Minimal Non-orthogonal Gate Decomposition for Qubits with Limited Control

Minimal Non-orthogonal Gate Decomposition for Qubits with Limited Control

Xiao-Ming Zhang Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, China, and City University of Hong Kong Shenzhen Research Institute, Shenzhen, Guangdong 518057, China    Jianan Li Department of Physics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China    Xin Wang Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, China, and City University of Hong Kong Shenzhen Research Institute, Shenzhen, Guangdong 518057, China    Man-Hong Yung Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China Shenzhen Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055, China Central Research Institute, Huawei Technologies, Shenzhen, 518129, China
July 19, 2019

In quantum control theory, a question of fundamental and practical interest is how an arbitrary unitary transformation can be decomposed into minimum number of elementary rotations for implementation, subject to various physical constraints. Examples include the singlet-triplet (ST) and exchange-only (EO) qubits in quantum-dot systems, and gate construction in the Solovay-Kitaev algorithm. For two important scenarios, we present complete solutions to the problems of optimal decomposition of single-qubit unitary gates with non-orthogonal rotations. For each unitary gate, we give the criteria for determining the minimal number of pieces, the explicit gate construction procedure, as well as a computer code for practical uses. Our results include an analytic explanation to the four-gate decomposition of EO qubits, previously determined numerically by Divincenzo et al [Nature, 408, 339 (2000)]. Furthermore, compared with the approaches of Ramon sequence and its variant [Phys. Rev. Lett., 118, 216802 (2017)], our method can reduce about 50% of gate time for ST qubits. Finally, our approach can be extended to solve the problem of optimal control of topological qubits, where gate construction is achieved through the braiding operations.

A universal gate set for quantum computation can be constructed by any two-qubit entangling gate, together with arbitrary single-qubit gates Nielsen and Chuang (2000). In the laboratory, elementary single-qubit gates are normally constructed by switching on and off an external field at certain times (i.e., a square pulse), resulting in a rotation of a Bloch vector along certain axis of the Bloch sphere. The question is, how to optimize the use of these elementary rotations to form arbitrary single-qubit gates? The question becomes crucial for quantum platforms where controls are limited, for example in quantum dot systems Petta et al. (2005); Maune et al. (2012). Consequently, a general rotation needs to be decomposed into a sequence of elementary rotations around non-parallel axes. In fact, this “piecewise” decomposition of general operations has inspired the development of composite pulses, which play an important role in quantum control on various types of qubits Wimperis (1994); Cummins et al. (2003); Wang et al. (2012); Bando et al. (2013); Kestner et al. (2013); Kosut et al. (2013); Wang et al. (2014).

Typically, one would like to reduce the complexity of gates: a long sequence of elementary gates implies the need of frequent switching of the applied field. Therefore, a minimal decomposition of single-qubit gates in terms of elementary rotation is of practical and fundamental interest in quantum computing. For the cases where the available elementary controls are rotations around two orthogonal axes, it is well known that arbitrary rotations can be constructed with a three-piece sequence alternating between the two axes Nielsen and Chuang (2000), for example the -- or -- sequence gat ().

However, in many systems, the available elementary rotations are non-orthogonal. For instance, for a singlet-triplet (ST) qubit Petta et al. (2005), the -rotation can be achieved via a magnetic field gradient Foletti et al. (2009); Brunner et al. (2011); Petersen et al. (2013); Wu et al. (2014), but a pure -rotation is hardly achievable as the magnetic field gradient has to be completely turned off during execution of a gate, which is impractical. As another example, control of an exchange-only (EO) qubit DiVincenzo et al. (2000); Laird et al. (2010) is only available via two rotation axes 120 apart from each other.

In the literature of quantum dots, much effort has been made to optimize gate sequences involving rotations around a pair of non-orthogonal axes DiVincenzo et al. (2000); Hanson and Burkard (2007); Ramon (2011); Zhang et al. (2016, 2017); Shim et al. (2013); Throckmorton et al. (2017). In particular, if the rotation axes along and are both available (with an angle 45), as is typically the case for an ST qubit, a Hadamard gate can convert an -rotation to a -rotation, so that an -- sequence can be replaced by a five-piece sequence, namely -Hadamard--Hadamard- Wang et al. (2012, 2014). Moreover, Ramon Ramon (2011) pointed out that if the acute angle between the two available axes (denoted as and ) is greater than , the Hadamard gate can be replaced by the rotational gate around to reduce the gate time Zhang et al. (2017); Throckmorton et al. (2017). However, the resulting Ramon sequence, namely ----, still contains five pieces of elementary gates. In the context of controlling quantum-dot qubits, it remains an outstanding problem whether a more efficient decomposition with non-orthogonal elementary rotations is possible.

In an early study of the EO qubit, Divincenzo et al. numerically found that four-piece sequences can be constructed for almost all quantum gates DiVincenzo et al. (2000), but no analytical explanation was given. On the other hand, in applying the Solovay-Kitaev theorem Nielsen and Chuang (2000), it was believed that an arbitrary gate can be decomposed into three pieces Nielsen and Chuang (2000); Kaye et al. (2007), but the problem turns out to be far more complicated. Furthermore, in quantum-dot systems including the ST qubit, instead of a pair of fixed rotational axes, it is also possible to access elementary rotational gates for a certain range of rotational angles. The problem of gate sequence optimization depends on the accessible range of rotational angles. Particularly, if the range covers the entire plane, only two pieces of elementary rotations are sufficient for the decomposition of any rotation Shim et al. (2013).

Here, we present a complete solution to the problem of minimal decomposition of single-qubit transformation, in terms of non-orthogonal elementary gates. For applications, we focus on two types of quantum-dot qubits, exemplified by the ST and EO qubit respectively; our results can also be extended to other quantum systems with limited control capability.

Definitions—A single-qubit rotation, , around the axis for an angle , can be generically described by,


where contains the Pauli matrices. We are interested in how a unitary gate (up to an overall phase factor) can be minimally decomposed into a sequence of elementary rotations , in a given set limited by physical constraints.

For convenience, we define the -power of a set to contain all combinations of products of elementary rotations, i.e., . Our task is to solve the following decomposition:


subject to the condition, . Here is referred to as “number of pieces”. Of course, for each the solution of satisfying the decomposition is not unique; in fact, there are infinitely many possible solutions.

The goal of this work is to determine the minimum value for any given unitary transformation . Furthermore, the explicit procedure (see Fig. 2) in constructing the minimum decomposition is also provided as a matlab code sm () for practical uses. Our main results are summarized as follows.

Main results for Type-I qubits—For Type-I, the rotation axes are allowed to vary in a limited range of a plane enclosed by the boundary rotation axes denoted by and ; the angle between the boundary axes are given by [see Fig. 1 (a)]. We define the set containing all possible elementary rotations by .

First of all, we obtain sufficient and necessary conditions sm () for the classes of unitary gates decomposable with one () and two () steps. Furthermore, for those unitary gates requiring at least , we show how the problem of non-orthogonal gate decomposition can be reduced to the case where the only allowed rotations axes are located at the boundaries and (without loss of generality), i.e., where the asterisks indicate angles. Therefore, we can focus on the elementary rotations formed by the joint set of rotations:


where and . This becomes essentially the same problem as the Type-II qubit to be discussed below.

More precisely, for , we divide our results into two parts, (i) and (ii) , summarized by the following theorem:

Theorem 1 (Bulk-to-boundary mapping)

(i) For , if a unitary gate can be decomposed to pieces, , it can always be decomposed into pieces with rotation axes at the boundary, i.e.,


(ii) for , one can always apply the orthogonal -- decomposition for any single-qubit unitary gate with pieces.

In the existing ST qubits literatures Ramon (2011); Zhang et al. (2017); Throckmorton et al. (2017), the single-qubit gates are typically decomposed into five or more pieces; our results show that as long as , all target rotation can be decomposed to four or even less number of pieces [see Eq. (10) below]. Specifically, when which is a typical experiment value Martins et al. (2016); Reed et al. (2016), we have found that for the set of 24 Clifford gates, gates can be realized with , and gates with . Furthermore, we have compared the performance of our method with previous alternative decompositions Ramon (2011); Zhang et al. (2017). The results indicate that our decomposition offers a significant improvement in reducing both gate time and gate error (see Fig. 1 (d) and Fig. 3).

Main results for Type-II qubits—For Type-II, only elementary rotations with two fixed axes are allowed, for example, and , where the angle between them are given by . The set containing all elementary rotations are given by [see Eq. (3) and Fig. 1 (b)]. For any given unitary gate and angle , we have solved the problem of minimal gate decomposition, in terms of a pair of inequalities [see Eq. (8) and (9)].

Figure 1: (a) Schematic description of Type-I qubits. Rotation axes are allowed to be chosen freely in a range (orange) bounded by and axes. (b) Schematic description of Type-II qubits. Rotation axes are fixed to be either or . (c) The minimum number of pieces for all possible rotations () for Type-I qubits. The color scale represents different when lies in the corresponding area. (d) Gate time comparison for ST qubits. Red: five-piece ”Ramon” sequence Ramon (2011); blue: revised schemes proposed by Zhang et al. Zhang et al. (2017); green: minimal decomposition proposed in this work. Target gates are , and we set .
Figure 2: Flow chart for constructing the minimum decomposition sequences for Type-I qubits and Type-II qubits. Eq.(S8)-(S9) are given in the supplemental material sm ().

From the experimental point of view, it is of interest to determine the optimal number of decomposition applicable for all possible unitary transformations, i.e.,


In principle, the values of for Type-I and Type-II qubits can be different, as they are subject to different physical constraints. However, we prove that the values of the are identical for both Type-I and Type-II qubits.

In particular, for EO qubits, where two available rotation axes are fixed with relative angle , our results imply that the minimum number of pieces is given by , which represents an analytic explanation to the numerical results obtained by Divincenzo et al in 2000 DiVincenzo et al. (2000).

Singlet-Triplet (ST) qubits— The ST qubit is constructed by double quantum dots in the following computational basis, and , and the Hamiltonian Petta et al. (2005) for a ST qubit is given by


where is the magnetic field gradient. In the laboratory, the value of is usually fixed. The term characterizes the exchange interaction that can be varied dynamically. However, the value of is bounded within a certain range, ; when exceeds the maximum value , the qubit behaves more like a charge qubit, where decoherence would be significantly increased Petta et al. (2005). In other words, if we let , the available rotations for the ST qubit is given by .

Exchange-only (EO) qubits.—On the other hand, the EO qubit is constructed by a coupled triple-quantum-dot system DiVincenzo et al. (2000). In the computational basis defined by and , the Hamiltonian in this subspace can be written as,


where and are coupling constants between the neighboring dots. However, it remains an experimental challenge to simultaneous apply both coupling, which means that either or should be non-zero at each moment of time. In other words, we assume only elementary rotations around or another axis can be applied, i.e., .

Details of Type-I qubits.—Here, the available elementary rotations are given by with rot (). Below, we will present all the cases where Eq.(2) can be satisfied for and with a certain value of (see proofs in supplementary materials sm ()). For , Eq. (2) can be satisfied, if and only if one of the following conditions are satisfied: (i) and , (ii) (identity rotation) or (iii) (around ). For , Eq. (2) can be satisfied, if and only if one of the following conditions are satisfied (i) , (ii) , (iii) , with , where we defined and . (iv) . In case (iv), the rotation axes can be chosen freely in the entire - plane; two pieces are sufficient, which is consistent with the result in Ref. Shim et al. (2013).

For , the results have been summarized in Theorem. 1. When (i) , it can be reduced to the Type-II with same apart, so the existence of -piece decomposition is determined by Eq. (8) and (9) below; when (ii) , decomposition with pieces always exist. The case of (ii) is obvious. We briefly sketch the proof procedure of case (i) here (see sm () for full detals):

Proof (Sketch) We define the product of two set , as . (i) we show that , which means that the product of any two rotations in is always equal to the product of a rotation at the boundary and another rotation in . (ii) This result implies that . (iii) we show that the product of any two rotations in can always be decomposed in the form of -- and --, i.e., .  

Optimal control of ST qubits— Here, we apply above results to the ST qubit described by Eq. (6). Since operations with is slow and may suffer from severe nuclear noise Zhang et al. (2017), to minimize the number of pieces while maintaining robustness and short gate time, we propose the following decomposition strategy. Given maximum coupling strength , we restrict , where . This ensures the axes can vary in a range with , and . For a given target rotation, we decompose it with or if such solutions exist. Otherwise, the decomposition with is realized with fixed axes at the boundary corresponding to and (and gate time are optimized).

Figure 3: Randomized Benchmarking for different decomposition schemes for ST qubits. Red: five-piece decomposition based ”Ramon” sequence Ramon (2011); blue: revised scheme proposed by Zhang et al. Zhang et al. (2017); green: minimal decomposition proposed in this work. We set , for all figures, and for (a), for (b), for (c) and for (d).

We compare our minimal decomposition scheme to (i) five-piece Ramon sequence Ramon (2011), realized by alternating couplings between and , and (ii) an alternative scheme Zhang et al. (2017) designed for avoiding operations for the case. Fig. 1 (d) shows the comparison of gate time for several target unitary gate with . Remarkably, our minimal decomposition scheme has on average and shorter gate time relative to Zhang et al. (2017) and Ramon (2011) respectively.

To further study the robustness, we perform randomized benchmarking with Gaussian static noise. The nuclear spin noise are drawn from , and the charge noise are drawn from . In Fig. 3 (a), we set and which are typical values of the noise in GaAs quantum dots Martins et al. (2016). The results of other noise level are given in Fig. 3 (b)-(d). It shows that the minimal decomposition schemes has substantial improvement in the robustness against nuclear spin noise.

Details for Type-II qubits— Different from Type-I, available elementary rotation now are given by with rot (). We present a set of constraints imposed to the rotation parameters for the decomposition described by Eq. (2) with (full proof is given in sm ()). The constraint is different when the number of pieces is an odd or an even. (i) For the odd-piece decomposition, i.e., , for some , the decomposition in Eq. (2) can be satisfied for a given rotation if and only if


where the value of is taken to be the minimum value between and .

Furthermore, the form of determines the resulting sequence. For the cases where , Eq. (2) can be constructed by the following sequence: ; if , Eq. (2) can be constructed in the form of .

(ii) For the even-piece decomposition where , the decomposition in Eq. (2) can be satisfied for a given rotation, if and only if


where is taken to be the minimum of . The other variables are defined as follows: , , and .

Minimum number of pieces for all possible — It is known that Wimperis (1994); Bando et al. (2013), for Type-II qubits, all rotations can be decomposed to pieces, if and only if (see sm () for alternative proof), which implies


From Theorem. 1, when , is the same for both Type-I and Type-II qubits. Moreover, when , criteria (iv) for indicates that . Therefore, we can conclude that Eq. (10) also holds for Type-I qubit. An illustration of is given in Fig. 1 (c).

To conclude, we have studied the minimal decomposition for two types of qubits: rotation axes are restricted in a range of a plane (Type-I), and rotation axes are fixed at two directions (Type-II). We also present an explicit procedure for minimally applying the elementary gates for an arbitrary single-qubit transformation. Furthermore, we discuss the implications of minimal decomposition for ST qubit, providing numerical evidences showing the effectiveness and robustness of our decomposition. Finally, we provide a code online sm () for experimentalists, who just need to input a target rotation; the code will generate the explicit minimal decomposition. The combination of our work with dynamical decoupling Wimperis (1994); Wang et al. (2012, 2014) or geometric control Duan et al. (2001); Liu et al. (2018); Yan et al. (2018) may be interesting in the future.


We thank Chengxian Zhang for helpful discussion. This work is supported by the National Natural Science Foundation of China (No. 11604277, 11875160), the Guangdong Innovative and Entrepreneurial Research Team Program (No. 2016ZT06D348), the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 21300116, CityU 11303617), Natural Science Foundation of Guangdong Province (2017B030308003), and the Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20170412152620376, JCYJ20170817105046702, ZDSYS201703031659262).


  • Nielsen and Chuang (2000) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
  • Petta et al. (2005) J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,  and A. C. Gossard, Science 309, 2180 (2005).
  • Maune et al. (2012) B. M. Maune, M. G. Borselli, B. Huang, T. D. Ladd, P. W. Deelman, K. S. Holabird, A. A. Kiselev, I. Alvarado-Rodriguez, R. S. Ross, A. E. Schmitz, M. Sokolich, C. A. Watson, M. F. Gyure,  and A. T. Hunter, Nature 481, 344 (2012).
  • Wimperis (1994) S. Wimperis, J. Magn. Reson., Ser. A 109, 221 (1994).
  • Cummins et al. (2003) H. K. Cummins, G. Llewellyn,  and J. A. Jones, Phys. Rev. A 67, 042308 (2003).
  • Wang et al. (2012) X. Wang, L. S. Bishop, J. P. Kestner, E. Barnes, K. Sun,  and S. Das Sarma, Nat. Commun. 3, 997 (2012).
  • Bando et al. (2013) M. Bando, T. Ichikawa, Y. Kondo,  and M. Nakahara, J. Phys. Soc. Jpn. 82, 014004 (2013).
  • Kestner et al. (2013) J. P. Kestner, X. Wang, L. S. Bishop, E. Barnes,  and S. Das Sarma, Phys. Rev. Lett. 110, 140502 (2013).
  • Kosut et al. (2013) R. L. Kosut, M. D. Grace,  and C. Brif, Phys. Rev. A 88, 052326 (2013).
  • Wang et al. (2014) X. Wang, L. S. Bishop, E. Barnes, J. P. Kestner,  and S. Das Sarma, Phys. Rev. A 89, 022310 (2014).
  • (11) The “--” sequence means ; the “-Hadamard--Hadamard-” sequence means .
  • Foletti et al. (2009) S. Foletti, H. Bluhm, D. Mahalu, V. Umansky,  and A. Yacoby, Nature Phys. 5, 903 (2009).
  • Brunner et al. (2011) R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladrière, T. Kubo, K. Yoshida, T. Taniyama, Y. Tokura,  and S. Tarucha, Phys. Rev. Lett. 107, 146801 (2011).
  • Petersen et al. (2013) G. Petersen, E. A. Hoffmann, D. Schuh, W. Wegscheider, G. Giedke,  and S. Ludwig, Phys. Rev. Lett. 110, 177602 (2013).
  • Wu et al. (2014) X. Wu, D. R. Ward, J. R. Prance, D. Kim, J. K. Gamble, R. T. Mohr, Z. Shi, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith,  and M. A. Eriksson, Proc. Natl. Acad. Sci. U.S.A. 111, 11938 (2014).
  • DiVincenzo et al. (2000) D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard,  and K. B. Whaley, Nature 408, 339 (2000).
  • Laird et al. (2010) E. A. Laird, J. M. Taylor, D. P. DiVincenzo, C. M. Marcus, M. P. Hanson,  and A. C. Gossard, Phys. Rev. B 82, 075403 (2010).
  • Hanson and Burkard (2007) R. Hanson and G. Burkard, Phys. Rev. Lett. 98, 050502 (2007).
  • Ramon (2011) G. Ramon, Phys. Rev. B 84, 155329 (2011).
  • Zhang et al. (2016) C. Zhang, X.-C. Yang,  and X. Wang, Phys. Rev. A 94, 042323 (2016).
  • Zhang et al. (2017) C. Zhang, R. E. Throckmorton, X.-C. Yang, X. Wang, E. Barnes,  and S. Das Sarma, Phys. Rev. Lett. 118, 216802 (2017).
  • Shim et al. (2013) Y.-P. Shim, J. Fei, S. Oh, X. Hu,  and M. Friesen, arXiv preprint arXiv:1303.0297  (2013).
  • Throckmorton et al. (2017) R. E. Throckmorton, C. Zhang, X.-C. Yang, X. Wang, E. Barnes,  and S. Das Sarma, Phys. Rev. B 96, 195424 (2017).
  • Kaye et al. (2007) P. Kaye, R. Laflamme, M. Mosca, et al.An introduction to quantum computing (Oxford University Press, 2007).
  • (25) See Supplemental Material and for code.
  • Martins et al. (2016) F. Martins, F. K. Malinowski, P. D. Nissen, E. Barnes, S. Fallahi, G. C. Gardner, M. J. Manfra, C. M. Marcus,  and F. Kuemmeth, Phys. Rev. Lett. 116, 116801 (2016).
  • Reed et al. (2016) M. D. Reed, B. M. Maune, R. W. Andrews, M. G. Borselli, K. Eng, M. P. Jura, A. A. Kiselev, T. D. Ladd, S. T. Merkel, I. Milosavljevic, E. J. Pritchett, M. T. Rakher, R. S. Ross, A. E. Schmitz, A. Smith, J. A. Wright, M. F. Gyure,  and A. T. Hunter, Phys. Rev. Lett. 116, 110402 (2016).
  • (28) If available elementary rotation axes line in different range, one can make a coordinate transformation such that they coincide with Type-I or for Type-II. For example, brings to , and brings to .
  • Duan et al. (2001) L.-M. Duan, J. I. Cirac,  and P. Zoller, Science 292, 1695 (2001).
  • Liu et al. (2018) B.-J. Liu, X.-K. Song, Z.-Y. Xue, X. Wang,  and M.-H. Yung, arXiv preprint arXiv:1806.07904  (2018).
  • Yan et al. (2018) T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, M.-H. Yung, et al.arXiv preprint arXiv:1804.08142  (2018).

Supplementary material

In this Supplemental Material we provide necessary proofs to claims made in the main text. We will discuss the two types of qubits mentioned in the main text in order: Type I which axes are allowed to vary in a range, and Type II which with two fixed axes. We also provide some instruction about our matlab code for constructing explicit minimal decomposition sequences.

Appendix A Definition

To facilitate the discussions, for with , we parametrize it as:


where , , unless otherwise specified. For clarity, we represent all target unitary transformation as . Inversely, given , one can calculate angles as follows, which are important for the actual construction of the decomposition:

If ,


If ,


If ,


Furthermore, we define the set for all possible rotations as:

For both for Type I and for Type II qubits, we define several sets of rotation with :
and rotation with :

Furthermore, given two set , we define the product of them as:


and for a set , we define the -power of it as


Appendix B Type I: Axes restricted in a range

In this section, we are given axes that are allowed to vary in a range: , where , with . We will give the condition for decompositions to exist, and discuss how these decompositions can be constructed or reduced to a Type II qubit case.

b.1 Lemmas

We first provide several useful lemmas. To begin with, we show that arbitrary rotations can be decomposed into a -rotation and another rotation with axis in the - plane.

Lemma 1

Given any , there exist certain , such that



Case I: or

Eq. (S-8) can be satisfied by taking and .

Case II: and

It can be verified that Eq. (S-8) can be uniquely constructed as



In the following, we discuss the decomposition of the product of two rotations in .

Lemma 2

given , with , and , there exist unique value of , and unique , such that


and .


Existence of and :

Let , and define


According to Lemma 1, there exist certain , , such that




we have , and . And combining Eq. (S-11), Eq. (S-12), and Eq. (S-9), after some calculation, one can verify that


Then, we apply a transformation on Eq. (S-12) , which then becomes


where . So obviously, .


We denote . According to Eq. (S-11), and Eq. (S-12), we have


Therefore, we have , which means .




for some , and . We can denote


One can find that


And since , we have


So Eq. (S-18) becomes


Since , and , we have


Therefore, the values of , are unique.


Lemma 3

Given , , with , there exist certain , and , such that

(i) if


(ii) if


Case I :

According to Lemma 2, we can define the following implicit functions , that satisfy


where , and . From Lemma 2, the above implicit functions have the following properties: 
(1) , are single-value functions (uniqueness);  

To prove case I of Lemma 3, we only need to show that . We first evaluate the continuity and monotonicity of . For an independent value , we always have


which can be rewritten as:


We note that


Since , when , we have


And since , , we have


Therefore, and are continuous. Since [property (2)], and [property (3)], according to the intermediate value theorem of continuous function, we have .

Moreover, from Eq. (S-30) , we know that if . Since , and , we have . Therefore, , and (i) of Lemma 3 hold true.

Case II:

In this case, we first let , one can verify that . Since , according to case I of Lemma 3, there exist certain , , such that


which is equivalent to