Hollow Triangle Incompatibility And Bell Nonlocality

In the last section we have provided an example of incompatible measure-ments that cannot lead to Bell violations (assuming dichotomic measuremeasure-ments on the second observer). Our “minimal” example requires 12-well chosen measurementsm and one could ask the minimal number of measurements required to show this incompatible but Bell local phenomenon. As stated before, it was shown in [9] that every pair of dichotomic measurements can lead to a CHSH [16] Bell inequality violation, hence we need at least three dichotomic measurements to obtain the desired phenomenon.

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BELL NONLOCALITY 60

Consider the set of three dichotomic POVMs (acting onC²) given by the following positive operators

M_0|x⁰ (η):= ¹

2(I+ησ_x) (4.14) forx=1, 2, 3, whereσ₁,σ₂,σ₃are the Pauli matrices, and 0≤η≤1. The other POVM element is then M1|x(η)⁰ = I−M⁰_0|x(η). This set of POVMs should be understood as noisy Pauli measurements. The set is jointly measurable if and only ifη≤1/√

3, although any pair of POVMs is jointly measurable forη≤1/√

2 [72] (see also [82]). Hence in the range 1/√

3<η≤1/√ 2, the set{M⁰_a|x(η)}forms ahollow triangle: it is pairwise jointly measurable but not fully jointly measurable.

We can also find different hollow triangle measurements with Pauli meas-urements by considering different kind of noise. For that, we now define

M⁰⁰_0|x(η):= ^η

2(I+σx) (4.15) for x=1, 2, 3 and 0≤η≤1, and M⁰⁰_1|x(η) = I−M⁰⁰_0|x(η). To determine the range of the parameterηfor which the above set of POVMs is pairwise jointly measurable, and fully jointly measurable, we use the SDP techniques of ref [9].

We find that the set{M⁰⁰_a|x(η)}is a hollow triangle for 0.4226≤η≤0.5858.

We now investigate whether the above hollow triangles can lead to Bell inequality violation when second part has dichotomic measurements. The most general class of Bell inequalities to be considered here are of the form

I=

∑

∑

γxy AxBy

+

∑

αxhAxi+

∑

βy By

≤1 (4.16) where

AxBy

= p(a=b|xy)−p(a6=b|xy); (4.17) hAxi= p(0|x)−p(1|x),

By

= p(0|y)−p(1|y).

All (tight) Bell inequalities of the above form forn ≤5 can be obtained by facet enumeration algorithms like PORTA [20] and cdd [21], we also remark the online database for Bell inequalities [83]. And using a robust numerical method based on SDP we can find the smallest value of the parameterηfor which a given inequality can be violated using the set of POVMs (4.14) and (4.15). The technical details of these methods are presented in the next section and the results are summarized in table4.2.

Notably, for the set M⁰ represented in equation (4.14) we could not find a violation in the range 1/√

3≤η≤1/√

2 where the set{M⁰_a|x(η)}is a hollow triangle. In fact, no violation was found forη≤0.7786, whereas pairwise joint measurability is achieved for η≤ 1/√

2' 0.7071, thus leaving a large gap.

Note also that pairwise joint measurability implies violation of the CHSH

CHAPTER4. MEASUREMENT INCOMPATIBILITY VS

BELL NONLOCALITY 61

{M⁰_a|x(η)} {M⁰⁰_a|x(η)}

Pairwise JM (CHSH violation) 1/√

2≈0.7071 0.5858

Triplewise JM 1/√

3≈0.5774 0.4226 Bell violation (n=3): I3322 0.8037 0.6635 Bell violation (n=4): I₃₄₂₂¹ 0.8522 0.7913

I₃₄₂₂² 0.8323 0.5636

I₃₄₂₂³ 0.8188 0.6795

Bell violation (n=5): I3522 0.7786 0.5636

Table4.1: Bell inequality violation with incompatible POVMs. Specifically, we consider the sets given in equations (4.14) and (4.15). For each set, we determine the smallest value of the parameterη, such that the set becomes Jointly Measurable (JM), and achieve Bell inequality violation. We consider tight Bell inequalities with up ton = 5 measurements for Bob (see section 4.2.1). Note that pairwise incompatibility (of two-outcome measurements) is equivalent to violation of the CHSH Bell inequality.

inequality here, since we have POVMs with binary outcomes [9]. We thus conjecture that there is a threshold value η^∗ > _1/√

3, such that all hollow triangles with 1/√

3<η≤η^∗ do not violate any Bell inequality.

From the above, one may actually wonder whether Bell inequality violation is possible at all using set of POVMs forming a hollow triangle. For the setM⁰⁰ represented in equation (4.15), Bell nonlocality can be obtained by considering a Bell inequality withn=4 measurements for Bob and forη >0.5636 (see Table4.2.)

In additional to our numerical approach, we also show that a large class of Bell inequalities of the form (4.16) (for arbitraryn) cannot be violated using the hollow triangle presented in eq. (4.14). Note that for the set of POVMs (4.14), we have thathAxi=ηtr(σ_xρ_A)and

AxBy

= ηtr(σ_x⊗Mb|yρ_AB)for x=1, 2, 3. Hence we can write the Bell polynomial as

I=ηI˜+ (₁−η)

∑

y

βy By

(4.18) where ˜I is the Bell expressionIevaluated for projective (Pauli) measurements on Alice’s side. Note that ˜I≤I_C2, whereI_C2 denotes the maximal value ofI for qubit strategies. Hence no Bell inequality violation is possible when

∑

|βy| ≤ ¹−ηI_C2

1−η , (4.19)

given thatηI_C2 ≤1. Notably this includes all full correlation Bell inequalities (αx = βy = _0), i.e. XOR games, for which it is known that the amount of violation is upper bounded for qubit strategies. More precisely, one has that I_C2 ≤K3[26,43,84] whereK3≤1.5163 is the Grothendieck constant of order 3[85]. Hence, for 1/√

3<η<1/K3'0.6595 we get that the hollow triangle (4.14) cannot violate any full correlation Bell inequality.

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4.2.1 Technical Methods

Here we present how to compute the maximal quantum violation β of a two-party Bell inequality defined by the vector of coefficientscab|xy, given a fixed set of measurements operatorsMa|xfor Alice. This method is implicit at [86] and here we describe it in full detail. Let us writeσ_b|y=tr_B(ρ_ABI⊗M_b|x) for the assemblage created on Alice’s side by Bob’s measurements on the state ρAB. With this, we have the conditional probabilitiesp(ab|xy) =tr(M_a|xσ_b|y) and we have to maximize ∑a,b,x,yp(ab|xy) for fixed c_ab|xy and M_a|x. The following SDP program is a relaxation of the above problem:

β^∗≡max

∑

a,b,x,y

c_ab|xytr(M_a|xσ_b|y) for fixedMa|x,cab|xy

subject toσ_b|y≥0,

∑

b

tr(σ_b|y) =1 ∀y.

(4.20)

It is well known that one can always find a quantum stateρABand quantum measurements Mb|yfor Bob which attain the maximumβ^∗[2,87,88]. Hence, β=β^∗, and the above SDP provides the exact quantum bound ofβfor a fixed set of Alice’s measurements M_a|xon the Bell inequality defined by coefficients c_ab|xy.

To study the (possible) violation of Bell inequalities with the hollow triangle measurements (4.14) and (4.15) we consider tight Bell inequalities of form4.16. Forn≤4, all of them were known, and we have found all tight inequalities for the casen=5 with the help of facet enumeration algorithms (see section 1.2). Forn=3, there is a single tight Bell inequality (besides CHSH) known asI₃₃₂₂[89].

I3322=−hA1i − hA2i+hB1i+hB2i (4.21) +hA₁B₁i+hA₁B₂i+hA₁B₃i

+hA2B1i+hA2B2i − hA2B3i +hA3B1i − hA3B2i ≤4

using the notation of equation4.16. Forn=4, there are3new tight inequalit-ies. Forn=5, we characterized completely the polytope and found a single new tight Bell inequality:

I3522=hB1i+hB2i − hA1B1i+hA1B2i (4.22)

− hA₁B₄i+hA₁B₅i+hA₂B₁i +hA2B2i+hA2B3i − hA2B5i

+hA3B1i − hA3B3i − hA3B4i − hA3B5i ≤6.