Non-Inertial Response of Correlations:
From Scalar Bell Observables to an Extended Correlation Tensor

In a standard Bell experiment [1, 2], the correlation between two binary outcomes is represented by a number $E(\bm{a},\bm{b})$ for selected analyzer directions $\bm{a}$ and $\bm{b}$. Several such numbers form the CHSH combination. A single Bell observable, however, is not a complete description of the two-particle correlation structure; it is only one scalar projection of that structure.

This distinction becomes important when the local measurement systems rotate, accelerate, or undergo controlled non-inertial modulation. A variation of the recorded correlation may result from a change of the local measurement frames, from propagation and mode transformations, or from a response of the state and its intrinsic correlation structure. A single value of the CHSH parameter does not separate these contributions.

Our aim is to formulate Bell observables as projections of an extended correlation tensor and to define an experimentally reconstructible response of that tensor to non-inertial modulation. The ordinary two-qubit correlation matrix is well known in quantum-information theory [5]. The additional object introduced here is its non-inertial susceptibility, including the amplitude and phase of the response to a periodic perturbation.

In the non-inertial description, the complete state of a pair may include a common phase variable $\lambda$ and a random metric structure. Let $A(a,\lambda)$ and $B(b,\lambda)$ denote local projection amplitudes. The physical axis of linear photon polarization is projective and has period $\pi$, but the phase variable of the correlation state completes the full cycle $0\leq\lambda<2\pi$. We therefore define

$$M_{AB}^{(\gamma)}(a,b)=\frac{1}{\pi}\int_{0}^{2\pi}A(a,\lambda)B(b,\lambda)\,d\lambda.$$

(1)

For the projection amplitudes

$$A(a,\lambda)=\cos(a-\lambda),\qquad B(b,\lambda)=\cos(b-\lambda),$$

(2)

the identity $\cos x\cos y=\tfrac{1}{2}[\cos(x-y)+\cos(x+y)]$ gives

$$M_{AB}^{(\gamma)}(a,b)=+\cos(a-b).$$

(3)

Thus coincident phase directions are positively correlated, $M_{AB}^{(\gamma)}(a,a)=+1$. The integration interval belongs to the phase of the complete correlation state and must not be confused with the projective periodicity of a linear-polarizer axis. On the equator of the Poincare sphere, $a=2\theta_{A}$ and $b=2\theta_{B}$, and hence

$$E_{\gamma}(\theta_{A},\theta_{B})=+\cos 2(\theta_{A}-\theta_{B}).$$

(4)

This positive photon sector is the baseline used below. It is distinct from the negative correlation of a fermionic spin singlet.

The complete non-inertial state is defined in an extended phase space $\Gamma_{\rm ni}=(\lambda,p_{\lambda})$. For two indistinguishable particles, an exchange is represented by a closed path $C_{\rm ex}$ in the reduced two-particle configuration space. Such a space is multiply connected and its path integral admits the two one-dimensional exchange factors $+1$ and $-1$ [4]. We write the corresponding holonomy as

$$\mathcal{H}_{\varepsilon}(C_{\rm ex})=\exp\!\left(i\oint_{C_{\rm ex}}\mathcal{A}\right)=\varepsilon,\qquad\varepsilon=\pm 1,$$

(5)

where $\mathcal{A}$ is the effective connection of the transported extended state. The symmetric photon sector carries $\varepsilon_{\gamma}=+1$, whereas the antisymmetric fermionic sector carries $\varepsilon_{f}=-1$. This statement does not attempt to rederive the relativistic spin–statistics theorem; it identifies the exchange character retained by the complete non-inertial phase–momentum sector.

Transport of the second local projection around the exchange loop gives

$$A_{\varepsilon}(a,\lambda)=\cos(a-\lambda),\qquad B_{\varepsilon}(b,\lambda)=\varepsilon\cos(b-\lambda).$$

(6)

Consequently,

$$M_{AB}^{(\varepsilon)}(a,b)=\frac{1}{\pi}\int_{0}^{2\pi}A_{\varepsilon}(a,\lambda)B_{\varepsilon}(b,\lambda)\,d\lambda=\varepsilon\cos(a-b).$$

(7)

Thus the positive photon and negative fermionic cosine laws follow from one phase average and the two exchange holonomies:

$$M_{AB}^{(\gamma)}=+\cos(a-b),\qquad M_{AB}^{(f)}=-\cos(a-b).$$

(8)

The standard Bell states used later are representations of these two tensor sectors, rather than the origin of the signs in Eq. (8).

Let the local outcomes be $A,B=\pm 1$. For fixed settings,

$$E(\bm{a},\bm{b})=\sum_{A,B=\pm 1}AB\,P(A,B\mid\bm{a},\bm{b}).$$

(9)

For two-level systems, the local observables are

$$\hat{A}(\bm{a})=a_{i}\hat{\sigma}_{i},\qquad\hat{B}(\bm{b})=b_{j}\hat{\sigma}_{j},$$

(10)

and hence

$$E(\bm{a},\bm{b})=\operatorname{Tr}\!\left[\hat{\rho}\,\hat{A}(\bm{a})\otimes\hat{B}(\bm{b})\right].$$

(11)

The CHSH parameter is

$$S=E(\bm{a},\bm{b})+E(\bm{a},\bm{b}^{\prime})+E(\bm{a}^{\prime},\bm{b})-E(\bm{a}^{\prime},\bm{b}^{\prime}).$$

(12)

It combines four scalar projections but does not reconstruct the complete tensor structure.

For unbiased local marginals, the signed non-inertial correlation $E_{\varepsilon}(a,b)=\varepsilon\cos(a-b)$ determines the binary joint probabilities

	$\displaystyle P_{++}=P_{--}$	$\displaystyle=\frac{1}{4}[1+\varepsilon\cos(a-b)],$
	$\displaystyle P_{+-}=P_{-+}$	$\displaystyle=\frac{1}{4}[1-\varepsilon\cos(a-b)].$		(13)

They are nonnegative, normalized, and reproduce the Bell observable,

$$\sum_{r,s=\pm 1}rs\,P_{rs}(a,b)=\varepsilon\cos(a-b).$$

(14)

Equation (14) supplies the explicit bridge from the continuous projection amplitudes to binary coincidence outcomes.

The same construction directly recovers the Tsirelson bound [3]. Introduce the unit phase vector $\bm{n}(a)=(\cos a,\sin a)$, so that $E_{\varepsilon}(a,b)=\varepsilon\bm{n}(a)\cdot\bm{n}(b)$. The CHSH combination becomes

$\displaystyle S_{\varepsilon}=\varepsilon\{$

$\displaystyle\bm{n}(a)\cdot[\bm{n}(b)+\bm{n}(b^{\prime})]+\bm{n}(a^{\prime})\cdot[\bm{n}(b)-\bm{n}(b^{\prime})]\}.$

(15)

Maximization over $a$ and $a^{\prime}$ gives

$$|S_{\varepsilon}|\leq|\bm{n}(b)+\bm{n}(b^{\prime})|+|\bm{n}(b)-\bm{n}(b^{\prime})|.$$

(16)

If the angle between $\bm{n}(b)$ and $\bm{n}(b^{\prime})$ is $\delta$, then

$$|S_{\varepsilon}|\leq 2\left(\left|\cos\frac{\delta}{2}\right|+\left|\sin\frac{\delta}{2}\right|\right)\leq 2\sqrt{2}.$$

(17)

For $a=0$, $a^{\prime}=\pi/2$, $b=\pi/4$, and $b^{\prime}=-\pi/4$, the two sectors give

$$S_{\gamma}=+2\sqrt{2},\quad S_{f}=-2\sqrt{2},\quad|S_{\gamma}|=|S_{f}|=2\sqrt{2}.$$

(18)

Thus the exchange sign reverses the oriented CHSH combination without changing its maximal absolute magnitude. For physical photon polarizers the phase angles are doubled, giving the familiar settings $0^{\circ}$, $45^{\circ}$, $22.5^{\circ}$, and $-22.5^{\circ}$.

Define the two-qubit correlation matrix

$$C_{ij}=\operatorname{Tr}\!\left[\hat{\rho}(\hat{\sigma}_{i}\otimes\hat{\sigma}_{j})\right]=\left\langle\hat{\sigma}_{i}\otimes\hat{\sigma}_{j}\right\rangle.$$

(19)

Then

$$E(\bm{a},\bm{b})=a_{i}C_{ij}b_{j}=\bm{a}^{\mathsf{T}}C\bm{b}.$$

(20)

The variation of the measured scalar separates three contributions:

$$\delta E=(\delta\bm{a})^{\mathsf{T}}C\bm{b}+\bm{a}^{\mathsf{T}}(\delta C)\bm{b}+\bm{a}^{\mathsf{T}}C(\delta\bm{b}).$$

(21)

The first and third terms describe changes of the local directions; the middle term describes a change of the correlation matrix itself. In orthonormal local bases, its elements are obtained from

$$C_{ij}=E(\bm{e}_{i},\bm{e}_{j}).$$

(22)

Let $\xi^{\alpha}$ denote controlled motion parameters, including orientation, angular velocity, acceleration, jerk, modulation phase, or stochastic variables. We introduce

$$\mathcal{C}_{ij}(\bm{\xi})=\operatorname{Tr}\!\left[\hat{\rho}(\bm{\xi})\,\hat{\Sigma}_{i}^{(A)}(\bm{\xi})\otimes\hat{\Sigma}_{j}^{(B)}(\bm{\xi})\right],$$

(26)

and the corresponding scalar projection

$$E_{\rm ni}(\bm{a},\bm{b};\bm{\xi})=a_{i}\mathcal{C}_{ij}(\bm{\xi})b_{j}.$$

(27)

The complete tensor is

$$\mathcal{T}(\bm{\xi})=\begin{pmatrix}1&\bm{v}^{\mathsf{T}}(\bm{\xi})\\ \bm{u}(\bm{\xi})&\mathcal{C}(\bm{\xi})\end{pmatrix}.$$

(28)

To separate motion of the measurement bases from an intrinsic response, write

$$\mathcal{C}_{\rm lab}(\bm{\xi})=R_{A}(\bm{\xi})\mathcal{C}_{\rm int}(\bm{\xi})R_{B}^{\mathsf{T}}(\bm{\xi}).$$

(29)

After compensation of the calibrated frame rotations, the residual tensor is

$$\Delta\mathcal{C}_{\mathrm{res}}(\bm{\xi})=R_{A}^{-1}\mathcal{C}_{\rm lab}R_{B}^{-\mathsf{T}}-C^{(0)}.$$

(30)

A nonzero residual is not by itself evidence for new physics: decoherence, channel imperfections, drift, and standard relativistic transformations must first be included in the baseline prediction.

For photons, use normalized Stokes operators,

$$C_{\mu\nu}^{(\gamma)}=\langle\hat{S}_{\mu}^{(A)}\otimes\hat{S}_{\nu}^{(B)}\rangle,\qquad\mu,\nu=1,2,3.$$

(31)

With $\hat{S}_{0}=I$, the complete two-photon Stokes tensor has indices $\mu,\nu=0,1,2,3$. The positive linear-polarization correlation is realized by

$$\lvert\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(\lvert H\rangle_{A}\lvert H\rangle_{B}+\lvert V\rangle_{A}\lvert V\rangle_{B}).$$

(32)

In the Stokes-axis order $(H/V,D/A,R/L)$,

$$C^{(\gamma,0)}=\operatorname{diag}(1,1,-1),\qquad\mathcal{T}^{(\gamma,0)}=\operatorname{diag}(1,1,1,-1).$$

(33)

The vanishing local marginals express the absence of polarization of either photon separately. The two positive entries describe positive correlations of linear polarization; the circular component has the sign fixed by the chosen Bell state and Stokes-axis convention. A linear analyzer at angle $\theta$ corresponds to

$$\bm{n}(\theta)=(\cos 2\theta,\sin 2\theta,0),$$

(34)

and therefore

$$E_{0}^{(\gamma)}(\theta_{A},\theta_{B})=\bm{n}^{\mathsf{T}}(\theta_{A})C^{(\gamma,0)}\bm{n}(\theta_{B})=+\cos 2(\theta_{A}-\theta_{B}).$$

(35)

The double angle expresses the equivalence of the physical axes $\theta$ and $\theta+\pi$; it does not reduce the full integration interval of the phase variable $\lambda$. The complete $3\times 3$ block requires linear, diagonal, and circular polarization projections. A fit to a single cosine law does not replace tensor reconstruction.

For

$$\lvert\Psi^{-}\rangle=\frac{1}{\sqrt{2}}(\lvert\uparrow\downarrow\rangle-\lvert\downarrow\uparrow\rangle),$$

(36)

one has

$$C_{ij}^{(f,0)}=-\delta_{ij},\qquad\mathcal{T}^{(f,0)}=\operatorname{diag}(1,-1,-1,-1).$$

(37)

For coplanar spin analyzers,

$$E_{0}^{(f)}(\alpha_{A},\alpha_{B})=-\bm{a}\cdot\bm{b}=-\cos(\alpha_{A}-\alpha_{B}).$$

(38)

The single angle for spin, compared with the double angle for linear photon polarization, reflects the difference between an oriented spin direction and an unoriented polarization axis.

In a non-inertial setting,

$$\mathcal{T}^{(f)}(\bm{\xi})=\begin{pmatrix}1&\bm{v}^{\mathsf{T}}(\bm{\xi})\\ \bm{u}(\bm{\xi})&-I+\Delta C^{(f)}(\bm{\xi})\end{pmatrix}.$$

(39)

For relativistic massive particles, the spin transformation is momentum dependent. At fixed momenta,

$$C_{\rm lab}^{(f)}=R_{W}^{(A)}(\bm{p}_{A})C_{\rm int}^{(f)}R_{W}^{(B)\mathsf{T}}(\bm{p}_{B}),$$

(40)

where $R_{W}$ denotes the Wigner rotation. If momenta are unresolved, the observed tensor is averaged over the joint distribution,

$$\overline{C}_{ij}^{(f)}=\int d^{3}p_{A}\,d^{3}p_{B}\,f(\bm{p}_{A},\bm{p}_{B})C_{ij}^{(f)}(\bm{p}_{A},\bm{p}_{B}).$$

(41)

Wigner rotations and spin–momentum averaging belong to the standard relativistic background and must be compensated before a residual non-inertial response is inferred [6, 7, 8].

Near the unperturbed state,

$$\mathcal{C}_{ij}(\bm{\xi})=C_{ij}^{(0)}+\chi_{ij\alpha}\xi^{\alpha}+\frac{1}{2}\chi^{(2)}_{ij\alpha\beta}\xi^{\alpha}\xi^{\beta}+\cdots,$$

(42)

with

$$\chi_{ij\alpha}=\left.\frac{\partial\mathcal{C}_{ij}}{\partial\xi^{\alpha}}\right|_{\bm{\xi}=0}.$$

(43)

For the full tensor,

$$X_{\mu\nu\alpha}=\left.\frac{\partial\mathcal{T}_{\mu\nu}}{\partial\xi^{\alpha}}\right|_{\bm{\xi}=0},$$

(44)

where $X_{i0\alpha}$ and $X_{0j\alpha}$ describe induced local polarization and $X_{ij\alpha}=\chi_{ij\alpha}$ describes the correlation response.

For harmonic modulation, $\xi^{\alpha}(t)=\xi_{0}^{\alpha}\cos\Omega t$, a delayed response is described by a complex susceptibility,

$$\delta\mathcal{C}_{ij}(\Omega)=\chi_{ij\alpha}(\Omega)\xi^{\alpha}(\Omega).$$

(45)

Its real and imaginary parts represent in-phase and quadrature responses. The observable scalar susceptibility is

$$\chi_{E}^{(\alpha)}(\bm{a},\bm{b};\Omega)=a_{i}\chi_{ij\alpha}(\Omega)b_{j}.$$

(46)

The fermionic and photon baselines must be expanded separately. For the isotropic fermionic singlet, a minimal residual correction is

$$\Delta C_{ij}^{(f),\mathrm{res}}=-\eta_{f}\delta_{ij}+q_{f}\left(m_{i}m_{j}-l_{i}l_{j}\right)+\kappa_{f}\epsilon_{ijk}n_{k},$$

(47)

where $\eta_{f}$ changes the overall spin anticorrelation, $q_{f}$ describes a symmetric anisotropy, and $\kappa_{f}$ is an antisymmetric rotational response.

For photons propagating parallel to the rotation axis, $\bm{n}\parallel\bm{k}$, choose transverse Stokes basis vectors $\bm{e}_{1}=\bm{m}$ and $\bm{e}_{2}=\bm{l}$. The unperturbed linear block is $+I_{2}$, not $-I_{2}$. Its minimal response is therefore

$$\mathcal{C}_{\perp}^{(\gamma)}(\Omega)=\begin{pmatrix}1-\eta_{\gamma}+q_{\gamma}&-\kappa_{\gamma}\\ \kappa_{\gamma}&1-\eta_{\gamma}-q_{\gamma}\end{pmatrix}.$$

(48)

Here $\eta_{\gamma}$ describes a reduction of the positive photon correlation, $q_{\gamma}$ is the symmetric linear-polarization anisotropy, and $\kappa_{\gamma}$ is the antisymmetric rotational component. A nonzero $\kappa_{\gamma}$ can imitate a small relative analyzer rotation; changes of $\eta_{\gamma}$, $q_{\gamma}$, or of the singular values after kinematic compensation provide stronger evidence for a modified correlation structure. At low frequency one may write

	$\displaystyle\eta_{\gamma}(\Omega)$	$\displaystyle=\eta_{\gamma 0}+\eta_{\gamma 2}\Omega^{2}+\cdots,$
	$\displaystyle q_{\gamma}(\Omega)$	$\displaystyle=q_{\gamma 0}+q_{\gamma 2}\Omega^{2}+\cdots,$
	$\displaystyle\kappa_{\gamma}(\Omega)$	$\displaystyle=i\kappa_{\gamma 1}\Omega+i\kappa_{\gamma 3}\Omega^{3}+\cdots.$		(49)

This is a phenomenological tensor and frequency structure, not a numerical prediction. A microscopic theory must determine which coefficients are nonzero and their magnitude.

The distinction $E_{\gamma}=+\cos\theta$ versus $E_{f}=-\cos\theta$ concerns the sign and geometry of the photon and fermion correlation sectors. It does not alter Bell’s theorem and does not assume that non-inertial motion necessarily changes quantum nonlocality. Its purpose is to distinguish a change of scalar projections from a change of the reconstructed correlation object. A variation of $E$ or $S$ alone may result entirely from a rotation of local bases. Singular values of the compensated correlation matrix and an independently reconstructed density matrix provide stronger tests.

Three outcomes are possible: a purely kinematic response; a residual fully accounted for by standard optical or relativistic transformations; or an additional reproducible residual. Only the third requires an extended dynamical interpretation. A candidate signal must be phase locked to the motion, reproducible in independent runs, scale consistently with $\phi_{0}$ and $\Omega$, transform correctly under rotation reversal, and remain after all calibrated optical, mechanical, and relativistic backgrounds are included.

The photon correlation obtained by averaging two projection amplitudes over the complete phase interval $0\leq\lambda<2\pi$ is $M_{AB}^{(\gamma)}(a,b)=+\cos(a-b)$. For physical linear-polarizer axes this becomes $E_{\gamma}=+\cos 2(\theta_{A}-\theta_{B})$. The fermionic singlet remains $E_{f}=-\cos(\alpha_{A}-\alpha_{B})$ because the complete phase–momentum exchange loop carries the negative holonomy. The full phase interval and the projective periodicity of linear polarization are therefore distinct geometric elements and must not be conflated.

The associated unbiased binary probabilities reproduce these signed cosine laws and lead to opposite signed optimal CHSH combinations with the same absolute Tsirelson value. The non-inertial construction therefore remains inside the quantum correlation bound while retaining an operational distinction between the two exchange sectors.

A Bell observable is a scalar projection of a complete two-particle correlation tensor. For the photon state $\lvert\Phi^{+}\rangle$ the baseline Stokes block is $C^{(\gamma,0)}=\operatorname{diag}(1,1,-1)$, whereas for the fermionic singlet $C^{(f,0)}=-I$. For a modulated system we introduced the extended tensor $\mathcal{T}_{\mu\nu}(\bm{\xi})$ and its susceptibility $X_{\mu\nu\alpha}(\Omega)$. Phase-synchronous Stokes tomography, comparison with equivalent optical modulation, and kinematic compensation define a null test for an additional non-inertial response. The construction replaces isolated scalar variations by a reconstructible dynamical tensor and provides an experimentally testable framework for Bell correlations in non-inertial reference frames.