Maximal violation of n-locality inequalities in a star-shaped quantum network

# Maximal violation of n-locality inequalities in a star-shaped quantum network

Francesco Andreoli Dipartimento di Fisica - Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy    Gonzalo Carvacho Dipartimento di Fisica - Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy    Luca Santodonato Dipartimento di Fisica - Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy    Rafael Chaves International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil    Fabio Sciarrino Dipartimento di Fisica - Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy
July 5, 2019
###### Abstract

Bell’s theorem was a cornerstone for our understanding of quantum theory, and the establishment of Bell non-locality played a crucial role in the development of quantum information. Recently, its extension to complex networks has been attracting a growing attention, but a deep characterization of quantum behaviour is still missing for this novel context. In this work we analyze quantum correlations arising in the bilocality scenario, that is a tripartite quantum network where the correlations between the parties are mediated by two independent sources of states. First, we prove that non-bilocal correlations witnessed through a Bell-state measurement in the central node of the network form a subset of those obtainable by means of a separable measurement. This leads us to derive the maximal violation of the bilocality inequality that can be achieved by arbitrary two-qubit quantum states and arbitrary projective separable measurements. We then analyze in details the relation between the violation of the bilocality inequality and the CHSH inequality. Finally, we show how our method can be extended to -locality scenario consisting of two-qubit quantum states distributed among nodes of a star-shaped network.

## I Introduction

Since its establishment in the early decades of the last century, quantum theory has been elevated to the status of the “most precisely tested and most successful theory in the history of science” Kleppner and Jackiw (2000). And yet, many of its consequences have puzzled – and still do– most of the physicists confronted to it. At the heart of many of the counter-intuitive features of quantum mechanics is quantum entanglement Horodecki et al. (2009), nowadays a crucial resource in quantum information and computation Nielsen and Chuang (2002) but that also plays a central role in the foundations of the theory. For instance, as shown by the celebrated Bell’s theoremBell (1964), quantum correlations between distant parts of an entangled system can violate Bell inequalities, thus precluding its explanation by any local hidden variable (LHV) model, the phenomenon known as quantum non-locality.

Given its fundamental importance and practical applications in the most varied tasks of quantum information Brunner et al. (2014), not surprisingly many generalizations of Bell’s theorem have been pursued over the years. Bell’s original scenario involves two distant parties that upon receiving their shares of a joint physical system can measure one out of possible dichotomic observables. Natural generalizations of this simple scenario include more measurements per party Collins and Gisin (2004) and sequential measurements Gallego et al. (2014), more measurement outcomes Collins et al. (2002), more parties Mermin (1990); Werner and Wolf (2001) and also stronger notions of quantum non-locality Svetlichny (1987); Gallego et al. (2012); Bancal et al. (2013); Chaves et al. (2016). All these different generalizations share the common feature that the correlations between the distant parties are assumed to be mediated by a single common source of states (see, for instance, Fig. 1a). However, as it is often in quantum networks Kimble (2008), the correlations between the distant nodes is not given by a single source but by many independent sources which distribute entanglement in a non-trivial way across the whole network and generate strong correlations among its nodes (Figs. 1b-d). Surprisingly, in spite of its clear relevance, such networked scenario is far less explored.

The simplest networked scenario is provided by entanglement swapping Zukowski et al. (1993), where two distant parties, Alice and Charlie, share entangled states with a central node Bob (see Fig. 1b). Upon measuring in an entangled basis and conditioning on his outcomes, Bob can generate entanglement and non-local correlations among the two other distant parties even though they had no direct interactions. To contrast classical and quantum correlation in this scenario, it is natural to consider classical models consisting of two independent hidden variables (Figs. 1b), the so-called bilocality assumption Branciard et al. (2010, 2012). The bilocality scenario and generalizations to networks with an increasing number of independent sources of states (Figs. 1d), the so called n-locality scenario Tavakoli et al. (2014); Mukherjee et al. (2015); Chaves (2016); Rosset et al. (2016); Tavakoli (2016a, b); Paul et al. (2017); Tavakoli et al. (2017) allow for the emergence of a new kind of non-local correlations. For instance, correlations that appear classical according to usual LHV models can display non-classicality if the independence of the sources is taken into account, a result experimentally demonstrated in Carvacho et al. (2016); Saunders et al. (2016). However, previous works on the topic have mostly focused on developing new tools for the derivation of inequalities characterizing such scenarios and much less attention has been given to understand what are the quantum correlations that can be achieved in such networks.

That is precisely the aim of the present work. We consider in details the bilocality scenario and the bilocality inequality derived in Branciard et al. (2010, 2012) and characterize the non-bilocal behavior of general qubit quantum states when the parties perform different kinds of projective measurements. First of all we show that the correlations arising in an entanglement swapping scenario, i.e. when Bob performs a Bell-state measurement (BSM), form a strict subclass of those correlations which can be achieved by performing separable measurements in all stations. Focusing on this wider class of correlations, we derive a theorem characterizing the maximal violation of the bilocality inequality Branciard et al. (2010, 2012) that can be achieved from a general two-qubit quantum states shared among the parties. This leads us to obtain a characterization for the violation of the bilocality inequality in relation to the violation of the CHSH inequality Clauser et al. (1969). Finally we show how our maximization method can be extended to the star network case Tavakoli et al. (2014), a -partite generalization of the bilocality scenario, deriving thus the maximum violation of the n-locality inequality that can be extracted from this network.

## Ii Scenario

In the following we will mostly consider the bilocality scenario, which classical description in terms of directed acyclic graphs (DAGs) is shown in Fig. 1-b. It consists of three spatially separated parties (Alice, Bob and Charlie) whose correlations are mediated by two independent sources of states. In the quantum case, Bob shares two pairs of entangled particles, one with Alice and another with Charlie. Upon receiving their particles Alice, Bob and Charlie perform measurements labelled by the random variables , and obtaining, respectively, the measurement outcomes , and . The difference between Bob and the other parties is the fact that the first has in his possession two particles and thus can perform a larger set of measurements including, in particular, measurements in an entangled basis.

Any probability distribution compatible with the bilocality assumption (i.e. independence of the sources) can be decomposed as

 p(a,b,c|x,y,z)=∫dλ1dλ2p(λ1)p(λ2)p(a|x,λ1)p(b|y,λ1,λ2)p(c|z,λ2). (1)

In particular, if we consider that each party measures two possible dichotomic observables (), it follows that any bilocal hidden variable (BLHV) model described by Eq. 1 must fulfill the bilocality inequality

 B=√|I|+√|J|≤1, (2)

with

 I=14∑x,z=0,1⟨AxB0Cz⟩,J=14∑x,z=0,1(−1)x+z⟨AxB1Cz⟩, (3)

and where

 ⟨AxByCz⟩=∑a,b,c=0,1(−1)a+b+cp(a,b,c|x,y,z). (4)

As shown in Branciard et al. (2010, 2012), if we impose the same causal structure to quantum mechanics (e.g. in an entanglement swapping experiment) we can nonetheless violate the bilocality inequality (even though the data might be compatible with LHV models), thus showing the existence of a new form of quantum non-locality called quantum non-bilocality.

To that aim let us consider the entanglement swapping scenario with an overall quantum state , with . We can choose the measurements operators for the different parties in the following way. Stations A and C perform single qubit measurements defined by

 Ax=σz+(−1)xσx√2,Cz=σz+(−1)zσx√2. (5)

Station B, instead, performs a complete BSM, assigning to the two bits the values

 00for|ϕ+⟩,01for|ϕ−⟩,10for|ψ+⟩,11%for|ψ−⟩. (6)

The binary measurement is then defined such that it returns , with respect to the value of . This leads to

 ⟨AxByCz⟩=∑a,b0,b1,c=0,1(−1)a+by+cp(a,b0,b1,c|x,z)=∑a,by,c=0,1(−1)a+by+cp(a,by,c|x,z)≡∑a,b,c=0,1(−1)a+b+cp(a,b,c|x,y,z), (7)

where, in the last steps, we made explicit use of the marginalization of probability over .
With these state and measurements, the quantum mechanical correlations achieve a value , which violates the bilocality inequality and thus proves quantum non-bilocality.

## Iii Results

### iii.1 Non-bilocal correlations with separable measurements

As reproduced above, in an entanglement swapping scenario QM can exhibit correlations which cannot be reproduced by any BLHV model. In turn, it was recently proved Rosset et al. (2016) that an equivalent form of the bilocality inequality (Eq. 2), can be violated by QM in the case where all parties only perform single qubit measurements (i.e. and linear combinations). Here we will prove that, given the bilocality inequality (Eq. 2), the non-bilocal correlations arising in an entanglement swapping scenario are a strict subclass of those obtainable by means of separable measurements.

The core of the bilocality parameter is the evaluation of the expected value (Eq. 4), that in the quantum case is given by

 =Tr[(Ax⊗By⊗Cz)(ϱAB⊗ϱBC)]. (8)

For the entanglement swapping scenario we can summarize the measurements in stations A and C by

 Ax=(1−x)A0+xA1x=0,1,Cz=(1−z)C0+zC1z=0,1, (9)

where and are general single qubit projective measurements with eigenvalues and . When dealing with station B, it is suitable to consider its operatorial definition which is implicit in Eq. 7. Indeed we can consider that is the outcome of our measurement, leading to values shown in Table 1.

The quantum mechanical description of the operator (in an entanglement swapping scenario) is thus given by

 By=|ϕ+⟩⟨ϕ+|+(1−2y)|ϕ−⟩⟨ϕ−|+(2y−1)|ψ+⟩⟨ψ+|−|ψ−⟩⟨ψ−| (10)

which relates each value of with its correct set of outcomes. This leads to the following theorem.

###### Theorem 1 (Non-bilocal correlations and separable measurements).

Given the general set of separable measurements

 By=(1−y)∑ijλijσi⊗σj+y∑klδklσk⊗σl, (11)

QM predictions for the bilocality parameter which arise in an entanglement swapping scenario (where Bob performs the measurement described in Eq. 10) are completely equivalent to those obtainable by performing a strict subclass of Eq. 11, i.e.

 {B}B.S.M.⊂{B}SEP.M.. (12)
###### Proof.

Let us write the Bell basis of a two qubit Hilbert space in terms of the computational basis (). From Eq. 10, we obtain

 By=|ϕ+⟩⟨ϕ+|+(1−2y)|ϕ−⟩⟨ϕ−|+(2y−1)|ψ+⟩⟨ψ+|−|ψ−⟩⟨ψ−|=(1−y)(|00⟩⟨00|−|01⟩⟨01|−|10⟩⟨10|+|11⟩⟨11|)+y(|00⟩⟨11|+|01⟩⟨10|+|10⟩⟨01|+|11⟩⟨00|)=(1−y)σz⊗σz+yσx⊗σx. (13)

This shows that the entanglement swapping scenario is equivalent to the one where station only performs the two separable measurements and , which form a strict subclass of the general set of separable measurements given by Eq. 11. Moreover if we consider a rotated Bell basis, then we obtain

 B′y=U†AB⊗U†BCByUAB⊗UBC=(1−y)U†ABσxUAB⊗U†BCσxUBC+yU†ABσzUAB⊗U†BCσzUBC=(1−y)→a⋅→σ⊗→c⋅→σ+y→a′⋅→σ⊗→c′⋅→σ (14)

where and () are orthogonal unitary vectors. Due to the constraints and , this case still represents a strict subset of Eq. 11. ∎

As it turns out, this theorem has strong implications in our understanding of the non-bilocal behavior of QM. Indeed, it shows how the entanglement swapping scenario is not capable of exploring the whole set of quantum non-bilocal correlations, since it is totally equivalent to a subclass of Bob’s separable measurements. As we will show next, a better characterization of quantum correlations within the bilocality context must thus in principle take into account the general form of Bob’s separable measurements, especially when dealing with different types of quantum states.

### iii.2 Non-bilocality maximization criterion

We will now explore the maximization of the bilocality inequality considering that Bob performs the separable measurements described by Eq. 11. It is convenient to consider that station B as a unique station composed of the two substations and , which perform single qubit measurements on one of the qubits belonging to the entangled state shared, respectively, with station A or C (see Fig. 1-c).
Let perform a general single qubit measurement and similarly for , and . We can define these measurements as

 Station A⟶→ax⋅→σ,Station B⟶→bAy⋅→σ⊗→bCy⋅→σ,Station C⟶→cz⋅→σ, (15)

where . Let us now define a general 2-qubit quantum state density matrix as

 ϱ=14(I⊗I+→r⋅→σ⊗I+I⊗→s⋅→σ+3∑n,m=1tnmσn⊗σm). (16)

The coefficients can be used to define a real matrix that lead to the following result:

###### Lemma 1 (Bilocality Parameter with Separable Measurements).

Given the set of general separable measurements described in Eq. 15 and defined the general quantum state accordingly to Eq. 16, the bilocality parameter is given by

 (17)
###### Proof.

Let us consider two operators in the form and a two qubit quantum state described by Eq. 16. We can write

 ⟨O1⊗O2⟩ϱ=Tr[(O1⊗O2)ϱ]=Tr[∑j,k=1,2,3(vj1vk2σj⊗σk)ϱ]=∑j,k=1,2,3vj1vk2tjk=→v1⋅(Tϱ→v2), (18)

where we made use of the properties of the Pauli matrices . Given the set of separable measurements described in Eq. 15, and the definitions of and (showed in Eq. 3), the proof comes from a direct application of Eq. 18 to the quantum mechanical expectation value:

 (19)

Next we proceed with the maximization of the parameter over all possible measurement choices, that is, the maximum violation of bilocality we can achieve with a given set of quantum states. To that aim, we introduce the following Lemma.

###### Lemma 2.

Given a square matrix and defined the two symmetric matrices and , each non-null eigenvalue of is also an eigenvalue of , and vice versa.

###### Proof.

Let be an eigenvalue of

 MTM→v=λ→v. (20)

If we must have . We can then apply the operator from the left, obtaining

 MMT(M→v)=λ(M→v), (21)

which shows that is an eigenvector of with eigenvalue .
The opposite statement can be analogously proved. ∎

We can now enunciate the main result of this section.

###### Theorem 2 (Bilocality Parameter Maximization).

Given the set of general separable measurements described in Eq. 15, the maximum bilocality parameter that can be extracted from a quantum state can be written as

 Bmax=√√tA1tC1+√tA2tC2, (22)

where and ( and ) are the two greater (and positive) eigenvalues of the matrix (), with and .

###### Proof.

We will prove Theorem 2, following a scheme similar to the one used by Horodecki Horodecki1995 () for the CHSH inequality. Let us introduce the two pairs of mutually orthogonal vectors

 (→a0+→a1)=2cosα→nA&(→a0−→a1)=2sinα→n′A,(→c0+→c1)=2cosγ→nC&(→c0−→c1)=2sinγ→n′C, (23)

and let us apply Eq. 23 to Eq. 17

 (24)

where the maximization is done over the variables and . We can choose and so that they maximize the scalar product. Defining

 ||M→v||2=M→v⋅M→v=→v⋅MTM→v, (25)

and remembering that and are unitary vectors, we obtain

 Bmax=max(√||TTϱAB→nA||||TϱBC→nC||∣∣cosαcosγ∣∣+√||TTϱAB→n′A||||TϱBC→n′C||∣∣sinαsinγ∣∣). (26)

Next we have to choose the optimum variables variables and . This leads to the set of equations

 ∂B(α,γ)∂α=12√||TTϱAB→n′A||||TϱBC→n′C||∣∣sin(α)sin(γ)∣∣cot(α)−12√||TTϱAB→n′A||||TϱBC→n′C||∣∣cos(α)cos(γ)∣∣tan(α)=0,∂B(α,γ)∂γ=12√||TTϱAB→n′A||||TϱBC→n′C||∣∣sin(α)sin(γ)∣∣cot(γ)−12√||TTϱAB→n′A||||TϱBC→n′C||∣∣cos(α)cos(γ)∣∣tan(γ)=0. (27)

This system of equations admits only solutions constrained by

 tan(α)2=tan(γ)2↔γ=±α+nπ,n∈Z, (28)

 Bmax=max(|cosα|√||TTϱAB→nA||||TϱBC→nC||+|sinα|√||TTϱAB→n′A||||TϱBC→n′C||)=max(\par√||TTϱAB→nA||||TϱBC→nC||+||TTϱAB→n′A||||TϱBC→n′C||) (29)

Next, we must take into account the constraints and . Since these two couples of vectors are, however, independent, we can proceed with a first maximization which deals only with the two set of variables and . Since is a symmetric matrix, it is diagonalizable. Let us call and its eigenvalues and let us write and in an eigenvector basis. If we define and , our problem can be written in terms of Lagrange multipliers related to the maximization of a function , given the constraints

 f(→nA,→n′A)=k1√∑i=1,2,3λi(niA)2+k2√∑i=1,2,3λi(n′iA)2,g1(→nA)=→nA⋅→nA−1,g2(→n′A)=→n′A⋅→n′A−1,g3(→nA,→n′A)=→nA⋅→n′A, (30)

where we considered that finding the values that maximize is equivalent to find these values for . Let us now introduce the scaled vectors and . We obtain

 f(→ηA,→η′A)=√∑i=1,2,3λi(ηiA)2+√∑i=1,2,3λi(η′iA)2,g1(→ηA)=→ηA⋅→ηA−(k1)2,g2(→η′A)=→η′A⋅→η′A−(k2)2,g3(→ηA,→η′A)=→ηA⋅→η′A, (31)

whose solution is given by vectors with two null components, out of three. If we define and if , the solution related to the maximal value is then given by

 fmax=k1√λ1+k2√λ2 (32)

 Bmax=max→nC,→n′C(√||TϱBC→nC||√tA1+||TϱBC→n′C||√tA2), (33)

where we made use of the Lemma 2.
The maximization over the last two variables leads to an analogous Lagrange multipliers problem with similar solutions, thus proving the theorem. ∎

This theorem generalizes the results of Mukherjee2016 () (which dealt with some particular classes of quantum states in the entanglement swapping scenario) to the more generic case of any quantum state in the separable measurements scenario (which, in a bilocality context, includes the correlations obtained through entanglement swapping). It represents an extension of the Horodecki criterion Horodecki1995 () to the bilocality scenario, taking into account the general class of separable measurements which can be performed in station B. Our result thus shows that as far as we are concerned with the optimal violations of the bilocality inequality provided by given quantum states, separable measurements or a BSM (in the right basis) are fully equivalent.

### iii.3 The relation between the non-bilocality and non-locality of sources

We will now characterize quantum non-bilocal behaviour with respect to the usual non-locality of the states shared between A, B and B, C. Let us start from Eq. 22 and separately consider Bell non-locality of the states and . We can quantify it by evaluating the greatest CHSH inequality violation that can be obtained with these states. Let us define the CHSH inequality as

 SUV≡12|⟨U0V0+U0V1+U1V0−U1V1⟩|≤1. (34)

If we apply the criterion by Horodecki et al. Horodecki1995 (), we obtain

 SABmax=√tA1+tA2,SBCmax=√tC1+tC2, (35)

where we defined and accordingly to Eq. 22. From a direct comparison of 22 and 35 we can write

###### Proposition 1.
 SABmax≤1&SBCmax≤1⟶Bmax≤1. (36)
###### Proof.

Applying the Cauchy-Schwarz inequality we obtain

 B2max≤SABmaxSBCmax≤1. (37)

This result shows that if the two sources cannot violate the CHSH inequality then they will also not violate the bilocality inequality. Thus, in this sense, if our interest is to check the non-classical behaviour of sources of states, it is just enough to check for CHSH violations (at least if Bob performs a BSM or separable measurements). Notwithstanding, we highlight that this does not mean that the bilocality inequality is useless, since there are probability distributions that violate the bilocality inequality but nonetheless are local according to a LHV model and thus cannot violate any usual Bell inequality.

Next we consider the reverse case: is it possible to have quantum states that can violate the CHSH inequality but cannot violate the bilocality inequality? That turns out to be the case. To illustrate this phenomenon, we start considering two Werner states in the form . In this case, indeed, in order to have a non-local behaviour between A and B (B and C) we must have () while it is sufficient to have in order to witness non-bilocality. This example shows that on one hand it might be impossible to violate the bilocality inequality although one of or is Bell non-local (for instance and ). It also shows that, when one witnesses non-locality for only one of the two states, it can be possible, at the same time, to have non-bilocality by considering the entire network (for instance and ).

Another possibility is the one described by the following Proposition

###### Proposition 2.

Given a tripartite scenario

 ∃ϱAB&ϱBCsuch thatSABmax>1,SBCmax>1&Bmax≤1. (38)
###### Proof.

We will prove this point with an example. Let us take

 (39)

where we defined as

 ϱ(v,λ)=v|ψ−⟩⟨ψ−|+(1−v)[λ|ψ−⟩⟨ψ−|+|ψ+⟩⟨ψ+|2+(1−λ)I4]. (40)

For these two quantum states one can check that

 tA1=1,tA2=0.04,tC1=0.64,tC2=0.49, (41)

 SABmax≃1.02,SBCmax≃1.06,Bmax≃0.97. (42)

This shows how it is possible to have non-local quantum states which nonetheless cannot violate the bilocality inequality (with separable measurements).

All these statements provide a well-defined picture of the relation between the CHSH inequality and the bilocality inequality in respect to the quantum states . We indeed derived all the possible cases of quantum non-local correlations which may be seen between couples of nodes, or in the whole network (according to the CHSH and bilocality inequalities). This characterization is shown in Fig. 2, in terms of a Venn diagram.

We finally notice that if A and B share a maximally entangled state while B and C share a generic quantum state, then it is easier to obtain a bilocality violation in the tripartite network rather than a CHSH violation between the nodes and . Indeed it is possible to derive

 Bmax(|Φ+⟩⟨Φ+|⊗ϱBC)=√√tC1+√tC2≥√tC1+tC2=SBCmax, (43)

where we made use of the following Lemma

###### Lemma 3.

Given the parameters and defined in Eq. 22, it holds

 0≤tA1,tA2,tC1,tC2≤1 (44)
###### Proof.

This proof will be divided in two main points.

1) .
As discussed in Makhlin2002 (), if we apply a local unitary to the initial quantum state , the matrix will transform accordingly to

 Tϱ⟶U1TϱUT2. (45)

According to the Singular Decomposition Theorem, it is always possible to choose and such that is diagonal, thus demonstrating point 1.

It is important to stress that we can always rotate our Hilbert space in a way that so we can take without loss of generality.

2) If is diagonal, then the eigenvalues of are less or equal to 1.
It was shown in Horodecki1995 () that, for every quantum state , we have regardless to the basis chosen for our Hilbert space. If is diagonal then and its eigenvalues can be written as .

Given the definitions of and ( and ) described in Eq. 22, the lemma is proved. ∎

### iii.4 Extension to the star network scenario

We now generalize the results of Theorem 2, to the case of a n-partite star network. This network is the natural extension of the bilocality scenario, and it is composed of sources sharing a quantum state between one of the stations and a central node B (see Fig. 1-d). The bilocality scenario corresponds to the particular case where . The classical description of correlations in this scenario is characterized by the probability decomposition

 p({ai}i=1,n,b|{xi}i=1,n,y)=∫(n∏i=1dλip(λi)p(ai|xi,λi))p(b|y,{λi}i=1,n). (46)

As shown in Tavakoli et al. (2014), assuming binary inputs and outputs in all the stations, the following n-locality inequality holds

 Nstar=|I|1/n+|J|1/n≤1, (47)

where

 (48)

We will now derive a theorem showing the maximal value of parameter that can be obtained by separable measurements on the central node and given arbitrary bipartite states shared between the central node and the parties.

###### Theorem 3 (Optimal violation of the n-locality inequality).

Given single qubit projective measurements and defined the generic quantum state accordingly to Eq. 16, the maximal value of is given by

 Nmaxstar= ⎷(n∏i=1tAi1)1/n+(n∏i=1tAi2)1/n, (49)

where and are the two greater (and positive) eigenvalues of the matrix with .

###### Proof.

In our single qubit measurements scheme the operator can be written as

 By=n⨂i=1Biy=n⨂i=1→biy⋅→σ. (50)

As pointed out in Tavakoli et al. (2014), this allows us to write

 (51)

 Nstar=|n∏i=112(→ai0+→ai1)⋅TϱAiB→bi0|1/n+|n∏i=112(→ai0−→ai1)⋅TϱAiB→bi1|1/n. (52)

Introducing the pairs of mutually orthogonal vectors

 (→ai0+→ai1)=2cosαi→ni&(→a0−→a1)=2sinαi→n′i, (53)

allows us to write

 Nstar=|n∏i=1cosαi→ni⋅TϱAiB→bi0|1/n+|n∏i=1sinαi→n′i⋅TϱAiB→bi1|1/n. (54)

We can choose the parameters so that they maximize the scalar products. We obtain

 Nmaxstar=max(|n∏i=1cosαi||TTϱAiB→ni|||1/n+|n∏i=1sinαi||TTϱAiB→n′i|||1/n). (55)

We can now proceed to the maximization over the parameters . Let us define the function

 K(α1,...αn)=|λ1n∏i=1cosαi|1/n+|λ2n∏i=1sinαi|1/n. (56)

We can write

 ∂K(α1,...αn)∂αj=|λ2∏ni=1sinαi|1/nncotαj−|λ1∏ni=1cosαi|1/nntanαj=0, (57)

which, similarly to Eq. 27, admits only solutions constrained by

 tan(αj)2=tan(αk)2↔αj=±αk+nπ,n∈Z∀j,k. (58)

 K(α1,...αn)max=maxα(|λ1/n1cosα|+|λ1/n2sinα|)=√λ2/n1+λ2/n2, (59)

which allows us to write

 Nmaxstar=max√|n∏i=1||TTϱAiB→ni|||2/n+|n∏i=1||TTϱAiB→n′i|||2/n. (60)

Let us now define

 k1=|n∏i=2||TTϱAiB→ni|||,k2=|n∏i=2||TTϱAiB→n′i|||, (61)

we have that

 Nmaxsta