Symmetric geometric measure and dynamics of quantum discord

# Symmetric geometric measure and dynamics of quantum discord

Mingjun Shi Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China    Fengjian Jiang Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China Huangshan University, Huangshan, Anhui 245021, People’s Republic of China    Jiangfeng Du Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
###### Abstract

A symmetric measure of quantum correlation based on the Hilbert-Schmidt distance is presented in this paper. For two-qubit states, we simplify considerably the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering dissipative noise. Sudden changes occur, even twice, in the time evolution of quantum correlation. There is certain correspondence between the evolution of quantum correlation in the systems and that in the environments, and the quantum correlation in the systems will be transferred into the environments completely and asymptotically.

###### pacs:
03.65.Ta, 03.67.–a

## I introduction

Quantum systems can be correlated in ways inaccessible to classical objects. There are quantum states that cannot be prepared with the help of local operations and classical communication, or that cannot be represented as a mixture of product states. These states are called entangled states. Entanglement is certainly a kind of quantum correlation, and moreover is by far the most famous and best studied kind of quantum correlation Horodecki et al. (2009). One reason for this situation is the fact that quantum entanglement plays an important role in much of the research of quantum information science Nielsen and Chuang (2000). However, quantum entanglement is not the only aspect of nonclassicality of correlations and does not account for all nonclassical properties of quantum correlations. A typical example is that quantum nonlocality can arise without entanglement Bennett et al. (1999). More importantly, although it is well known that entanglement is essential for certain kinds of quantum-information tasks like teleportation Bennett et al. (1993) and super-dense coding Bennett and Wiesner (1992), there is no definite answer as to whether all quantum algorithms that outperform their best known classical counterparts require entanglement as a resource. Indeed, there are several instances where we see a quantum improvement in the absence or near absence of entanglement (see e.g. Braunstein et al. (1999); Meyer (2000); Vidal (2003); Biham et al. (2004)). In particular, no entanglement is present in the computational model referred to as “the power of one qubit” with the acronym DQC1 Knill and Laflamme (1998); Laflamme et al. (2002). Despite this, the mixed separable states can create an advantage for computational tasks over their classical counterparts Laflamme et al. (2002); Datta et al. (2005); *Datta.PhysRevA.75.042310.2007.

On the other hand, many works have been devoted to understanding and quantifying the quantum correlations beyond entanglement Henderson and Vedral (2001); Ollivier and Zurek (2001); Oppenheim et al. (2002); Koashi and Winter (2004); Groisman et al. (2005); Horodecki et al. (2005); Luo (2008); Rodríguez-Rosario et al. (2008); *Shabani.PhysRevLett.102.100402.2009; Piani et al. (2008); *Piani.PhysRevLett.102.250503.2009; Modi et al. (2010); Dakić et al. (2010); Adesso and Datta (2010); Luo and Fu (2011); Cavalcanti et al. (2011); *Madhok.PhysRevA.83.032323.2011; Streltsov et al. (2011a); *Piani.PhysRevLett.106.220403.2011. As the total correlation can be split into a classical part and a quantum part Henderson and Vedral (2001), various measures of quantum correlations are proposed by considering different notions of classicality and operational means to quantify quantumness. Amongst them, quantum discord Ollivier and Zurek (2001) has attracted much attention. Quantum discord has been defined as the mismatch between two quantum analogues of classically equivalent expression of the mutual information. It can be also expressed as the difference between total correlation, measured by quantum mutual information, and the classical correlation defined in Henderson and Vedral (2001). The notion of quantum discord goes beyond entanglement: separable states can have nonzero discord. In particular, it is believed that quantum discord is the figure of merit for the DQC1 model of quantum computation Datta et al. (2008). Generally speaking, quantum discord plays an important role in quantum information processing Merali (2011).

In order to obtain the classical correlation (and thereby the quantum discord) of a bipartite quantum system, one has to perform measurement on one subsystem to extract the information about the other subsystem, i.e., locally accessible information. Hence this sort of one-sided measurements implies that classical correlation and quantum discord are not symmetric under the permutation of subsystems. As a measure of correlation, whether classical or nonclassical, one would expect that it is symmetric. Some symmetric measures of correlations have been proposed in Refs. Terhal et al. (2002); Horodecki et al. (2005); Luo (2008); Modi et al. (2010). The key step in obtaining these measures is to consider the classical-classical (CC) states. The state is called a CC state, if it can be expressed as the mixture of locally distinguishable states, namely, , where is a joint probability distribution and local states and span an orthonormal basis. The set of product states is the subset of the set of CC states, while the set of CC states in turn is the subset of the set of separable states. A CC state is in fact the embedding of a classical probability distribution in the formalism of quantum theory and as such has no quantumness. Given a state , one can find the closest CC state, , to it. It is natural to regard this minimal distance as the measure of quantum correlation. Such a measure has a transparent geometrical meaning. The distance can be measured with trace distance, Hilbert-Schmidt distance, Bures distance, relative entropy and so on. In Modi et al. (2010), the authors used the relative entropy as the distance measure to provide the unified view of quantum and classical correlation.

We present in this paper a symmetric geometric measure of quantum correlation based on Hilbert-Schmidt (HS) distance. By applying von Neumann measurement on each subsystem, any bipartite state will become a CC state, which we call a measurement-induced CC (MICC) state. Given a bipartite state , we define the quantum correlation, denoted by , as the squared HS distance between and the closest MICC state. The evaluation of requires an optimization procedure over the set of all local von Neumann measurements on two subsystems, and thus attacking the general case is a formidable task. However, for 2-qubit states, we are able to simplify the optimization over two-sided measurements to that over one-sided measurements. In other words, the number of measurement parameters over which the optimization procedure is performed is reduced from four to two. Thus we are able to evaluate the geometric measure of quantum correlation by the efficient numerical method. We further find the exact analytical expressions for some special states.

Moveover, we will discuss the dynamics of quantum correlation, quantified by the geometric measure , when quantum states undergo a noisy channel. It has been shown that the evolution of quantum correlation, whether in terms of quantum entanglement or of quantum discord, may behave in a “sudden” way. Quantum entanglement can evolve to sudden death or birth Yu and Eberly (2004); *Eberly.Sci.316.555.2007; *Yu.Sci.323.598.2009. The decoherence regime of quantum discord may change suddenly to that of classical correlation Mazzola et al. (2010). Also, the dynamics of quantum discord has attracted much attention Maziero et al. (2009); *Maziero.PhysRevA.81.022116.2010; Werlang et al. (2009); Wang et al. (2010); *Fanchini.PhysRevA.81.052107.2010. This “sudden” behavior has not been thoroughly understood yet. It is natural to enquire as to whether and how such a geometric measure of quantum correlation will exhibit some sort of sudden changes. To this end, we consider the case that a 2-qubit state is effected by the action of two independent non-unital channels, e.g., amplitude damping (AD) channel, and calculate analytically , the time evolution of the geometric measure developed in this paper. We observe the nontrivial phenomenon that the function may be neither monotone nor smooth. In other words, during the time evolution, the quantum correlation can increase under the influence of AD channel, and the rate of evolution can exhibit sudden changes at some critical times. Despite these novel aspects, the asymptotical behavior of is in accordance with the coherence decay. In addition, we see that the quantum correlation in the system qubits is completely transferred to the environments after sufficiently long time.

## Ii Definitions and notations

This section is a prelude providing the definitions and notations that will be used throughout the whole paper.

### ii.1 Hilbert-Schmidt distance

For any 2-qubit state shared by two parties and , let’s define a matrix as with the elements given by for , where is identity matrix and () are usual Pauli matrices. We write as

 R=(1→y→xTT), (1)

where and are the Bloch vectors (in row form) of reduced state and respectively, and is a matrix and usually called correlation matrix. The superscript denotes transposition.

Hilbert-Schmidt (H-S) norm (also called Frobenius norm) is defined as for any bounded operator on complex-dimensional Hilbert space. It follows that the H-S distance between two operators and is given by

 DHS(M,N)=√Tr[(M−N)(M†−N†)]. (2)

In the following, we will omit the subscript .

Let and be the matrix associated with 2-qubit states and respectively. The H-S distance between and can be expressed in terms of the elements of matrix:

 D2(ρ,τ)=14[|→xρ−→xτ|2+|→yρ−→yτ|2+∥Tρ−Tτ∥2]. (3)

Now suppose that we perform local von Neumann measurements on both qubits and . The measurement operators for and are given by

 ΠA±=12(\mathbbm1±→k⋅→σ),ΠB±=12(\mathbbm1±→ℓ⋅→σ),

respectively, where and are unit vectors in three-dimensional real space. After measurements, we obtain a MICC state , that is,

 χ=∑i,j=+,−(ΠAi⊗ΠBj)ρ(ΠAi⊗ΠBj) (4)

Let and . Both of them are real symmetric matrices. The matrix of can be written as

 Rχ=(1→yLK→xTKTL). (5)

From (3), the squared distance between and is given by

 D2(ρ,χ)=14[|→x−→xK|2+|→y−→yL|2+∥T−KTL∥2]=14{x2+y2+∥T∥2−[→xK→xT+→yL→yT+Tr(TLTTK)]},

where and so forth, and in the last line we have used and . Let and . We rewrite as

 (6)

If only one qubit, say , is measured, the resulting state is a classical-quantum (CQ) state. A general CQ state has the form of . In the case we are considering, the measurement-induced CQ state is expressed as

 ρ→=(ΠA+⊗\mathbbm1)ρ(ΠA+⊗\mathbbm1)+(ΠA−⊗\mathbbm1)ρ(ΠA−⊗\mathbbm1). (7)

The corresponding matrix is

 Rρ→=(1→yK→xTKT). (8)

It follows that the squared distance between and is given by

 D2(ρ,ρ→)=14[Tr(X+TTT)−Tr(XK+TTTK)]. (9)

If only qubit is measured, the squared distance can be defined similarly.

### ii.2 Quantum channel

We will give only a brief introduction to the quantum channel, and refer the reader to books, say, Nielsen and Chuang (2000); Bengtsson and Życzkowski (2006) for a detailed analysis.

A quantum channel is a trace preserving completely positive (TPCP) map. A linear map is completely positive if and only if it is of the form

 ρ⟶ρ′=∑iKiρK†i, (10)

where are the operators on the state space of the system and are known as Kraus operator. Eq. (10) is called operator-sum representation of a quantum operation. A completely positive map is called trace-preserving if and only if

 ∑iK†iKi=\mathbbm1. (11)

If a quantum channel leaves the maximal mixed state invariant, it is called a bistochastic map or unital channel, that is, . Otherwise, the channel is non-unital.

The action of a quantum channel on a quantum system can be described by a unitary evolution on an extended system, the system plus the environment , followed by a partial trace over . This description is called the unitary representation of the quantum channel. This representation is not unique since many different unitary evolutions will lead to the same effect of the channel.

Now let us consider a concrete quantum channel, amplitude damping (AD) channel. The AD channel is used to describe the evolution of the system state in the presence of a dissipative environment. In this process the system interacts with a thermal bath at zero temperature. This process could be described as spontaneous emission of a two-state atom (system ) coupled with the vacuum modes of the ambient electromagnetic field (environment ) which leads the atom state to the ground state (see, e.g., Breuer and Petruccione (2002) for more in-depth discussion).

Considering the behavior of a two-level atom in a -mode cavity, the interaction between them gives rise to the phenomenological map

 |0S⟩|0E⟩⟶|0S⟩|0E⟩, (12) |1S⟩|0E⟩⟶γ(t)|1S⟩|0E⟩+√1−γ2(t)|0S⟩|1E⟩. (13)

Here, is the ground state and is the excited state of the atom, while is the vacuum state of the cavity and describe the cavity state with only one excitation distributed over all modes. The amplitude converges to in the limit of . The parameter is usually called coupling strength. Then the atom and environment evolve as an effective 2-qubit system.

From (12) and (13), we can write the Kraus operators of the AD channel:

 K0=(100γ),K1=(0√1−γ200). (14)

The evolution of the system state is then given by .

Let us further consider the case of a 2-qubit system being affected by their independent dissipative environments. We assume that at time the system-plus-environment state is described by

 ρABA′B′=ρAB⊗|0A′0B′⟩⟨0A′0B′|, (15)

where is the vacuum state of two environments.

By means of Eqs. (12) and (13), we can work out , the total state of at time . Tracing out environments and , we obtain the system state , which can be expressed in the operator-sum representation,

 ρAB(t)=1∑i,j=0(Ki⊗Kj)ρAB(K†i⊗K†j). (16)

On the other hand, the reduced environment state is given by

 ρA′B′(t)=TrABρABA′B′(t). (17)

## Iii Geometric measures of quantum correlation

As stated in the Introduction, we can define the quantum correlation in a 2-qubit state as the geometric distance, measured by HS norm, between and the closest MICC state . We will present below a detailed analysis in this direction.

### iii.1 One-sided measure

To begin with, we discuss a simpler case in which only one-sided measurement is performed on qubit . The geometric measure of quantum correlation based on such a sort of one-sided measurements has been proposed in Dakić et al. (2010) and discussed in Luo and Fu (2010). We recover these results below for completeness.

Given a 2-qubit state , one-sided von Neumann measurement on induces the CQ state given by (7). With the squared distance given by (9), the geometric measure of quantum correlation is defined as

 G→(ρ)=minD2(ρ,ρ→), (18)

where the minimization is performed over all von Neumann measurements on qubit .

For convenience, we use Dirac bra-ket notation to express the vectors in real three-dimensional space, that is, and . We will use alternatively both notations in the present work.

It can be seen from (9) that to find the minimal value of is equivalent to calculate the maximal value of . By noting that , we have that among all unit ,

 ⟨k|X+TTT|k⟩max=λmax, (19)

where is the largest eigenvalue of . Thus we have

 G→(ρ)=14[Tr(X+TTT)−λmax]. (20)

It is exactly the result given in Dakić et al. (2010). Similarly, if qubit is measured, we have

 G←(ρ)=14[Tr(Y+TTT)−κmax], (21)

with the largest eigenvalue of . Generally, .

In Dakić et al. (2010), the authors have used general, rather than measurement-induced, CQ state in the derivation of (20). Subsequently, it is pointed out in Luo and Fu (2010) that such a geometric measure is essentially measurement-oriented, meaning that the optimal CQ state in the most general sense is indeed the sort of measurement-induced CQ state.

### iii.2 Two-sided measure

Now we proceed to the case of two-sided measurements, which will lead us to a symmetric measure of quantum correlation. Two-sided measurements will result in MICC states. The squared HS distance between the state and the corresponding MICC state has been given by (6). We define the geometric measure of quantum correlation, based on two-sided measurement, as

 G(ρ)=min{|l⟩,|ℓ⟩}D2(ρ,χ), (22)

where the minimization is performed over all , or equivalently, over all measurement directions . As a CC state does not incorporate any quantum correlation, the two-sided quantifier can be regarded as a more strict geometric measure of quantum correlation.

By referring to (6), we need only focus on the maximal value of for all allowed and . Let’s define the following two matrices:

 M=X+TLTT+⟨ℓ|Y|ℓ⟩\mathbbm13, (23) N=Y+TTKT+⟨k|X|k⟩\mathbbm13, (24)

where denotes the identity matrix. It follows that

 Tr(XK+YL+TLTTK)=⟨k|M|k⟩=⟨ℓ|N|ℓ⟩. (25)

Then the problem is to calculate the maximal value of or over all unit vectors .

By noting that the eigenvalue of is equal to the sum of and the eigenvalue of , we see that the eigenvalue of is the function of , which we denote by . Let be the maximal one over all unit vector . It follows that the maximal value of is exactly the . Similarly, let be the eigenvalue of and the maximal one. The maximal value of is . Also, we have . The directions of the optimal measurement, denoted by and , are the eigenvectors of and referring to the eigenvalues and , respectively. Thus we see that the problem of two-sided optimization is indeed reduced to that of the one-sided optimization.

To proceed further, we will use the following lemma, which can be proved by direct calculation.

###### Lemma 1.

For any two vectors and (not necessarily normalized) in , the largest eigenvalue of the matrix is given by

 λ=12[a2+b2+√(a2−b2)2+4⟨a|b⟩2], (26)

with and . The corresponding normalized eigenvector is

where and are the unit vector along and respectively, and is the normalization factor.

Let and . It follows from the Lemma 1 that

 λM(→ℓ)=12[2⟨ℓ|Y|ℓ⟩+x2+ℓ′2+√(x2−ℓ′2)2+4⟨x|ℓ′⟩2] (27) λN(→k)=12[2⟨k|X|k⟩+y2+k′2+√(y2−k′2)2+4⟨y|k′⟩2]. (28)

The remaining problem is to find the maximal values or . The geometric measure is then given by

 G(ρ)=14[x2+y2+∥T∥2−λmaxM(N)]. (29)

Referring to (27), we see that involves two parameters, that is, two angles indicating the direction of . It is not difficult to attain the by efficient numerical method. For some special states, the exact results are available, which will be stated as follows.

#### iii.2.1 States with two-sided maximally mixed marginals

In this case, . We can always transform the correlation matrix into the diagonal form by local unitary operations. Eq. (27) becomes . The maximum is given by .

In this case there is no difference between the one-sided measure and the two-sided measure .

#### iii.2.2 States with one-sided maximally mixed marginal

In this case, only one reduced state, say , is maximally mixed. Then . It follows from (27) that and the is given by the largest eigenvalue of the matrix . On the other hand, if is maximally mixed, we can consider with . Similar results are easily obtained.

#### iii.2.3 X states with the identical local purity

We call a 2-qubit state the X state if the only nonzero elements in the density matrix lie along the diagonal or skew diagonal. By local unitary transformations, the entries in the matrix can always be written as

 →x=(0,0,x3),→y=(0,0,y3), T=diag(t1,t2,t3).

The exact expression of for any X state is in fact available. However, the derivation is too lengthy and too tedious. So we present the results referring to a restricted class of X states, that is, the X states with , i.e., with the identical local purity. The concrete analytical results are presented in Appendix.

## Iv Dynamics of quantum correlation

Now we are in the position to analyze the dynamics of quantum correlation, in terms of the geometric measure developed in this paper. We focus on a non-unital channel, AD channel. It is assumed that the identical AD channel is applied on each qubit independently and simultaneously. In order for the analysis to be precise, we take the system’s initial state as the X state with the same local purity. The time evolution of is given by (16). The environment state can be attained by considering the total state and then tracing out and . We see that at any time both and are the sort of the X state with the identical local purity, which allows for an analytical computation of the geometric quantum correlation .

Let us consider the following two examples.

Suppose that two 2-qubit X states and are given by, in terms of the elements in the matrix,

 ρABI:{x3=y3=0.7949,T=diag(0.4705,−0.5277,0.8947), (30)

and

 ρABII:{x3=y3=0.6479,T=diag(0.3926,−0.0772,0.0360), (31)

respectively. The coupling strength of the AD channel is set to be . We calculate and (or simply and ) with , and plot the results in Fig. 1. We observe several nontrivial aspects. Figure 1: Plots of the quantum correlation evolutions G(t) (solid line) and G′(t) (dashed line) for states ρABI (upper panel) and ρABII (lower panel) respectively. See text for details.

(i) The time evolution of quantum correlation, whether for system qubits or for the environments , is neither monotone nor smooth. There exist critical times on which the evolution rate exhibits a sudden change in behavior. The sudden change may even happen twice. More importantly, the quantum correlation in the system state can increase during the time evolution, even if the state is suffering a dissipative process.

(ii) By comparing with , we see that there is a symmetry between them. Roughly speaking, the evolution of from to is similar to the “inverse” evolution of , namely from to . Precisely, we have the following relationship: , , , and , where and () are the critical time for and , respectively (see Fig.1). In other words, there is a correspondence at the initial, critical and asymptotical point. A similar situation occurs in the comparison of and .

(iii) The fact that means that the initial quantum correlation between the system qubits is transferred completely to the asymptotical quantum correlation between the ancillary qubits.

Let us make some remarks about the above observations and related topics.

For a 2-qubit system interacting with two independent environments, the dynamics and the sudden change behavior of quantum discord have been analyzed in Ref. Maziero et al. (2009); Mazzola et al. (2010); Maziero et al. (2010). Subsequently, these issues are discussed in Lu et al. (2010) in terms of another quantifier of quantum correlation, one-sided HS distance measure Dakić et al. (2010). The results we present here are referring to the two-sided (i.e., symmetric) HS distance measure. One significant phenomenon is that quantum correlation can increase under the influence of the separate independent dissipative environments.

Recently, some authors show that the quantum discord can increase under a local amplitude damping channel with a CC input state Ciccarello and Giovannetti (2011). A more general conclusion is drawn in Streltsov et al. (2011b), that is, any local channel which is non-unital and not semi-classical can in principle create quantum correlations, independently of the considered measure, out of a CC state. Our work provides a concrete evolution process, in particular, with respect to the state , in which we see a rapid increase of the quantum correlation in system qubits. It should be noted that the input state is a separable but not a CC state. Meanwhile, we also see that the quantum correlation of the environments undergoes a similar evolution: increases in the beginning and then changes suddenly to decrease.

It should be mentioned that quantum discord can increase in the non-Markovian environments Wang et al. (2010); Fanchini et al. (2010), whereas the AD channel corresponds to a Markovian process.

Concerning the asymptotical behavior of , we see that after the second sudden change (if existing). This behavior is qualitatively identical to the decay of the skew diagonal elements of the density matrix , e.g., . It means that the asymptotical evolution of quantum correlation is closely related the coherence decay (in the case discussed above, they are qualitatively identical). Our discussion, from the viewpoint of symmetric distance quantifier, provides an evidence for the robustness of quantum discord Werlang et al. (2009) and the claim that “almost all quantum states have nonclassical correlations” Ferraro et al. (2010).

## V Conclusion

In conclusion, we have introduced in this paper a symmetric geometric measure of quantum correlation. By performing two-sided von Neumann measurements on bipartite state, we obtain a MICC state. The geometric measure is defined as the HS distance between the given state and the closest MICC state. For 2-qubit system, we simplify the optimization procedure considerably, that is, the two-sided optimization is reduced to the one-sided one. Hence the numerical computation can be performed efficiently. Moreover, the analytical results are available for some special class of 2-qubit states.

Using this quantifier, we have studied the dynamics of quantum correlation under the action of AD channel. We present the nontrivial aspects which may be exhibited during the time evolution: (i) the quantum correlation can increase; (ii) the quantum correlation can change suddenly, even twice. As for the environments, we see that the quantum correlation therein increases from zero at the beginning, and then evolves asymptotically to the value of the initial quantum correlation in the system qubits, after one or two sudden changes.

The geometric measure developed in this paper provides a symmetric viewpoint to study the quantum correlation. It is also a measurement-oriented measure, since the minimal HS distance is referring to the MICC, rather general CC, states. Then a question follows: Is there more rigorous quantifier, which is referring to the general CC states? This question has been solved in the case of one-sided geometric measure Luo and Fu (2010). However, it remains open for the two-sided measure.

###### Acknowledgements.
This work was supported by National Nature Science Foundation of China, the CAS, and the National Fundamental Research Program 2007CB925200.

## Appendix A G for X state with the same local purity

Let with . We assume that all is nonzero for the sake of simplicity. Referring to (27), we rewrite as

 λM(→ℓ)=12[r2+ℓ′2+√(r2−ℓ′2)2+4r2t23ℓ23]+r2ℓ23. (32)

Define the function as

 F(→ℓ)=(x2−ℓ′2)2+4r2t23ℓ23. (33)

In the following, we discuss two cases separately: and .

### a.1 Case of t1=t2

When , we have . So both functions and depend only on . By taking derivative of with respect to , and solving the equation for , we have the following results.

If and , we have

 ℓ23=−r2+t1(t1−t3)(t1−t3)2. (34)

If and , we have

 ℓ23=−r2+t1(t1+t3)(t1+t3)2. (35)

Some remarks are needed here.

(i) The conditions for (34) and (35) come from two considerations: one is the non-negativity of ; the other is the requirement of .

(ii) In solving the equation , it is assumed that . In fact, if , we have and . It follows that . We will see below that cannot be the maximal value of .

(iii) Eqs. (34) and (35) require that and , respectively. In fact, if , we have and . It follows from (32) that

 λM(→ℓ)=12[r2+t2+√(r2−t2)2+4r2t2ℓ23]+r2ℓ23.

It is a monotone increasing function for . Then when . This result can be contained in (39). Similarly for the case of . This analysis also holds for the derivation process in the next Case.

Inserting (34) and (35) into (32) respectively, we get two candidates for :

 λ(1)M=r2[2t1(t1−t3)−r2](t1−t3)2, (36) λ(2)M=r2[2t1(t1+t3)−r2](t1+t3)2. (37)

We have to take the end points of into consideration, i.e., and . When , we have . Let’s define

 λ(3)M=max{r2,t21}. (38)

When , we have

 λ(4)M=2r2+t23. (39)

Combining the above analysis, we conclude as follows.

If and , the maximal is given by

 λmaxM=max{λ(1)M,2r2+t23,t21}. (40)

If and , the maximal is given by

 λmaxM=max{λ(2)M,2r2+t23,t21}. (41)

### a.2 Case of t1≠t2

Let’s prove the following lemma.

###### Lemma 2.

If , then at least one of is zero.

With given by (32), we introduce Lagrange multiplier and take partial derivative of with respect to . Three equations follow.

 ℓ1[t21−t21(r2−ℓ′2)√F+2μ]=0, (42) ℓ2[t22−t22(r2−ℓ′2)√F+2μ]=0, (43) ℓ3[t23+t23(r2+ℓ′2)√F+2r2+2μ]=0. (44)

Assume that all are nonzero. If so, we can delete in each equation. By noting that all are nonzero (as assumed at the beginning of this Subsection), we see from (42) and (43) that the Lagrange multiplier must be zero. Then Eqs. (42) or (43) reduce to . It follows that , which contradicts the assumption. Lemma 2 is proved.

Subsequently, we will discuss one by one the measurements allowed by Lemma 2. In each case, we obtain a candidate for . The largest one is what we want.

##### If ℓ1=0 and ℓ2,ℓ3≠0.

It follows that , and that both and are the functions of only. By the approach similar to that presented in Section A.1, we have the following results.

If and , we have

 λ(5)M=r2[2t2(t2−t3)−r2](t2−t3)2. (45)

If and , we have

 λ(6)M=r2[2t2(t2+t3)−r2](t2+t3)2. (46)
##### If ℓ2=0 and ℓ1,ℓ3≠0.

In this case, . The results are very similar to that presented in the last paragraph, that is:

If , we have

 λ(7)M=r2[2t1(t1−t3)−r2](t1−t3)2. (47)

If , we have

 λ(8)M=r2[2t1(t1+t3)−r2](t1+t3)2. (48)

Note that and have the same form as and (given in (36) and (37)) respectively.

##### If ℓ3=0.

Here it is not required that both and are nonzero. Inserting into the expression of , i.e., Eq. (32), we have

 λM(→ℓ)=12[r2+(t21ℓ21+t22ℓ22)−|r2−(t21ℓ21+t22ℓ22)|].

The maximal value of the above expression is given by

 λ(9)M=max{r2,t21,t22}. (49)
##### If ℓ3=1.

In this case, . It easily follows that

 λ(10)M=2r2+t23. (50)

It is not difficult to see that the above four cases cover all allowed measurements. We summary the results obtained in this Subsection in Table 1.

Now let’s use an example to show how to calculate . Given an X state with , we write its matrix. If we see that and , then we face the following possibilities:

• If and , then .

• If and , then .

• If and , then .

• If and , then
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