No Disturbance Without Uncertainty as a Physical Principle
Liang-Liang Sun Xiang Zhou Sixia Yu



Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei,
Anhui 230026


Finding physical principles lying behind quantum mechanics is essential to understand various quantum features, e.g., the quantum correlations, in a theory-independent manner. Here we propose such a principle, namely, no disturbance without uncertainty, stating that the disturbance caused by a measurement to a subsequent incompatible measurement is no larger than the uncertainty of the first measurement, equipped with suitable theory-independent measures for disturbance and uncertainty. When applied to local systems in a multipartite scenario, our principle  imposes such a strong constraint on non-signaling correlations that quantum correlations can be recovered in many cases: i.) it accounts for   the Tsirelson's bound; ii.) it provides the so far tightest boundary for  a  family of the noisy super-nonlocal box with 3 parameters,  and iii.) it rules out an almost quantum correlation from quantum correlations by which   all the previous principles fail, as well as the celebrated quantum criterion due to Navascues, Pironio, and Acín. Our results pave the way to understand nonlocality exhibited in  quantum correlations from local principles. 

98.80.-k, 98.70.Vc

 No Disturbance Without Uncertainty as a Physical Principle
    Liang-Liang Sun^*, Xiang Zhou[Co-first author], Sixia Yu[email: yusixia@ustc.edu.cn]
    December 30, 2023
========================================================================================




Introduction—   Quantum  non-locality   is the most pronounced nonclassical feature of quantum mechanics (QM) <cit.>.  Understanding  quantum non-locality   is a fundamental  and challenging task that aims, on the one hand, at searching for  physical principles lying behind  its formalism <cit.>, on the other hand, at searching for correlation criteria, i.e., the conditions under which a set of observed correlations admit a quantum mechanical description, that witness the boundaries of quantum correlations emerging from the mathematical structure of QM <cit.>.

The boundary between quantum  and local realistic correlations, i.e., to answer the question of being local or nonlocal, is relatively well characterized by the violations of Bell inequalities.  Yet it is notorious difficult to characterize  the boundary between   quantum and  post-quantum correlations, i.e., to answer the question of why certain amount of non-locality is exhibited in quantum correlations.
Even in the simplest Bell scenario the sufficient correlation criterion is inaccessible. Tsirelson, Landau, and Masanes (TLM) derived the first correlation criterion, which becomes sufficient in the case of correlations with unbiased marginals. Later on,   Navascues, Pironio, and
Acín (NPA) generalized the TLM condition to the general correlations <cit.> and proposed a systematic method using a heirachy of semidefinite positive programs (SDP) to push the boundaries of quantum correlations. NPA criterion is reproduced when the SDP method stops at the first level <cit.>.


In comparison, the  theory-independent line of research,  initiated by Popescu and Rohrlich <cit.>,  focuses on  finding the physical principles that can account for quantum correlations without resorting to quantum  formalism. Generally, the theory-independent approach is less efficient   than the correlation criterion approach in witnessing the quantum correlations, but they provide us deep understandings on the physical principles lying behind QM <cit.>. Van Dam found the first essential discrepancy between QM and  Popescu and Rohrlich (PR) box, an extremal nonlocal correlation, and noted that the PR box requires only a single bit of classical communication for  all distributed computation tasks <cit.>.  Since then various physical principles such as Information Causality (IC)<cit.>, Global Exclusive  <cit.>,  Macroscopic Locality <cit.>, and Local Orthogonality <cit.>  have been proposed.  These principles are able to  recover the Tsirelson's bound but not the boundaries of  quantum  correlations. For example, none of them can  single out quantum correlations   from  almost quantum correlations (AQC),  which is strictly larger than the set of quantum correlations <cit.>.

All those  principles mentioned above are based on multipartite scenario and constraints  are directly imposed on correlations. Recently, a local principle, namely, Uncertainty-Complementarity Balance Relation (UCBR) <cit.>,  has been proposed as an underlying physical principle for quantum correlations. Instead of Bell scenario, sequential measurement scenario is considered in which  incompatible measurements  are performed in certain order on local systems.
 The balance relation establishes quantitative connections between uncertainty, complementarity and nonlocality to account for the Tsirelson's bound <cit.>. Even though some addition conditions are assumed in order to derive the Tsirelson bound,  the UCBR principle has shown us a way to understand quantum correlations from some local properties of subsystem.

In this Letter,  we pursue this line of research further by introducing a different local principle, namely, No Disturbance without Uncertainty (NDWU), which claims that for two incompatible measurements performed sequentially on a local system, the disturbance caused by the first measurement to the second measurement is no larger than the uncertainty of the first measurement, with properly chosen theory-independent measures for disturbance and uncertainty. Together with non-signaling conditions, our NDWU principle enables us to derive correlation criterion  that can characterize quantum correlations in general probabilistic theories (GPT).  In the simplest Bell's scenario our NDWU principle outperforms all the criteria found so far as it can account for the Tsirelson's bound, recover the  quantum boundaries for a family of the noisy non-signaling boxes, and single out  AQC  from quantum correlations, in which case our local principle  outperforms both the correlation criteria such as quantum NPA, TLM criteria and those based on correlation principles.





No disturbance without uncertainty — 
 We  shall work in the framework of a general probabilistic theory in which the concepts of physical state and sharp measurement (observable) are well-defined.  Generally, a system can be prepared in different states which can be any mathematical structure that enable us to predict the statistics of all possible measurements performed on the system or can be operationally taken  as a black box  with  measurement setting as input and  outcome as output.  A sharp measurement is accurate and repeatable.  More precisely,  a sharp measurement A on a physical state 𝒮 would return a definite value denoted by a, dependent on which  the system is brought into a unique state 𝒜^a, which is independent of the original state, with a probability p_𝒮(a|A). It is repeatable since the same outcome a will be obtained if we perform the same measurement subsequently.
We assume that there are more than one sharp measurement.

 Uncertainty and disturbance in sequentially performing two incompatible measurements are two characteristics features in a general nonlocal theory <cit.> and can be quantified in a theory-independent way. Recently, a balance relation between uncertainty and complementary  <cit.> has been established in a GPT. Inspired by the UCBR, here we also consider a scenario of sequential measurements A_0→ A_1 in which a sharp measurement A_0 is followed by another sharp measurement A_1. Since two sharp measurements are involved, it is necessary to consider the  transfer probabilities
  
    γ_A_1^a';𝒜_0^a:=p_𝒜_0^a(a^'|A_1)
  of obtaining outcome a' by measurement A_1 after the first sharp measurement A_0 has been performed on the initial state yielding an outcome a and leaving the system at state 𝒜_0^a. We note that the transfer probabilities do not depend on the initial state since the state 𝒜_0^a does not.
Before presenting our principle we need theory-independent measures to quantify the uncertainty and  disturbance:


  * The uncertainty of a measurement A_0, giving rise to probability distribution {p(a|A_0)}, is quantified by

    Δ_A_0:=√(1-∑_ap(a|A_0)^2).



  * The disturbed uncertainty of a measurement A_0, which is performed after measurement A_1, is quantified by a sum of  uncertainties of measurement A_0

    Δ _A_0|A_1= ∑_a'Δ_A_0;𝒜_1^a':=∑_a'√(1-∑_aγ^2_A_0^a;𝒜_1^a')

over all possible states after measurement A_1.


  * The disturbance  caused by  measurement A_0 on the subsequent measurement A_1 is quantified by

    D_A_0→ A_1:=∑ _a^'|p(a^'|A_1)-p(a^'|A_0→ A_1)|,

where {p(a|A_1)} is the statistics obtained by measuring A_1 on the original state and

    p(a^'|A_0→ A_1)=∑_a p(a|A_0)  γ_A_1^a';𝒜_0^a

 is the disturbed statistics obtained by measuring A_1 after measurement A_0.


Theorem 1    On a quantum system with finite levels, if two sharp measurements are performed in order A_0→ A_1  then it holds the following uncertainty disturbance relation
 
    Δ_A_0Δ_A_0| A_1≥ D_A_0→ A_1.


The  proof is  presented in the supplemental materials (SM). We note that this uncertainty disturbance relation is formulated in a theory-independent manner as all quantities involved, namely, uncertainty and disturbance, are quantified without referring to quantum theory. This means that it is possible for us to impose relation Eq.(<ref>) in a  GPT as physical principle satisfied by all pairs of sharp measurements and the resulting theory shall include quantum theory as a subset. As a qualitative reading, a nonzero disturbance (right hand side) caused by measurement A_0 to the subsequent measurement A_1 requires a nonzero uncertainty in measuring A_0 both in the original state and in the disturbed states resulting from measuring A_1 first. Thus we shall refer this qualitative readings from relation Eq.(<ref>) in a general probabilistic theory to as the principle of  no disturbance without uncertainty (NDWU) and the quantitative uncertainty disturbance relation Eq.(<ref>) as NDWU relation.


For an example let us consider the simplest physical system with sharp measurements having only two outcomes. We denote by ⟨ A_0⟩=p_0|A_0-p_1|A_0 and ⟨ A_1⟩=p_0|A_1-p_1|A_1 the expectation values of the  two-outcome measurements A_0  and A_1, respectively. Furthermore it can be shown that the transfer probabilities are symmetric
γ_A_1^a';𝒜_0^a=γ_A_0^a;𝒜_1^a' (see SM)
and, considering normalization, there is only a single independent parameter,  which is taken to be
 c=2γ_A_1^0;𝒜_0^0-1, among eight of them. The NDWU relation Eq.(<ref>) in this case becomes (see SM)
 
    ⟨ A_0⟩^2+⟨ A_1⟩^2+c^2-2c⟨ A_0⟩⟨ A_1⟩≤ 1.







Correlation boundaries from Local Properties —  We now consider the simplest Bell  scenario in which there are two physical systems prepared in some joint state distributed to two space-like separated observers Alice and Bob. Each observer can choose to perform two alternative sharp measurements {A_ν} and {B_μ} (ν,μ=0,1)  on their local systems with two outcomes labeled by a,b=0,1, resulting a  joint probability distribution, or correlation,  {p(ab|νμ)}. Furthermore we assume the non-signaling conditions, i.e., the local statistics of each observer does not depend on the measurement settings of  the other observer.

 When measurement settings are prefixed to be independent of any shared randomness the  correlations are referred to as pure correlation (PC).
When Alice and Bob  preshare some random variable,  the state they share as well as the measurements they choose may depend on this random variable. The correlations obtained in this way are referred to as mixed correlation (MC). For example, in quantum theory, PC are of form

    p_ pure(ab|νμ)=Tr(ρ M_A^a_ν⊗ M_B^b_μ)

while  MC are of form
 
    p_ mixed(ab|ν,μ)=∑_λP(λ) Tr(ρ_λ M^λ_A^a_ν⊗ M^λ_B^b_μ)

with preshared random variable λ distributed according to some probability distribution P(λ).
Quantum MC  can be generated  by   convex combinations of quantum PC <cit.>. We note that the set of PC in general might not be convex while the set of MC is by definition the convex hull of the set of all possible PC.  As a consequence the set of PC   determines the corresponding MC uniquely  but not vise versa so that PC encodes more information on the structure of underlying theory than MC. Also  the commonly tested correlation in laboratory is PC rather than MC.  Thus PC are more fundamental than mixed ones.  So far, all the correlation criteria such as  SDP method, NPA criterion, and the  criteria arising from principles such as Global Exclusive,  Macroscopic Locality and Local Orthogonality are intrinsic convex thus they are not cable of characterizing   PC <cit.>.


 Since two observers are space-like, the same joint probability p(ab|νμ) can be obtained by two different orders of measurements: Bob measures first or Alice measures first. Suppose that Bob or Alice measures first with measurement B_μ or A_ν yielding outcome b or a with probability p(a|ν)=∑_bp(ab|νμ) and p(b|μ)=∑_ap(ab|νμ), which are well defined due to non-signaling conditions, respectively. For the other observer, a conditional state ω_b|μ or ω_a|ν is  prepared accordingly, which determines the statistics of all the measurements concerned via the conditional probability

    p_ω_b|μ(a|ν)=p(ab|νμ)/p(b|μ),  
    p_ω_a|ν(b|μ)=p(ab|νμ)/p(a|ν).



In the study of  general nonlocal correlations, it is a fundamental  question what correlation can arise from the subsystems exhibiting some local properties.     There has been some interesting results regarding to the understanding of  correlation from  perspectives of local properties:  any non-local theory ensures the local properties of intrinsic uncertainty, complementarity and  non-clone principle <cit.>; the violation of Clauser-Horne-Shimony-Holt (CHSH) is upper-bounded by a function of a balance strength characteristic for the underlying theory <cit.>.  In following, we shall assume that the local system obeys the NDWU relation, i.e., in any given state (especially in those four effective conditional states) of each local system for any two sharp measurements it holds NDWU relation Eq.(<ref>). This imposes strong constraints on the correlations:

Theorem 2  In a non-signaling theory, if for each subsystem the NDWU relation Eq.(<ref>) holds, then the  correlation {p(ab|νμ)}  satisfy

    max_𝒮∈𝔅{⟨ A_0⟩_𝒮⟨ A_1⟩_𝒮-√(1-⟨ A_0⟩_𝒮^2)√(1-⟨ A_1⟩_𝒮^2)} 0.75cm
    ≤min_𝒮∈𝔅{⟨ A_0⟩_𝒮⟨ A_1⟩_𝒮+
    √(1-⟨ A_0⟩_𝒮^2)√(1-⟨ A_1⟩_𝒮^2)}, 
    max_𝒮∈𝔄{⟨ B_0⟩_𝒮⟨ B_1⟩_𝒮-
    √(1-⟨ B_0⟩_𝒮^2)√(1-⟨ B_1⟩_𝒮^2)} 0.75cm
    ≤min_𝒮∈𝔄{⟨ B_0⟩_𝒮⟨ B_1⟩_𝒮+
    √(1-⟨ B_0⟩_𝒮^2)√(1-⟨ B_1⟩_𝒮^2)},

 where 𝔅={ω_b|μ| b,μ=0,1}, 𝔄 ={ω_a|ν| a,ν=0,1}, and

    ⟨ A_ν⟩_ω_b|μ=p(0b|νμ)-p(1b|νμ)/p(0b|νμ)+p(1b|νμ)=⟨ A_ν⟩+(-1)^b⟨ A_ν B_μ⟩/1+(-1)^b⟨ B_μ⟩,
    ⟨ B_μ⟩_ω_a|ν=p(a0|νμ)-p(a1|νμ)/p(a0|νμ)+p(a1|νμ)=⟨ B_μ⟩+(-1)^a⟨ A_ν B_μ⟩/1+(-1)^a⟨ A_ν⟩.


The proof is presented in SM.  We note that the above result provides a quantum correlation criterion since the NDWU principle holds for QM.  We shall refer the constraints in Theorem 2 to as NDWU criterion. In the simplest Bell scenario QM violates the CHSH inequality and the maximum  violation is known as the  Tsirelson bound. As the first application we shall reproduce the Tsirelson bound:

Theorem 3 In a non-signaling theory respecting the no disturbance without uncertainty principle it holds

    :=∑_a,b,μ,ν=0^1(-1)^a+b+μνp(ab|νμ)≤2√(2).



Recovering quantum correlation boundary — 
Any correlation violates Tsirelson's bound is not quantum mechanical and  there exist correlations that satisfy Tsirelson's bound but still not quantum mechanical.  This is not surprising since there are eight independent variables in the correlation and a single bound cannot fully characterize its behaviors.
 The NDWU criterion aims to characterize the boundaries of a general quantum correlation. We now use it to recover quantum boundary for a family of noisy PR-boxes.

 In the simplest Bell scenario, the full set of non-signaling  boxes  includes  eight extremal non-local boxes
 
    P^τσλ_NL(ab|AB)={   1/2  if  a⊕ b=AB⊕τ A ⊕σ B ⊕λ
       0      otherwise,
     .

 with τ, σ, λ∈{0,1}, and  16 local deterministic boxes
 
    P^τσλς_L(ab|AB)={    1   if  a= τ A ⊕σ,    b=λ B ⊕ς
       0      otherwise.
     .

We consider the following 3-parameter family of noisy non-signaling box:
  
    PR_αβτ=α PR +β PR'+τ L +I/4(1-α- β-τ).

 where PR=P^000_NL, PR'=P^010_NL, and L=P^0000_L.
From Tsirelson's bound we see that those correlations with parameters outside the region max{4α+2τ, 4β+2τ}≤ 2√(2), which is  bounded by two blue planes in Fig.1a, cannot be quantum mechanical.




We shall now apply our NDWU criterion provided by Theorem 2 to this family of correlations to derive  the  allowed regions of α,  β, and τ for a quantum PC.
Two constraints   Eq.(<ref>) and Eq.(<ref>) end up with the following three-dimensional boundary (see SM)


    √(1-[α+β/1-τ]^2)√(1-[α-β/1-τ]^2)-
    |α^2-β^2|/(1-τ)^2 8cm
    ≥(α+β+2τ)(|α-β|+2τ)/(1+τ)^2
    -
    √(1-[α+β+2τ/1+τ]^2)√(1-[|α-β|+2τ/1+τ]^2),


which is plotted in Fig.1a. We can see that NDWU criterion is strictly  tighter than the NPA criterion,  the most powerful   criterion we have until now.   In the case of  τ=0, Eq.(<ref>) reduces to α^2+β^2 ≤1/2, referred to as boundary 1 here, which coincides with the boundary given by NPA criterion and IC, being a sufficient quantum correlation boundary<cit.>. In the case of β=0, Eq.(<ref>) becomes
 
    (α+2τ)^2/(1+τ)^2+α^2/(1-τ)^2≤ 1,
 which is referred to as boundary 2, NPA criterion implies 
    |3arcsinα+τ-τ^2/(1-β^2)-arcsinα-τ-τ^2/1-τ^2|≤π,
 while  IC implies (α+τ)^2+τ^2 < 1. These boundaries are compared in Fig.1b and it is clear that the most tight boundary is provided by our NDWU criterion.

Note that among all the criteria witnessing the quantum boundaries, only the boundary from NDWU criterion is non-convex, and it is the first PC criterion as far as we know.
   The  boundary 2 from NDWU criterion can generate  MC boundary as  β+√(2)α=1 (by a convex mixing of  two extremal points {β=1, α=0  } and {β=0, α=√(2)/2} in boundary 2) as shown in Fig.1b.   The SDP method  is a convex criterion and shares  the two extremal points in this case, thus it would not give a tighter boundary than this MC boundary <cit.>.  Regarding the powers of these methods in this case, the relation NDWU ≥ SDP ≥ NPA ≥  IC is shown in the figure.




Excluding Almost quantum correlation — The set of AQC  has been introduced to investigate to what extent the information-theoretic method is useful in bounding quantum correlation <cit.>.
  It has been found that AQC satisfies IC (with numeric evidence), and  all the current principles, thus these principles are fundamentally restricted  as they cannot exclude any AQC.

  An AQC that is not quantum mechanical and undetectable  even by the quantum NPA criterion <cit.> reads
  
    {9/20,2/11,2/11, 9/20,22/125, √(2)/9,37/700, 22/125}

which are eight probabilities of measuring observables 
    (A_0, A_1, B_0, B_1, A_0B_0, A_1B_0, A_0B_1, A_1B_1),
 respectively, with 1 as outcome. As there are  eight independent probabilities in the correlations in the simplest Bell scenario, we can recover all the correlations {p(ab|νμ)} so that the conditional expectation values {⟨ A_ν⟩_ω_b|μ} and {⟨ B_μ⟩_ω_a|ν} immediately. With a straightforward calculation, the right hand side of Eq.(<ref>) equals -0.25 while the left hand side equals 0.44, i.e., the condition Eq.(<ref>) is violated, i.e., the AQC above is singled out from PC.


We summarize the above comparison of NDWU criterion with  IC and NPA  criteria in Table I, showing that
our NDWU criterion outperforms both IC and NPA criteria. A fundamental question arises as whether NDWU criterion is a sufficient PC criterion.  Some evidences for  a positive answer  can be found in Ref <cit.>, where it is reported that if all the involved local systems and all the possible measurements performed on them allow a quantum mechanical description,   the correlations are also quantum. Differently,  the  correlation considered here deals with two dichotomic measurements only and four local conditional states on each side.  In this case NDWU principle is  sufficient to ensure  quantum description for these local measurements, from which it probably  follows that (no proof yet) NDWU criterion might also be a sufficient condition for quantum pure correlations.

Although our method has been demonstrated in details for the simplest Bell scenario, it should be emphasized that our method applies to most general scenario, e.g, multipartite and higher dimensional system with multi-settings.   The two cornerstones of our method, namely, the perspective  of understanding  correlation from  local properties of subsystems and the NDWU principle,  are applicable to general scenarios. Therefore our framework  could be taken as  a general method in characterizing  quantum correlations.

In conclusion, we have proposed no disturbance without uncertainty as a physical principle lying behind quantum theory.  As applications of this local principle to the non-signaling correlations, we have accounted for many quantum features such as the Tsirelson bound and  the quantum boundaries of a family of nonlocal boxes, and  ruled out  an almost quantum correlation from quantum correlation where other criteria all failed. In these applications, our NDWU criterion outperforms the celebrated quantum NPA criterion and  the previous information-theoretical results.




§ ACKNOWLEDGEMENT




 99


bel   N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani,   and S. Wehner, Bell nonlocality   Rev. Mod. Phys. 86, 419 (2014).


npa    M.Navascues, S. Pironio, and A. Acín, Bounding the set of quantum correlations, 	Phys. Rev. Lett. 98/ , 010401 (2007).
tir B. S. Cirel'son, Lett. Math. Phys. 4, 93 (1980).
sp S. Popescu,  Nonlocality beyond quantum mechanics,   Nat. Phys 10, 264C270 (2014).
ic  M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Zukowski. Information Causality as a Physical Principle.  Nature 461, 1101 (2009).
wvd W. van Dam. Implausible consequences of superstrong nonlocality. Nat. Comput 12:9-12 (2013).
cl A. Cabello.  Simple Explanation of the Quantum Violation of a Fundamental Inequality. Phys. Rev. Lett. 110, 060402 (2013).
by B. Yan. Quantum Correlations are Tightly Bound by the Exclusivity Principle. Phys. Rev. Lett. 110, 260406 (2013).
 ca  A. Cabello. Exclusivity principle and the quantum bound of the Bell inequality. Phys. Rev. A 90, 062125 (2014).
lv  L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).
sdp1  L. Masanes,  arxiv:quant-ph/0309137.
    sdp   M. Navascues, S. Pironio, and A. Acín,   New J. Phys. 10, 073013 (2008).
    dim M. Navascués, and T. VertesiPhys, Phys. Rev. Lett. 115, 020501 (2015).
hh   S. Ishizaka, Phys. Rev. A 97, 050102.




     tem C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, Phys. Rev. Lett. 111, 020403 (2013).
  acq M. Navascués, Y. Guryanova, M. J. Hoban,  and A. Acín, Nat. Commun 6, 6288 (2015).
  pr   S. Popescu  and D, Rohirlich. Quantum nonlocality as an axiom. Found. Phys. 24 379 (1994).
   tn1 S. Popescu,   Nonlocality beyond quantum mechanics.  Nature Phys, 10, 264 (2014).


 ml M. Navascués and H. Wunderlich, Proc. Royal Soc. A 466:881-890 (2009).
  lo  T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier and A. Acín, arXiv:1210.3018.
   meLiang-Liang Sun, Sixia Yu, Zeng-Bing Chen, arXiv:quant-ph/ 1808.06416.
     ran J. M. Donohue and E. Wolfe, Phys. Rev. A 92, 062120.
  gpt  P. Janotta, and H. Hinrichsen.  Generalized probability theories: what determines the structure of quantum theory? J. Phys. A: Math. Theor. 47,  323001 (2014).

   gs Li. Masanes, A. Acin, and N. Gisin. General properties of nonsignaling theories. Phys. Rev. A 73, 012112 (2006).




     15'  T. Vértesi and N. Brunner, Nat. Commun. 5, 5297 (2014).
 16'E. Wolfe and S. F. Yelin, Phys. Rev. A , 012123 (2012).
 17' K. T. Goh, J. Kaniewski, E. Wolfe, T. Vértesi, X. Wu, Y. Cai,
Y.-C. Liang, and V. Scarani, Phys. Rev. A 97, 022104 (2018).





    bou          J. Allcock, N. Brunner, M. Pawłowski, and V. Scarani, Recovering part of the boundary
 between quantum and nonquantum correlations from information causality. Phys. Rev. A 80, 040103 (2009).
 quan H. Barnum, S. Beigi, S. Boixo, M. B. Elliott, and S. Wehner, Phys. Rev. Lett. 104, 140401.




















§ SUPPLEMENTAL MATERIAL


Proof of  Theorem 1.—  Suppose that a finite-level quantum system is prepared in  state  ρ and two sharp measurements correspond to two orthonormal bases {|A^a_0⟩⟨ A^a_0|} and {|A^a'_1⟩⟨ A^a'_1|}. By denoting ⟨ A^a_0| A^a'_1⟩=Λ_a'a we have  expansion
|A^a'_1⟩=∑_aΛ_a'a|A^a_0⟩. As a result

    p(a'|A_1)=⟨ A^a'_1|ρ |A^a'_1⟩=∑_s,tΛ^*_a'sΛ_a't⟨ A^s_0|ρ| A^t_0⟩=∑_s≠ tΛ^*_a'sΛ_a't⟨ A^s_0|ρ| A^t_0⟩+p(a'|A_0→ A_1)

where p(a'|A_0→ A_1)=∑_ap(a|A_0)γ_A_1^a';𝒜_0^a with p(a|A_0)=⟨ A_0^a|ρ |A_0^a⟩ and transfer probabilities are symmetric

    γ_A_1^a';𝒜_0^a=γ_A_0^a;𝒜_1^a'=|⟨ A^a'_1|A^a_0⟩|^2.

Finally we calculate

    D_A_0→ A_1   =   ∑ _a'|p(a'|A_1)-p(a'|A_0→ A_1)|
       =   ∑_a'|∑_s≠ tΛ^*_a'sΛ_a't⟨ A^s_0|ρ| A^t_0⟩|
       ≤   ∑_a'√(∑_s≠ t|Λ^*_a'sΛ_a't|^2)√(∑_s≠ t|⟨ A^s_0|ρ| A^t_0⟩|^2)
       ≤   ∑_a'√(∑_s≠ tγ_A_0^t;𝒜_1^a'γ_A_0^s;𝒜_1^a')√(∑_s≠ tp(s|A_0)p(t|A_0))
       =   ∑ _a'√(1-∑_sγ_A_0^s;𝒜_1^a'^2)√(1-∑_tp(t|A_0)^2)
       =   Δ_A_0|A_1Δ_A_0.

Here the first inequality is due to Cauchy inequality, the second inequality is due to the fact that the state is positive semidefinite so that in the basis {|A_0^a⟩} it holds
|⟨ A_0^s|ρ|A_0^t⟩|^2≤⟨ A_0^s|ρ|A_0^s⟩⟨ A_0^t|ρ|A_0^t⟩.



 The transfer probabilities are symmetric, i.e., γ_A^a_0;𝒜^a^'_1=γ_A^a^'_1;𝒜^a_0, for two-outcome sharp measurements if the NDWU relation holds for all pairs of sharp measurements —  Considering normalization conditions, there are four independent transfer probabilities (which is independent of the initial state)

    γ_A^0_1;𝒜^0_0=1+c_1/2,   γ_A^0_1;𝒜^1_0=1+c_2/2,  γ_A^0_0;𝒜^0_1=1+c'_1/2,  γ_A^0_0;𝒜^1_1=1+c'_2/2.

It is symmetric if and only if c_1=-c_2=c_1'=-c_2'.
By denoting the expectation values ⟨ A_0⟩_𝒮=p_𝒮(0|A_0)-p_𝒮(1|A_0) and
 ⟨ A_1⟩_𝒮=p_𝒮(0|A_1)-p_𝒮(1|A_1) for two sharp measurements in an arbitrary state 𝒮, the NDWU relation
 Eq.(<ref>) becomes
 
    1/2(√(1-c_1'^2)+
    √(1-c_2'^2))√(1-⟨ A_0⟩_𝒮^2)≥|⟨ A_1⟩_𝒮-c_1+c_2/2-c_1-c_2/2⟨ A_0⟩_𝒮|

 for the sequential measurement scenario A_0→ A_1 while
  
    1/2(√(1-c_1^2)+
    √(1-c_2^2))√(1-⟨ A_1⟩_𝒮^2)≥|⟨ A_0⟩_𝒮-c_1'+c_2'/2-c_1'-c_2'/2⟨ A_1⟩_𝒮|

  for the sequential measurement scenario A_1→ A_0. Let us take at first the state 𝒮=𝒜_1^0, i.e., the state on which measurement A_1 gives a definite value 0. In this case ⟨ A_0⟩_𝒜_1^0=c_1' and ⟨ A_1⟩_𝒜_1^0=1 and noting that transfer probabilities are state independent, we obtain
  
    1-c_1+c_2/2-c_1-c_2/2c_1'   ≤   1/2(√(1-c_1'^2)+
    √(1-c_2'^2))√(1-c_1'^2)
       ≤   √(1-(|c_1'|+|c_2'|/2)^2)√(1-c_1'^2)
       ≤   1-|c_1'|+|c_2'|/2|c_1'|≤ 1-c_1'-c_2'/2c_1'

  where the first inequality is due to NDWU relation, the second inequality is due to the concavity of function √(1-x^2), and the third inequality is due to inequality √(1-x^2)√(1-y^2)≤ 1-|xy|.
 By taking the state to be 𝒮=𝒜_1^1, i.e., the state on which measurement A_1 gives a definite value 1, we have ⟨ A_0⟩_𝒜_1^1=c_2' and ⟨ A_1⟩_𝒜_1^1=-1 so that
   
    1+c_1+c_2/2+c_1-c_2/2c_2'   ≤   1/2(√(1-c_1'^2)+
    √(1-c_2'^2))√(1-c_2'^2)
       ≤   √(1-(|c_1'|+|c_2'|/2)^2)√(1-c_2'^2)
       ≤   1-|c_1'|+|c_2'|/2|c_2'|≤ 1+c_1'-c_2'/2c_2'.

 As a result we obtain 
    (c_1-c_2)(c_1'-c_2')≥ (|c_1'|+|c_2'|)^2≥ |c_1'-c_2'|^2≥0.

 Similarly from the A_1→ A_0 scenario we obtain
 
    (c_1-c_2)(c_1'-c_2')≥ (|c_1|+|c_2|)^2≥ |c_1-c_2|^2.

 If c_1'=c_2' or c_1=c_2 we have c_1'=c_1'=c_1=c_2=0 then we already have symmetric transfer probabilities. If c_1≠ c_2 and c_1'≠ c_2' we obtain
 both c_1-c_2≥ c_1'-c_2' and c_1'-c_2'≥ c_1-c_2, by noting they have the same sign, so that c_1-c_2= c_1'-c_2'. As a result we have both c_1+c_2≥0 and c_1+c_2≤ 0 giving the desired symmetry property of transfer probabilities c_1=-c_2=c_1'=-c_2'=c. The NDWU relation becomes
 
    √(1-c^2)√(1-⟨ A_0⟩^2)≥ |⟨ A_1⟩-c⟨ A_0⟩|

which is exactly NDWU relation Eq.(<ref>) for two-outcome sharp measurements.

 Proof of Theorem 2 —  By explicitly showing the dependence on the state the NDWU relation Eq.(<ref>) can be recast into the following equivalent form

    ⟨ A_0⟩_𝒮⟨ A_1⟩_𝒮-√(1-⟨ A_0⟩_𝒮^2)√(1-⟨ A_1⟩_𝒮^2)≤ c≤⟨ A_0⟩_𝒮⟨ A_1⟩_𝒮+√(1-⟨ A_0⟩_𝒮^2)√(1-⟨ A_1⟩_𝒮^2)

Because c is state-independent we shall have

    max_𝒮{⟨ A_0⟩_𝒮⟨ A_1⟩_𝒮-√(1- ⟨ A_0⟩_𝒮^2)√(1-⟨ A_1⟩_𝒮^2)}≤min_𝒮{⟨ A_0⟩_𝒮⟨ A_1⟩_𝒮+√(1-⟨ A_0⟩_𝒮^2)√(1-⟨ A_1⟩_𝒮^2)}

to ensure the existence of c. In the simplest Bell scenario, two measurements with two outcomes from Bob's side prepares four conditional states 𝔅={ω_b|μ} for Alice while two measurements with two outcomes from Alice's side prepares four conditional states 𝔄={ω_a|ν} for Bob, which lead to the NDWU conditions in Theorem 2.

 Proof of Theorem 3 — 
Note that NDWU relation Eq.(<ref>) can be recast to an equivalent form

    (⟨ A_0⟩+⟨ A_1⟩)^2/2(1+c)+(⟨ A_0⟩-⟨ A_1⟩)^2/2(1-c)≤ 1,

where c=2γ_A^0_0;𝒜^0_1-1.
As a result

    |⟨ A_0⟩_𝒮±⟨ A_1⟩_𝒮|≤√(2(1± c))

for an arbitrary state 𝒮, especially for those four conditional states ω_b|μ resulting from the measurement B_μ made by Bob with outcome b.
Now the Tsirelson bound can be derived:
 
    CHSH   =∑_a,b,ν,μ=0^1(-1)^a+b+μνp(ab|νμ)
       =∑_a,b,ν,μ=0^1(-1)^a+b+μνp(b|μ)p(ab|νμ)/p(b|μ)
       =∑_b,ν,μ=0^1(-1)^b+μνp(b|μ)⟨ A_ν⟩_ω_b|μ
       =∑_b,μ=0^1(-1)^bp(b|μ)(⟨ A_0⟩_ω_b|μ+(-1)^μ⟨ A_1⟩_ω_b|μ)
       ≤∑_ b,μ=0^1p(b|μ)|⟨ A_0⟩_ω_b|μ+(-1)^μ⟨ A_1⟩_ω_b|μ| 
       ≤∑_ b,μ=0^1p(b|μ)√(2(1+(-1)^μ c))
       =∑_μ=0^1 √(2(1+(-1)^μ c))≤ 2√(2).

where the footnote μ_b specify the conditional states yielding the expectations. The maximum in the fourth inequality is taken when γ_A^0_0;𝒜^0_1=1/2, which is just the condition for the maximum violation in QM.

Proof of three dimensional boundary Eq.(<ref>) —
Let 
    PR=1+(-1)^a+b+νμ/4,   PR^'=1+(-1)^a+b+νμ+μ/4,   L=(1+(-1)^a)(1+(-1)^b)/4

the 3-parameter family of non-signaling box becomes

    α PR+β PR^'+τ L+1/4(1-α-β-τ)=1+(-1)^aA_ν+(-1)^bB_μ+(-1)^a+bC_νμ/4

with α,β,τ≥ 0 and α+β+τ≤ 1 and
A_ν=B_μ=τ, C_νμ=(-1)^νμ(α+(-1)^μβ)+τ.
Define

    E_ν;μ_b=A_ν+(-1)^bC_νμ/1+(-1)^b B_μ,    D^±_μ_b=E_0;μ_bE_1;μ_b±√(1-(E_0;μ_b)^2)√(1-(E_1;μ_b)^2)

and

    F_μ; ν_a=B_μ+(-1)^aC_νμ/1+(-1)^aA_ν,   d_ν_a^±= F_0; ν_aF_1;ν_a±√(1-(F_0;ν_a)^2)√(1-(F_1;ν_a)^2)

We shall at first prove that the following boundary arises from NDWU constraint on Alice's side

    (2τ+α+β)^2/(1+τ)^2+(α-β)^2/(1-τ)^2≤ 1

and then prove that the boundary Eq.(<ref>) arises from NDWU constraints on Bob's side. Then we compare these two boundaries to arrive at Eq.(<ref>).
First, we consider the NDWU constraints from Alice's side.


  * By definition we have

    E_0;μ_b=τ+(-1)^b(α+(-1)^μβ+τ)/1+(-1)^bτ,   E_1;μ_b=τ+(-1)^b((-1)^μα+β+τ)/1+(-1)^bτ

or, by denoting α_±=β±α,

    E_ν;0_0=2τ+α_+/1+τ,   E_ν;0_1=-α_+/1-τ,   E_ν;1_1=(-1)^να_-/1-τ,   E_ν;1_0=2τ-(-1)^να_-/1+τ



  * Since 1=D_0_0^+=D_0_1^+=-D_1_1^+, we have

    D^+_m=min{ 1-2(E_ν;1_1)^2,D^+_1_0},   D^-_M=max{2(E_ν;0_0)^2-1,2(E_ν;0_1)^2-1,D^-_1_0}


  * Lower bound D_M=2(E_ν;0_0)^2-1.


  * From (1+τ)α_+=(1-τ)α_++2τα_+≤ (1-τ)(2τ+α_+) it follows
2(E_ν;0_0)^2-1≥ 2(E_ν;0_1)^2-1.

  * To prove
2(E_ν;0_0)^2-1≥ D_1_0^-
i.e.,

    2(2τ+α_+)^2-(1+τ)^2≥ 4τ^2-α_-^2-√((1+τ)^2-(2τ+α_-)^2)√((1+τ)^2-(2τ-α_-)^2)

or
f(Z)≥ 0 with Z=(1+τ)^2, a,b=2τ±α_- and

    f(ν)=√(ν-a)√(ν-b)-ν+C,    C=2(2τ+α_+)^2-4τ^2+α_-^2

being a non-decreasing function of ν since

    f^'(ν)=1/2(√(ν-a)/√(ν-b)+√(ν-b)/√(ν-a))-1≥ 0.


We can assume
Z≥ C, otherwise we have trivially f(ν)≥0. Thus we have f(Z)≥ f(C)=√(C-a)√(C-b)≥ 0.
We have only to check C≥ a,b, i.e.,

    2(2τ+α_+)^2-4τ^2+α_-^2- (2τ±α_-)^2=8τα_++α^2_+∓ 4τα_-≥ 0

which is obviously true since τ≥ 0 and α_+≥α_-.


  * Upper bound D_m^+= D_1_0^+, i.e.,

    (1+τ)^2-2λ^2α_-^2≤ 4τ^2-α_-^2+√((1+τ)^2-(2τ+α_-)^2)√((1+τ)^2-(2τ-α_-)^2)

with λ=1+τ/1-τ. We can assume

    1+τ^2+6τ/(1-τ)^2α_-^2=(2λ^2-1)α_-^2≤ (1+τ)^2-4τ^2=(1-τ)(1+3τ)

i.e., we have (1-τ)^3≥α_-^2.
Compute

    ((1+τ)^2-(2τ+α_-)^2)((1+τ)^2-(2τ-α_-)^2)-( (1+τ)^2-2λ^2α_-^2+α_-^2-4τ^2)^2


    =(1+τ)^2(2(2λ^2-1)α_-^2+8τ^2-8τ^2-2α_-^2)-4λ^2α_-^2(λ^2α_-^2-α_-^2+4τ^2)


    =4(1+τ)^2(λ^2-1)α_-^2-4λ^2α_-^2(λ^2α_-^2-α_-^2+4τ^2)


    =4α_-^2((λ^2-1)((1+τ)^2-λ^2α_-^2)-4λ^2τ^2)=16τα_-^2λ^2/(1-τ)^2((1-τ)^3-α_-^2)≥ 0



And then we consider the NDWU constraints from Bob's side. We assume α_-≥ 0 i.e., β≥α in what follows. By definition we have

    F_μ;ν_a=τ+(-1)^a(τ+(-1)^νμ(α+(-1)^μβ))/1+(-1)^aτ

leading to

    F_0;ν_0=2τ+α_+/1+τ:=p,   F_0;ν_1=-α_+/1-τ:=-u,

and

    F_1;0_0=2τ-α_-/1+τ:=q,   F_1;1_0=2τ+α_-/1+τ:=q^',    F_1;0_1=α_-/1-τ=v,   F_1;1_1=-α_-/1-τ=-v.

First we note that function f(x,y)=xy-√(1-x^2)√(1-y^2) is non-decreasing for x,y≥0 since

    ∂_x f(x,y)=y+x√(1-y^2)/√(1-x^2)≥0,  ∂_y f(x,y)=x+y√(1-x^2)/√(1-y^2)≥0

and the function g(x,y)=xy+√(1-x^2)√(1-y^2)  is non-decreasing for x≥ y since

    ∂_y g(x,y)=√(1-x^2)(x/√(1-x^2)-y/√(1-y^2))≥ 0   (x≥ y)




  * Lower bound: max{d^-_ν_a}=d^-_1_0 because


  * d^-_1_1=f(u,v)≥ f(-u,v)= d^-_0_1 since u,v≥ 0 as α_-≥ 0.

  * d^-_1_0=f(p,q^')≥ d^-_0_0=f(p,q).

  * d^-_1_0=f(p,q^')≥ f(p,v) ≥ f(u,v)=d^-_1_1
because p≥ u and q^'≥ v as

    (1-τ)(2τ+α_±)-(1+τ)α_±=2τ(1-τ-α_±)≥ 0



  * Upper bound: min{d^+_ν_a}=d^+_0_1


  * d^+_1_1=g(u,v)≥ g(-u,v)= d^+_0_1

  * d^+_1_0=g(p,q^')≥ g(p,q)= d^+_0_0 since p≥ q,q^'

  * Need to show d^+_0_0≥ d^+_0_1 i.e.,
g(p,q)=pq+p̅q̅≥ -uv+u̅v̅:=c.
As a result Δ:= x^2+y^2-2 x y c=2(1-c)( x+ y)-(1-c)^2 where

    t=1-τ/1+τ,    x=1-u,    y=1+v,    p=1-t  x,   q=1-t  y

Then either c≤ pq which gives trivially the result or c≥ pq in which case g(p,q)≥ c becomes

    0≥ p^2+q^2+c^2-1-2pqc=t^2Δ-2t(1-c)( x+ y)+(1-c)^2

or equivalently

    t_c=1-c/2 x+2 y-1+c≤ t≤ 1

which is ensured by c≥ pq=1-t( x+ y)+t^2 x y, i.e., t_-≤ t≤ t_+ where

    t_±= x+ y±√(( x- y)^2+4 x y c)/2 x  y

as long as we can show
t_c≤ t_-. Noting that

    x+ y-1+c= x+ y-1-(1- x)( y-1)+u̅v̅= x y+u̅v̅≥ 0
 we have

    1/t_c=2 x+2 y/1-c-1≥ x+ y/1-c≥ x+ y+√(( x+ y)^2-4 x y(1-c))/2(1-c)=1/t_-.





In sum, we have d^+_m=d^+_0_1 and d^-_M=d^-_1_0 in the case of α_-≥0. Similarly we have d^+_m=d^+_1_0 and d^-_M=d^-_0_1 in the case of α_-≤ 0 which gives rise to bound Eq.(<ref>).









Last,  we shall compare two boundaries arising from Alice and Bob. Assume still α_-≥0 and
we shall prove that Alice's boundary d_0_1^+≥ d_1_0^-, i.e., p̅q̅^'+u̅v̅-p q^'- uv≥ 0 infers Bob's boundary p^2+v^2≤1 or 1≥ t≥ (1-v̅)/ x. We have q^'=1-tz with z=2- y (z≥  x) so that v= y-1=1-z. In terms of z we have
 again  either  
 
  * u̅v̅-uv=c≥ pq^', i.e., c≥ (1-tx)(1-tz)=1-t(x+z)+t^2xz. If (x+z)^2<4(1-c)xz then we have c<pq^'. Thus we should have (x+z)^2≥ 4(1-c)xz giving rise to
 
    T_+=x+z+√((x+z)^2-4(1-c)xz)/2xz≥ t≥x+z-√((x+z)^2-4(1-c)xz)/2xz:=T_-

  or
 
  * c≤ pq^' with
p^2+q^'^2+c^2-2pq^' c≤ 1.
 i.e., t≥ T_+ or t≤ T_-, and

    0≥(1-c)^2-2t( x+z)(1-c)+t^2( x^2+z^2-2 x zc)

i.e.,

    t_p:=1-c/ x+z-√(2(1+c) x z)≥ t ≥1-c/ x+z+√(2(1+c) x z):=t_m


  * From c=u̅v̅-u v it follows that

    |1-c-x-z|=√(2(1+c)xz)

leading to t_m=1 if 1-c≥ x+z (trivially infering Bob's boundary) and t_p=1 if 1-c≤ x+z.


Since

    t_m=1-c/ x+z+√(2(1+c) x z)≤2(1-c)/2(x+z)≤2(1-c)/ x+z+√((x+z)^2-4(1-c) x z)= T_-
 we obtain finally Alice's boundary as 
    1≥ t≥  t_m∩ 1-c≤ x+z.

We have only to show that,  given 1-c≤ x+z,

    t_m=1-c/ 2(x+z)-1+c≥1-v̅/x

or

    (1-c)(x+1-v̅)-2(x+z)(1-v̅)≥0

which is plotted in Fig.2 as a function of u and v which is clearly non-negative.





