No-causal order constraint in spacelike events on quantum measurement postulate
Yeong Deok Han
We have studied how relativistic causality limits on the generalization of quantum probability assignment, which can give more nonlocal (superquantum) correlations than quantum correlations. We have shown that 'no causal order' condition between two spacelike separate events is stronger than no faster-than-light-signaling condition as the requirement of relativistic causality. Using no causal order condition we have proved that Born rule is the only probability assignment to quantum measurement on an arbitrary entangled quantum state described by the Hilbert space language, consistent with relativistic causality. We have also shown that all superquantum correlations, which preserve relativistic causality, can be understood as hypothetical correlations given by applying modified quantum probability assignments to quantum measurements on a Bell state.
QPRRelativityvC.tex
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\begin{document}

%\title{No-causal order constraint in spacelike events on quantum measurement postulate}
\title{No causal order and quantum probability assignment}%with modified Born rule}

\author{Yeong Deok Han}
\affiliation{Department of Game Contents, Woosuk University, Wanju, Cheonbuk, 565-701, Republic of Korea}
\author{Taeseung Choi}
\email{tschoi@swu.ac.kr}
\affiliation{ Division of General Education, Seoul Women's University, Seoul 139-774, Republic of Korea} %\textbackslash\textbackslash



%
\date{\today}

\begin{abstract}
We have studied how relativistic causality limits on the generalization of quantum probability assignment, which 
can give more nonlocal (superquantum) correlations than quantum correlations. 
We have shown that 'no causal order' condition between two spacelike separate events is stronger than no 
faster-than-light-signaling condition as the requirement of relativistic causality. 
Using no causal order condition 
we have proved that Born rule is the only probability assignment to quantum measurement on an arbitrary entangled 
quantum state described by the Hilbert space language, consistent with relativistic causality. 
We have also shown that all superquantum correlations, which preserve 
relativistic causality, can be understood as hypothetical correlations given 
by applying modified quantum probability assignments to quantum measurements on a Bell state.    
\end{abstract}

\pacs{ 03.65.Ud, 03.65.Ta, 03.67.-a}% PACS, the Physics and Astronomy
                             % Classification Scheme.
%\keywords{Suggested keywords}%Use showkeys class option if keyword
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\maketitle

\section{Introduction}

The compatibility between nonlocality in quantum mechanics and causality in special relativity is a 
fundamental question and not fully understood yet. 
Quantum nonlocality manifests itself in joint probability distributions between local measurement results 
on an entangled state shared between two space-like separate parties, Alice and Bob \cite{Bell}. 
A local measurement on one state of entangled pair by Alice seems to influence the other state of Bob instantaneously. 
This means that the speed of the hypothetical influence of quantum nonlocality seems to be infinite. 
The important question is whether the hypothetical influence can be used as a tool of faster-than-light communication. 
Bancal et al. have shown that for any finite speed hypothetical influences, 
faster-than-light communication can be built \cite{Bancal}. 
According to their results, only when the collapse of the states by measurement is instantaneous, 
quantum nonlocality can be reconciled with relativistic causality.  
Recent experiments determined that the lower bound of the speed of the hypothetical influence has to exceed 
the speed of light by at least four orders of magnitude \cite{Salart, Yin}. 
Those results suggest that the superluminal signaling by using quantum nonlocality seems to be implausible. 
The origin of this impossibility, however, is not clearly understood yet. 
% It seems that there is no relativistic constraint in quantum mechanical framework, therefore, 
% the existence of faster-than-light signaling in quantum mechanics is not forbidden at least by principle. 

Another important question is whether the nonlocality of quantum correlation is maximum among nonlocal correlations, preserving relativistic 
causality. Popescu and Rohrlich found the surprising result that quantum nonlocality is not maximum \cite{Popescu1}. 
They have shown the existence of 'superquantum' correlations that are more nonlocal than 
quantum correlations.  
The nonlocality can be demonstrated by a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, 
in which the degree of a violation is represented by the CHSH value \cite{Clauser}.
The quantum upper bound of the CHSH value, which is known as Tsirelson's bound, is 
$\mathcal{B}_Q=2\sqrt{2}$ \cite{Tsirelson}. 
The maximum CHSH value, preserving relativistic causality, is $4$ for the Popescu-Rohrlich nonlocal box (PR box).  
At this stage we naturally ask the question; Does any post-quantum theory, which has superquantum correlations 
and preserves relativistic causality, exist in nature?
The properties of post-quantum theories are studied by several authors in an information theoretic view
\cite{Dam, Barrett, Brassard1, Linden1, Brunner, Brunner1}. Recent study for information causality has shown that 
the possibility of post-quantum theory is very low \cite{Pawlowski}. 

In this Letter we will investigate whether we can obtain a post-quantum theory or at least superquantum correlations, 
preserving relativistic causality, by modifying the standard quantum framework. 
The existence of faster-than-light signal will break relativistic causality, therefore, 
no faster-than-light (no-signaling) condition has been considered equivalent to relativistic causality. 
We have found a new condition to formulate the requirement that one cannot define a chronological order, 
agreed by all observers in an inertial frame, between spacelike separate events. 
We call this 'no causal order' condition. We will show that no causal order condition is a sufficient, but 
not a necessary, condition for no-signaling condition.
Hence the requirement to preserve relativistic causality must be no causal order condition. 
Let us suppose an input state collapses to an output state by quantum measurement. 
Then a quantum correlation, which can provide a nonlocality, will depend on the inner product structure of a 
Hilbert space and quantum measurement postulates. 
The change of inner product will change the norm of the state. The norm of the state is related with the probability 
of preserving the state in case of no interaction, therefore, the change of the norm of the state is not desirable. 
A probability assignment to measurement outcome is given by a positive real function of inner product 
between an input and an output states. 
Hence the generalization of quantum probability assignment, by using a general function of inner product, 
can change quantum correlations without modifying the Hilbert space structure in quantum physics.  
In general, new quantum correlations, generated by applying a generalized Born rule, would be inconsistent 
with relativistic causality.  
By using this fact, we will prove that Born rule is the only probability assignment 
for a quantum theory described by the Hilbert space language, 
consistent with relativistic causality. 
This fact implies that a post-quantum theory cannot be obtained 
at least by a modification of quantum probability assignment. 
 
We will also show that the joint probabilities of quantum measurements on a Bell state, which are obtained by 
applying a certain modified Born rule, can describe all the superquantum correlations. 
Our results suggest that a device with superquantum correlations can be understood as the hypothetical device consisted 
of a Bell state and the quantum measurement with the modified Born rule. 
Bell states are special in the sense that new quantum correlations of a Bell state obtained by the modified Born rule 
can automatically satisfy no causal order condition. Quantum correlations of other entangled states cannot 
satisfy no causal order condition in general. Quantum correlations of every entangled state 
residing in two-dimensional Hilbert space, however, can satisfy no causal order condition not for all measurements but 
only for a special measurement. 
The PR box is a mere assignment of joint probabilities, which shows the maximum superquantum correlation \cite{Popescu1}.  
Our formalism, however, provides the systematic way to obtain the nonisotropic nonlocal boxes with hypothetical superquantum 
correlations, which are helpful to study the nature of nonlocality related with communication complexity \cite{Dam1}.
  
First, we will establish a relationship between the two causality conditions, no-signaling condition and 
no causal order condition, in special relativity. 
No causal order condition is originated from the fact that one cannot assign a cause-and-effect relationship 
between two spacelike separate events, because one can always find two observers to whom the chronological sequence 
of two spacelike events is in reverse order. 
The significance of no causal order condition, however, has been overlooked. It seems that 
no causal order condition is considered to be equivalent to local measurement condition, which is weaker. 
We will prove that no causal order condition 
is a sufficient, but not a necessary, condition to satisfy no-signaling condition. 
%To prove this proposition, we will consider two nonlocal devices with binary inputs and outputs 
Suppose Alice and Bob share a nonlocal device with inputs and outputs. 
The local device of Alice (Bob) has an input $x$ ($y$) and an output $a$ ($b$). 
The device will provide joint probabilities $P_A(ab|xy)$ for Alice-first measurement 
and $P_B(ab|xy)$ for Bob-first measurement, respectively. 
%The subscripts $A$ and $B$ are introduced because we assume that the joint probabilities of binary input-output 
%devices can be different according to the chronological order between Alice's measurement and Bob's measurement. 
Alice-first measurement means that Alice's device is operated before the operation of Bob's device, 
and vice versa for Bob-first measurement. 
For any observer in its own inertial frame, the chronological order between Alice's measurement and Bob's 
measurement can be clearly defined, because the observer has its own global time. 
Here we suppose that Alice and Bob do not move relative to each other for simplicity. 
No causal order condition can be satisfied when the two probability distributions, 
$P_A(ab|xy)$ and $ P_B(ab|xy)$, are equal. 
On the other hand, local measurement condition requires $P_A(a|x)=P_B(a|x)$, where 
$P_A(a|x)=\sum_{b,y} P_A(ab|xy)/2$ and $P_B(a|x)=\sum_{b,y}P_B(ab|xy)/2$. 
Local measurement condition implies that results of 
a local measurement at one position must solely determined by local informations at that position. 
One can easily check that no causal order condition is stronger than local measurement condition. 
Now let us prove the proposition.

{\it Proposition.-} No causal order condition is a sufficient, but not a 
necessary, condition for no-signaling condition.

{\it Proof.-}  First we will show that breaking of no signaling condition derives breaking of no causal order condition. 
In order to send a signal by choosing one's input, there must be causal order 
between sender and receiver, because the event of sending a signal has to occur before the event of receiving that signal 
for all observers. 
%This implies that an existence of causal order is a necessary condition for superluminal signaling.    
%We will give more rigorous proof by using probability distributions of Alice-first and Bob-first measurements. 
To be more specific, let us assume that Bob send a signal by choosing one of his inputs and 
Alice can distinguish between two inputs of Bob by her local measurement in Bob-first measurement; 
This assumption requires 
\begin{eqnarray} 
\label{Eq:NS}
P_B(a|x,0) \neq P_B(a|x,1),
\end{eqnarray}
 where $P_B(a|x,0)=\sum_b P_B(ab|x0)$ and $P_B(a|x,1)=\sum_b P_B(ab|x1)$. 
On the other hand, no causal order condition requires 
\begin{eqnarray}
\label{Eq:NCO}
P_A(a|x,0)=P_B(a|x,0) \mbox{ and } P_A(a|x,1)=P_B(a|x,1).
\end{eqnarray} 
The two relations in Eqs. (\ref{Eq:NS}) and (\ref{Eq:NCO}) derives the inequality, $P_A(a|x,0) \neq P_A(a|x,1)$ in 
Alice-first measurement, which implies that Alice knows Bob's input before Bob's operation. 
This conclusion is definitely a contradiction. 
Therefore, no causal order condition is a sufficient condition for no-signaling condition. 

Next we will show that no causal order condition is not a necessary condition for no-signaling condition. 
We will prove this statement by providing an example of a nonlocal device, which satisfies no-signaling condition and violate 
no causal order condition. One example is the following; 
%Such a device must provide different nonlocal correlations 
%depending on the chronological order between Alice's and Bob's operations such as; 
The device operates as the PR box for Alice-first measurement with joint probabilities $P_A(ab|xy)=1/2$, 
if $a \oplus b=xy$, and $0$ otherwise, 
where $\oplus$ denotes an addition modulo $2$. On the other hand, the same device 
shows anti-PR box correlations for Bob-first measurement defined by $P_B(ab|xy)=0$, if $ a \oplus b=xy$, and 
$1/2$ otherwise. Hence this device certainly breaks no causal order condition. 
It is easy to check that PR box and anti-PR box both satisfy no-signaling 
conditions, which are $\sum_{a(b)} P_{A(B)}(ab|xy) =\sum_{a(b)} P_{A(B)}(ab|\tilde{x}{y})$. 
%and $\sum_b P_B(ab|xy) =\sum_b P_A(ab|x\tilde{y})$. 
%But no causal condition, which is $P_A(ab|xy) = P_B(ab|xy)$, is violated. 
A tilde operation is defined as $\tilde{x}=x+1$ ($\tilde{y}=y+1$) mod $2$. Q.E.D.\\
Note that local measurement condition $P_A(a|x)=P_B(a|x)$ is also satisfied. 
According to the Proposition, no causal order condition is stronger than no-signaling condition. Hence we will 
impose no causal order condition as the requirement of relativistic causality. 

Let us return to the study of the relativistic limit on the candidates of post-quantum theories 
generated by applying a modification of quantum probability postulates.  
In the standard quantum framework, a physical observable $\mathcal{A}$ is a linear Hermitian operator with 
real eigenvalues $\{ a_1, \cdots, a_d\}$ 
and mutually orthonormal eigenvectors $\{ |\phi_1\rangle, \cdots, |\phi_d\rangle\}$, 
where $d$ is the dimension of a Hilbert space.
Physical observables satisfy the following measurement postulates; i) an outcome of a measurement is always an eigenvalue of 
$\mathcal{A}$. ii) The probability of an outcome $a_k$ for an initial state $|\psi\rangle$ is obtained 
with $P(a_k)=|\langle \phi_k|\psi\rangle|^2$.
iii) The quantum state after the measurement that gives the outcome $a_k$ 
reduces to the corresponding eigenstate $|\phi_k\rangle$.
The modification of i) has nothing to do with nonlocal correlations. 
The modification of iii) will destroy the notion of orthogonality 
in a Hilbert space, because it involves the fact that 
each measurement outcome in sequential measurements can change measurement by measurement even for 
a free evolution. 
%In other words, it will destroy the consistency of a Hilbert space.
% That is, the modification of one of measurement postulates i) and iii) 
%will destroy the consistency of a Hilbert space description.  
%This implies that 
%these two postulates are accepted to be confirmed during our long history of eperiments. 
Hence we will focus on a possible generalization of the quantum probability assignment in measurement postulate ii), 
which is known as Born rule. 

We will consider a pair of two qubits because 
this is a minimal case, which shows nonlocal quantum correlations. 
The same generalization of Born rule will be applied to both Alice's and Bob's measurements, 
because physical laws must take the same form in all inertial frames. 
Quantum measurements can be locally defined, hence it is enough to define the generalized Born rule 
only in a two-dimensional Hilbert space, e.g., of Alice. 
Let an initial state be $|\psi\rangle = c|0\rangle_x +d |1\rangle_x$, where 
$|0\rangle_x$ and $|1\rangle_x$ are eigenvectors of an input (observable) $x$ which give outcomes $0$ and $1$, respectively. 
The generalized Born rule is defined by assigning an arbitrary non-negative function $H(|c|^2)$ of $|c|^2$ 
to $P_A(0|x)$, which is the probability of the outcome $0$ for the input $x$. 
The probability $P_A(1|x)$ for the outcome $1$ can be determined by the normalization of probability as $1-H(|c|^2)$. 
Note that Born rule corresponds to $H(|c|^2)=|c|^2$.

Now we will investigate whether the new nonlocal correlations, between a pair of spacelike separate qubits, 
generated by the generalized Born rule can satisfy no causal order condition. 
The general state for a pair of qubits is described by 
\begin{eqnarray}
|\psi\rangle &=& \alpha_1 |0\rangle_{x=0} |0\rangle_{y=0} + \alpha_2 |0\rangle_{x=0} |1\rangle_{y=0} \\ \nonumber 
&+& \alpha_3 |1\rangle_{x=0} |0\rangle_{y=0} + \alpha_4 |1\rangle_{x=0} |1\rangle_{y=0},
%|\psi\rangle &=& \alpha_1 |0\rangle_A |0\rangle_B + \alpha_2 |0\rangle_A |1\rangle_B 
%+ \alpha_3 |1\rangle_A |0\rangle_B + \alpha_4 |1\rangle_A |1\rangle_B,
\end{eqnarray} 
where $|\alpha_1|^2+|\alpha_2|^2+|\alpha_3|^2+|\alpha_4|^2=1$.
%where the subscripts $0$ and $0$ are measurement inputs of Alice and Bob, respectively. 
%The measurement inputs are shown in Fig. \ref{Fig:MAxis}.  
 %where thenotation of the state is the same as before.  
% The states $|x (y)\rangle$ and $|\tilde{x} (\tilde{y})\rangle$ are eigenstates for the measurement desciribed by 
% $\hat{x} (\hat{y})$ observable with eigenvalues $+$ and $-$, respectively. 
We will explain the way to obtain the probability $P_A(00|00)$ in Alice-first measurement. 
To consider Alice-first measurement, it is convenient to rewrite the state as 
\begin{eqnarray}
&&|\psi\rangle = \sqrt{C} |0\rangle_{x=0} |Y_1\rangle_{y=0}  + 
\sqrt{D} |1\rangle_{x=0} |Y_2\rangle_{y=0}, 
\end{eqnarray}
%&&|\psi\rangle = \sqrt{|\alpha_1|^2 +|\alpha_2|^2} |0\rangle_{x=0} |Y_1\rangle_{y=0}  + 
%\sqrt{|\alpha_3|^2 +|\alpha_4|^2} |1\rangle_{x=0} |Y_2\rangle_{y=0}, \\ \nonumber
where $ C=  |\alpha_1|^2 +|\alpha_2|^2$, $D= |\alpha_3|^2 +|\alpha_4|^2$, 
$|Y_1\rangle_{y=0}=\frac{1}{\sqrt{C}}(\alpha_1 |0\rangle_{y=0}+ \alpha_2|1\rangle_{y=0})$, and 
$|Y_2\rangle_{y=0}= \frac{1}{\sqrt{D}}(\alpha_3 |0\rangle_{y=0}+ \alpha_4|1\rangle_{y=0})$.
% are normalized states.
The state of Bob, after the $x=0$ measurement of Alice, will be projected either to 
$|Y_1\rangle_{y=0}$ with probability $H(|C|^2)$ or to $|Y_2\rangle_{y=0}$ with probability $1- H(|C|^2)$. 
Therefore, we can obtain
%Using successive measurement ${y}$ of Bob, the state finally will be projected to one of 
%$|0|x\rangle_A |0|y\rangle_B$, $|0|x\rangle_A |1|{y}\rangle_B$, $|1|{x}\rangle_A |0|y\rangle_B$ and 
%$|1|{x}\rangle_A |1|{y}\rangle_B$. 
%Let the projective measurements ${x}$ and ${y}$ be inputs $0$ of Alice and $0$ of Bob, respectively. 
%Then the probability of the projection to output state $|0\rangle_A|0\rangle_B$ for Alice-first measurement becomes
\begin{eqnarray}
P_A(00|00)=H({|\alpha_1|^2 +|\alpha_2|^2})
H\left(\frac{|\alpha_1|^2}{{|\alpha_1|^2 +|\alpha_2|^2}}\right).
\end{eqnarray}
%Here only the relavant terms are written to study the erquirement of no causal order condition. 
Similarly, the probability of Bob-first measurement for the same inputs and outputs can be obtained as% the projection to the same state $|0\rangle_A|0\rangle_B$ %becomes
\begin{eqnarray}
P_B(00|00) =H({|\alpha_1|^2 +|\alpha_3|^2}) 
H\left(\frac{|\alpha_1|^2}{{|\alpha_1|^2 +|\alpha_3|^2}}\right).
\end{eqnarray}

 
No causal order condition requires $P_A(00|00) = P_B(00|00)$. 
This condition must be satisfied for arbitrary $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$. 
By substituting $0$ for $\alpha_3$, we can obtain the relation, %the following condition
\begin{eqnarray}
\label{eq:NCO}
H(|\alpha_1|^2 +|\alpha_2|^2)H(\frac{|\alpha_1|^2}{|\alpha_1|^2 +|\alpha_2|^2})
= H(|\alpha_1|^2 ),
\end{eqnarray}
where $H(1)=1$ is used.  
Another relation is obtained as $|\alpha_1|$ and $|\alpha_2|$ are exchanged with each other in Eq. (\ref{eq:NCO}). 
 By adding those two relations and using the probability normalization,  
$H(|c|^2)$ is proven to be a linear function of $|c|^2$, that is, $H(|c|^2)=|c|^2$. 
Therefore, we have proven that no causal order condition derives Born rule as the only quantum probability assignment 
in a two-qubit Hilbert space. The proof in higher dimensional Hilbert space will be essentially the same 
because the Hilbert space of the minimal case can always be considered as the subspace of higher dimensional Hilbert space. 
Our derivation of Born rule implies that no post-quantum theory, preserving relativistic causality, 
can be obtained by generalizing Born rule. 
This proof is general in the sense that we assumed a general probability assignment by assigning 
an arbitrary non-negative function to an outcome probability of quantum measurement. 

\begin{figure}
\includegraphics[scale=0.7]{B.eps}\\%[scale=1.3]
\caption{(Color online)  The real line shows the monotonic increasing behavior of the CHSH value $\mathcal{B}$ 
on the exponent $m$ up to the maximum nonlocality of $4$. 
The dot represent the quantum Tsirelson's bound $2\sqrt{2}$. 
The point $\mathcal{B}_Q$ shows that the Tsirelson's bound is reproduced for the Born rule.}
\label{FIG:CHSHPRBox}
\end{figure}


We will show another important result that Bell states can generate all superquantum correlations by using 
the following modification of Born rule, called the modified Born rule:  
 \begin{eqnarray}
 \label{Eq:MBR}
H(|c|^2)= \frac{|c|^{\frac{m}{2}}}{|c|^{\frac{m}{2}}+|d|^{\frac{m}{2}}}, \mbox{ for real positive } m.
\end{eqnarray}
Suppose that Alice and Bob share the Bell state $
|\psi\rangle_{\mbox{\scriptsize{S}}} = \frac{1}{\sqrt{2}} \left( |0\rangle_{x=0} |0\rangle_{y=0} + 
 |1\rangle_{x=0} |1\rangle_{y=0}\right).$
We will put measurement inputs by the standard Bell-CHSH observables such as  
$\sigma_z$ ($x=0$) and $\sigma_x$ ($x=1$) for Alice and $(\sigma_z +\sigma_x)/\sqrt{2}$ ($y=0$) and 
$ (\sigma_x -\sigma_z)/\sqrt{2}$ ($y=1$) for Bob, where $\sigma_i$ are Pauli spin matrices \cite{Clauser}.
The probability distributions $P_{A(B)}(ab|xy)$, which are calculated by using the modified Born rule in Eq. (\ref{Eq:MBR}), 
can be shown to satisfy no causal order condition for arbitrary real positive $m$. 
 The CHSH value of the Bell states under the modified Born rule is obtained as 
 \begin{eqnarray}
 \label{Eq:CHSH}
\mathcal{B}=4\frac{\sqrt{2+\sqrt{2}}^m-\sqrt{2-\sqrt{2}}^m}{\sqrt{2+\sqrt{2}}^m + \sqrt{2-\sqrt{2}}^m}.
\end{eqnarray}
One can reproduce the PR Box by defining 
$P_{\scriptsize{PR}}(ab|xy) = \lim \limits_{m\to \infty} P_{A(B)}(ab|xy)$. 
Then all superquantum correlations are obtained by the formula in Eq. (\ref{Eq:CHSH}). 
Fig. \ref{FIG:CHSHPRBox} shows the monotonic increasing behavior of the CHSH value $\mathcal{B}$ up to 
the maximum value $4$ with increasing the power $m$. 


Note that no causal order condition in Eq. (\ref{eq:NCO}) cannot give any restriction 
on the functional form of $H(|c|^2)$ for $\alpha_2=0$. This implies that any two-dimensional entangled state has a 
possibility to satisfy the requirement of no causal order condition with the modified Born rule. 
In fact, every entangled state in a two-dimensional Hilbert space can be easily shown to generate a superquantum correlation, 
satisfying no causal order condition, under the modified Born rule when a special observable 
(e.g., corresponding to spin observable along a certain direction) is chosen. 
This result does not contradict with previous conclusion of non-existence of post-quantum theory because 
a consistent physical theory must have a freedom on choosing a measurement observable. 
This means that superquantum correlations generated by our formalism are not physical but only hypothetical ones. 
%Every entangled state, which is represented in two dimensional Hilbert space, 
%can be easily shown to satisfy no causal order condition for special set of measurement axes depending on 
%the power $m$ and to give hypothetical superquantum correlations. 
That is, a two-dimensional entangled state with a proper set of entangled state and measurement observable 
can provide a hypothetical model for nonisotropic nonlocal box with superquantum correlations under the modified Born rule.  

%\begin{figure}
%\centering
%\label{Fig:MAxis}
%\includegraphics[scale=0.3]{Inputs.eps}\\%[scale=1.3]
%\caption{(Color online) The schematic diagram for measurement axis of Alice and Bob.}
%\label{fig:1}
%\end{figure}


%Here we will use the following generalized Born rule for the state $|\phi\rangle = c |0|0\rangle + d |1|0\rangle$ 
%outcome probabilites
%\begin{eqnarray}
%H_M(|c|^2)= \frac{|c|^m}{|c|^m+|d|^m} \mbox{ and }
%H_M(|d|^2)= \frac{|d|^m}{|c|^m+|d|^m}.
%\end{eqnarray}
%We will call this generalized Born rule 'modified Born rule' to distinguish from the former general function of $H(|C|^2$.





In conclusion, we have proven that Born rule is the only possible probability assignment to quantum measurement 
for a linear quantum theory under relativistic causality.  
Our proof strongly suggest that there is no post-quantum theory in nature, because 
the nonlinear extensions of quantum mechanics, proposed by Weinberg \cite{Weinberg}, 
can be used to produce unnatural communications including superluminal signaling \cite{Weinberg, Gisin1, Polchinski}. 
 Also, contrary to previous understanding, relativistic causality requires no causal order condition 
 for spacelike separate events, which is shown to be stronger than no-signal condition in this Letter. 
 No causal order condition made the derivation of Born rule general and simple in a two-qubit Hilbert space.  
 Even though post-quantum theories do not exist, we can find that 
 all superquantum correlations can be achieved by Bell states with the modified Born rule. 
 This means that superquantum correlations are hypothetical, that is, mathematical tools.   
% Bell states can satisfy no causal order condition even with the modified Born rule.  
As a result we have provided a way to understand both questions of ``Why is not quantum theory more nonlocal?'' and 
``Why have not the proposed superquantum correlations (e.g., PR correlation) 
been observed in nature?''. 

This work was supported by National Research Foundation of Korea Grant funded by the Korean
Government (2012-0003786).



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