposition of quantum states; Two qutrits; Quantum information; Entanglement invariants
pacs:
03.67.-a, 03.67.Mn, 03.65.-w

I Introduction

Decomposition of quantum states is an interesting topic in quantum information theory Thiang et al. (2009); Carteret et al. (2000); Acín et al. (2000). Given an arbitrary bipartite state, it is well-known that the Schmidt decomposition is always applicable Nielsen and Chuang (2000). For instance, under local unitary transformations any two-qubit state can be transformed into its Schmidt-form as .

Besides the Schmidt decomposition, other decompositions are possible. For example, in 2001, Abouraddy et al. have proposed an alternative decomposition for two-qubit pure states based on the maximally entangled state Abouraddy et al. (2001):

(1)

where , , is the two-qubit maximally entangled state, and is a factorizable state orthogonal to . They showed that such a decomposition always exists and is not unique, but the parameter is unique. In comparison to the Schmidt decomposition, the merit of the new kind of decomposition is that the parameter has a definite physical significance as the degree of entanglement of two qubits. In this work, we would like to generalize the alternative decomposition to a two-qutrit system based on the maximally entangled states in the full- and sub-spaces. To our knowledge, such a generalization has not been reported in the literature.

This paper is organized as follows: In section II, we make a brief review for the previous result of Abouraddy et al., but from a different viewpoint of entanglement invariants. In section III, we present a Theorem on the alternative decomposition of two-qutrit pure states, and also show its relation with the entanglement invariants. Conclusion and discussion are made in the last section.

Ii Brief Review of Entanglement Invariants and Previous Result of Abouraddy et al.

Let us consider a general pure state of two -dimensional quantum systems (two qudits), which takes of the following form:

(2)

where and are the orthonormal bases of the Hilbert spaces A and B respectively, and ’s are complex numbers satisfying the normalization condition .

Let denote the matrix whose matrix elements are given by . It has been shown that the following quantities are entanglement invariants under local unitary transformations Albeverio and Fei (2001):

(3)

Denote , since the reduced density matrices , , thus Eq. (3) can be also expressed as

(4)

For , one easily has , which is nothing but the normalization condition of the reduced density matrix of or . Therefore, for a two-qudit system, there are only nontrivial entanglement invariants.

After performing an appropriate local unitary transformation, one may transform the general state into its Schmidt-form as

(5)

where ’s are the Schmidt coefficients, which satisfy the normalization condition: =1. In the Schmidt representation, it is easy to obtain the entanglement invariants as

(6)

Now, the previous result of Abouraddy et al. can be re-expressed as the following theorem:

Theorem 1. Under local unitary transformations any two-qubit state can be always transformed into an alternative decomposition as

(7)

where is unique and is an entanglement invariant under the local unitary transformations.

By comparing Eq. (1) and Eq. (7), one notes that we have chosen the maximally entangled state of two-qubit as and the factorizable state as . Moreover, the phase factor in Eq. (1) can be eliminated further by a suitable transformation. Therefore the decomposition in Eq. (7) is unique for the pure states of a two-qubit system.

The standard way to prove Theorem 1 is owing to the local unitary transformations, which has been actually given in Ref. Abouraddy et al. (2001), namely, by acting the appropriate local unitary transformations on an arbitrary two-qubit pure state , then one obtains the decomposition (7). However, there is another equivalent way to prove Theorem 1, which is due to the entanglement invariants. Now we use the new approach to prove Theorem 1, the same approach will be used to prove the corresponding Theorem for the two-qutrit case.

Proof. On one hand, for the two-qubit state in the Schmidt-form

(8)

one has the entanglement invariants as

(9)

here , means that is expressed by the parameters and .

On the other hand, for the two-qubit state in the alternative decomposition as in Eq. (7), one has the matrices

(10)

Thus the corresponding entanglement invariants reads

(11)

here , means that is expressed by the parameters and .

Because an arbitrary two-qubit state can be transformed into the Schmidt decomposition under the local unitary transformation, if one can prove that for any given and , there always exists satisfying , (), then it implies that an arbitrary two-qubit state can be transformed into the alternative decomposition as shown in Eq. (7) under the local unitary transformation. Since is the normalization condition, one only need to study , this yields the following solution:

(12)

which means that an arbitrary two-qubit state can be transformed into the alternative decomposition (7) under the local unitary transformation if relation (12) is satisfied. This ends the proof.

By the way, it is easy to show that the determinants of matrices and are

(13)

therefore one has

(14)

One will find later that such a similar relation holds for the any two-qudit system.

Iii Entanglement Invariants of Two-Qutrit and the Alternative Decomposition

Under local unitary transformations an arbitrary two-qutrit pure state can be transformed into its Schmidt-form as

(15)

one has the entanglement invariants as

(16)

here , and is trivial as the normalization condition of a quantum state.

By expanding , one may get an interesting and useful relation:

(17)

with

(18)

Since and are entanglement invariants, thus is an entanglement invariant under local unitary transformation. reaches its maximum value when . We shall use such a useful relation to prove the Theorem 2 in this section.

Actually, the entanglement property of a two-qutrit system is completely characterized by two entanglement invariants and , or equivalently,

(19)

where the normalized entanglement invariants .


Figure 1: (Color online) In the coordinate, the factorizable state, such as , locates at the origin ; the maximally entangled state (or say the GHZ state) in the full-space of two-qutrit locates at the point , which is the point farthest from the origin; and the maximally entangled state in the sub-space of two-qutrit, such as , locates at the point .

In Fig.1, we have plots points for the two-qutrit state by randomly taking values of , , and , see the red region of figure, whose contour lines form a curved triangle . In the coordinate, one may observe that there are three special points: the first point is the origin , which corresponds to the factorizable states, such as ; the second is the point , which corresponds to the maximally entangled state (or say the GHZ state) in the full-space of two-qutrit, such as ; and the third is the point , which corresponds to the entangled state in the sub-space of two-qutrit, such as .

Inspired by the success of Theorem 1, we suggest the following decomposition for two-qutrit pure states:

(20)

and

(21)

Here is the maximally entangled state (or say the GHZ state) in the full-space of two-qutrit spanned by , is the maximally entangled state in the sub-space of two-qutrit spanned by , and is the factorizable state, they are mutually orthogonal, i.e., . ’s are some phase factors. However, five phases can be eliminated by the transformation , with and , thus there is only one phase factor is survival. In general, one may select the phase factor involved in is not zero. Consequently, one arrives at the alternative decomposition of two-qutrit pure states as follows: .

Our main result is the following Theorem.

Theorem 2. Under local unitary transformations any two-qutrit state can be always transformed into an alternative decomposition as

where is unique and is an entanglement invariant under the local unitary transformations.

Proof. Similarly, for the 2-qutrit pure quantum state in form (III), one can write the related matrices as

(23)

Its entanglement invariants are obtained immediately

(24)

From them, one can find the relation

(25)

On condition that the state in Eq. (III) is equivalent to the one in Eq. (15) under local unitary (LU) transformations, the parameter should satisfies

(26)

which always has a root in the interval for any value of . Then, the two nontrivial entanglement invariants can be replaced by

(27)

For a fixed value of , if the range of in Eq. (III) is the same as the one of in Eq. (III), one can conclude there exists a pure state in the form (III) equivalent the one (15) with any under LU transformations. Let us denote the minimum and maximum of as and . Based on the fact that the values of and vary continuously from their minimums to maximums, it is only to prove

(28)

for a given value of . In Appendix A, we show the two relations come into existence. Since an arbitrary two- qutrit pure state can be transformed into the form (III) under LU operation, it can always be decomposed as Eq. (III). This ends the proof.

Iv Conclusion and Discussion

In conclusion, we show that all 2-qutrit pure states can be rewritten as The method we have used is to verify the invariant space is as same as achieved by expression of Schimidt-form. The parameter is unique and it is an entanglement invariant under LU operations. The values of and can be derived from the relations in Eq. (III).

In this paper, we concerns us in the pure states of two-qutrit system. There are two natural extensions of this issue: (i) to decompose the pure states in a bipartite arbitrary-dimensional system, (ii) to decompose the pure states in a multipartite system. For the case (i), we can foretell a two-qudit state can be transformed as

(29)

where the parameters , and . And, here is a maximally entangled state in the sub-space , whose spacial case is shown in Eq. (III) for . Under locally phase transformations , the phases and can be eliminated. We have numerically verified that the entanglement invariants of the states (29) cover the the ones of Schmidt-form states (2) perfectly for . For (ii), the quantum correlation or entanglement in a multipartite state carry more nonclassical characteristics of quantum mechanics Linden et al. (2002); Svetlichny (1987). Many perspectives have been presented to attempt an understanding of the problem in recent studies Svetlichny (1987); Linden et al. (2002); Zhou (2008); Linden and Wootters (2002); Walck and Lyons (2008, 2009); Ghose et al. (2009). In our subsequent investigation, we hope to give a decomposition of a multipartite pure state, dividing it into sub-spaces which reflect the entanglement in different levels.

Acknowledgements.
FLZ is supported by NSF of China (Grant No. 11105097). JLC is supported by National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and NSF of China (Grant Nos. 10975075 and 11175089).

References

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Appendix A Equivalence of the Ranges of and

a. and . Firstly, for the Schmidt-decomposed state (15),we consider the extremal values of , when (or say ) is fixed. From the relations (III), one can obtain

(30)
(31)

Then the problem is transformed to derive extremal values of in Eq. (30) in the range , under the constraint that the values of and should be legitimate. Solving the Eq. (31), we find

(32)

or permutation. Therefore the constraint can be explicitly expressed as the discriminant

(33)

This leads to , where are two of the roots of the cubit equation . They are given by

(34)

where the angle satisfies . The minimal value of occurs when and or

(35)

Substituting the solutions of Eq. (35) into Eq. (31), one can find the result is only a permutation of the former case, e.g. and . In the same way, one can conclude that the maximal value of occurs when and . Uniformly, we write the minimum and maximum of as and with

(36)

b. and . For the pure states (III), when the parameter or say the entanglement invariant is fixed, can be expressed as the function of and

(37)

Because , the maximum value of happens at and the minimum one at .

When , the derivative on Eq. (A) leads to

(38)

One of its three roots lies in being

(39)

where . It corresponds to the minimal value of as

(40)

When , by completely the same analysis, we obtain the maximum

(41)

where

c. Comprising the Ranges. The relation leads to and consequently . Therefore we have

(42)

Let , one can obtain

(43)

Substituting them and Eq. (42) into Eqs. (A) and (A), we get the first relation in Eq. (III), . In the same process, the angle . Setting , we obtain

(44)

These relations in company with Eqs. (A) and (41) lead to , which is the second relation in Eq. (III).

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