LAQC: Quantum Correlations via Local Optimization

Updated 7 July 2026

Local-Available Quantum Correlations (LAQC) is a symmetric quantum-correlation quantifier defined via a two-stage local-basis optimization that minimizes relative-entropy to classical states.
Exact analytic expressions for LAQC have been derived for Bell-diagonal and various two-qubit X states, revealing distinct optimization branches based on state symmetry.
LAQC shows asymptotic decay under decoherence and exhibits unique redistribution features in network protocols, distinguishing its behavior from entanglement and discord.

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Local-Available Quantum Correlations (LAQC) are a symmetric quantum-correlation quantifier defined through a two-stage local-basis optimization. In the formulation attributed to Mundarain and de Guevara, one first chooses the local computational basis that minimizes the relative-entropy distance to the nearest classical state, and then evaluates the maximal mutual information obtainable in the basis complementary to that optimal basis. In later work on residual quantum correlations (RQC), LAQC is described as a maximal RQC obtained after fixing the local basis through the classical-correlation optimization [2204.07552, 2603.13648].

1. Formal definition and operational construction

For a bipartite state (\rho_{AB}), the measurement-induced classical state used in the LAQC construction is
[
X_\rho=\sum_{\mu\nu} R_{\mu\nu} |\mu,\nu\rangle\langle \mu,\nu|,\qquad R_{\mu\nu}=\langle \mu,\nu|\rho_{AB}|\mu,\nu\rangle .
]
The optimal local basis is the one minimizing
[
S(\rho_{AB}|X_\rho)=\min_{R_c}S(\rho_{AB}|X_\rho).
]
Once that basis is fixed, the complementary basis is generated by local (\pi/2)-rotations. In the notation used for non-symmetric (X) states,
[
|u^{{(n)}(m)\rangle=\frac{1}{\sqrt} 2}\left(|m\rangle_{\mathrm{opt}}+(-1)^m |1-m\rangle_{\mathrm{opt}}\right),
]
equivalently through
[
\theta_n \to \frac{\pi}{2},\qquad \phi_n \to \phi_n+\frac{\pi}{2}.
]
The LAQC quantifier is then the maximal mutual information in that complementary basis,
[
C(\rho_{AB})=\max_{\phi_1,\phi_2} I(\phi_1,\phi_2),
]
while other papers denote the same quantity by (L(\rho_{AB})) or (\mathcal L(\rho_{AB})) [2204.07552, 2107.00158, 2105.15166].

This construction separates two optimization problems. The first identifies the basis in which the state is as classical as possible. The second asks how much correlation remains available when both parties measure in a basis complementary to that classicalizing basis. The resulting quantity is explicitly basis-relative, but only after the optimal basis has been fixed by the relative-entropy criterion.

A closely related later formulation appears in the RQC framework, where one first defines a symmetric classical correlation (C_S(\rho_{AB})=\max I(p_{ij})) over local projective measurements, and then computes the residual mutual information in a mutually unbiased basis. Within that framework, LAQC is obtained by reversing the first optimization and using the basis that minimizes classical mutual information; for the (X)-state classes treated there, (L(\rho_{AB})\ge Q_S(\rho_{AB})) [2603.13648].

2. Exact analytical results for Bell-diagonal and (X)-state families

The earliest closed-form LAQC formulas in this corpus concern Bell-diagonal two-qubit states,
[
\rho_{BD}=\frac14\left(\mathbb{1}\otimes\mathbb{1}+\sum_{i=1}³ c_i\,\sigma_i\otimes\sigma_i\right).
]
For these states, the corrected LAQC formula is
[
\mathcal{L}(\rho_{BD})= \frac{1+C_M}{2}\log_2(1+C_M)+ \frac{1-C_M}{2}\log_2(1-C_M),
\qquad
C_M=\max{|c_1|,|c_2|,|c_3|},
]
and the corrected classical-correlation expression is
[
C(\rho_{BD})= \frac{1+c_m}{2}\log_2(1+c_m)+ \frac{1-c_m}{2}\log_2(1-c_m),
\qquad
c_m=\min{|c_1|,|c_2|,|c_3|}.
]
The 2021 comment established that a prior Bell-diagonal analysis had omitted optimization cases and had incorrectly treated generic Bell-diagonal states as invariant under the optimal-basis transformation; the comment emphasizes that this invariance holds only for Werner and Werner-like states [2105.15166].

For (X) states with local Bloch vectors of equal magnitude, exact closed forms were derived for the symmetric class (x_3=y_3) and the anti-symmetric class (x_3=-y_3). For symmetric (X) states,
[
C(\rho_{Xs})=\min(g_1,g_2,g_+), \qquad L(\rho_{Xs})=\max(g_1,g_2,g_+),
]
with
[
g_1 = 1+T_1\log_2(1+T_1)+(1-T_1)\log_2(1-T_1),
]
[
g_2 = 1+T_2\log_2(1+T_2)+(1-T_2)\log_2(1-T_2),
]
and
[
g_+ = \frac{1+T_3+2x_3}{2}\log_2!\left(\frac{1+T_3+2x_3}{(1+x_3)^2}\right) + \frac{1+T_3-2x_3}{2}\log_2!\left(\frac{1+T_3-2x_3}{(1-x_3)^2}\right) +\frac{1-T_3}{2}\log_2(1-T_3).
]
For anti-symmetric (X) states,
[
C(\rho_{Xas})=\min(g_1,g_2,g_-), \qquad L(\rho_{Xas})=\max(g_1,g_2,g_-),
]
with
[
g_- = \frac{1-T_3+2x_3}{2}\log_2!\left(\frac{1-T_3+2x_3}{(1+x_3)^2}\right) + \frac{1-T_3-2x_3}{2}\log_2!\left(\frac{1-T_3-2x_3}{(1-x_3)^2}\right) +\frac{1+T_3}{2}\log_2(1+T_3).
]
These formulas reduce the optimization to the three branches associated with ((\theta,\phi)=(0,\text{arbitrary})), ((\pi/2,0)), and ((\pi/2,\pi/2)) [2107.00158].

A later synthesis for general two-qubit (X) states gives the exact result as
[
L(\rho_X)=\max\left{\mathcal{F}1(T_1),\mathcal{F}_2(T_2),\mathcal{F}_3(x_3,y_3,T_3)\right},
]
or, in the RQC notation,
[
L(\rho_X)=\max{g_i(T{i0},T_{0i},T_{ii})}_{i=1,2,3}.
]
This expresses the full (X)-state problem as a branchwise maximization over three analytic candidates [2507.23142, 2603.13648].

Family	Condition	Exact LAQC form
Bell-diagonal	(\rho_{BD}=\frac14(\mathbb{1}\otimes\mathbb{1}+\sum_i c_i\sigma_i\otimes\sigma_i))	(\mathcal L(\rho_{BD})) with (C_M=\max{
Symmetric (X)	(x_3=y_3)	(L(\rho_{Xs})=\max(g_1,g_2,g_+))
Anti-symmetric (X)	(x_3=-y_3)	(L(\rho_{Xas})=\max(g_1,g_2,g_-))
Non-symmetric (X)	(	x_3
General (X)	arbitrary two-qubit (X) state	(L(\rho_X)=\max{\mathcal F_1,\mathcal F_2,\mathcal F_3})

These results establish that exact LAQC expressions are available for all two-qubit (X) states, although the computational structure differs sharply across subclasses.

3. Non-symmetric (X) states and completion of the two-qubit (X)-state program

The general two-qubit (X)-state density matrix considered in the non-symmetric analysis is
[
\rho_X= \begin{pmatrix} a & 0 & 0 & w\ 0 & b & z & 0\ 0 & z & c & 0\ w & 0 & 0 & d \end{pmatrix},
\qquad a,b,c,d\ge 0,\quad d=1-(a+b+c),
]
with positivity constraints
[
z<\sqrt{bc},\qquad w<\sqrt{ad}.
]
In Fano-Bloch form,
[
\rho_X=\frac14\left(\mathbb{1}\otimes \mathbb{1}+x_3\,\sigma_3\otimes \mathbb{1}+ y_3\,\mathbb{1}\otimes\sigma_3+\sum_{n=1}³ T_n\,\sigma_n\otimes\sigma_n\right),
]
where
[
x_3=a+b-c-d,\qquad y_3=a-b+c-d,
]
[
T_1=2(z+w),\qquad T_2=2(z-w),\qquad T_3=a-b-c+d.
]
The key structural criterion is the norm of the local Bloch vectors: symmetric and anti-symmetric (X) states satisfy (|x_3|=|y_3|), whereas non-symmetric (X) states satisfy (|x_3|\neq |y_3|). For the non-symmetric family treated in detail, this is equivalent to
[
a=d,\qquad b\neq c .
]
Such states are not invariant under subsystem exchange (A\leftrightarrow B), so the full angle set (\theta_1,\theta_2,\phi_1,\phi_2) must be retained in the optimization [2204.07552].

The central result is the closed formula
[
C(\rho_X^{ns}) = 1+\frac{1+T_2}{2}\log_2(1+T_2) +\frac{1-T_2}{2}\log_2(1-T_2).
]
The minimizing computational bases satisfy
[
\theta_i=(2k+1)\frac{\pi}{2},\qquad \theta_j=k\pi,
]
or the swapped choice
[
\theta_i=k\pi,\qquad \theta_j=(2k+1)\frac{\pi}{2},
\qquad k=0,1,
]
leading, up to irrelevant multiples of (\pi), to
[
\theta_1=\frac{\pi}{2},\ \theta_2=0
\qquad\text{or}\qquad
\theta_1=0,\ \theta_2=\frac{\pi}{2}.
]
The maximizing complementary-basis phases are
[
\vartheta_1=\pm \frac{\pi}{2},\qquad \vartheta_2=\pm \frac{\pi}{2}.
]
Consequently, the final LAQC depends only on (T_2), not on (x_3), (y_3), or (T_1,T_3). The special case (T_2=0) gives (C(\rho_X^{ns})=0), and the formula is well-defined over the full physical range (|T_2|\le 1) [2204.07552].

This result completes the analytical classification initiated for the symmetric and anti-symmetric subclasses. Together, the symmetric, anti-symmetric, and non-symmetric formulas provide exact analytical LAQC expressions for all two-qubit (X) states [2204.07552, 2107.00158].

4. Decoherence, dynamical behavior, and the monotonicity question

LAQC has been studied under several Markovian and non-Markovian channels. For Bell-diagonal and Werner states under Markovian depolarizing and phase-damping channels, the 2018 analysis found that LAQC decreases smoothly and vanishes only asymptotically, in qualitative contrast with concurrence, which can exhibit entanglement sudden death at finite noise strength. For Werner states, LAQC and discord were reported to vanish only asymptotically, with LAQC generally below discord [1803.02426].

For Werner states under amplitude damping with equal damping parameter on both qubits, the evolved state remains a symmetric (X) state with
[
x_3=y_3=p,\qquad T_1=T_2=-(1-p)z,\qquad T_3=p^2-(1-p)² z.
]
In that case, LAQC again exhibits no sudden death and decays asymptotically, whereas concurrence can show sudden death. The same paper states that this agrees with the previously observed behavior under depolarization and phase damping [2107.00158].

A distinct amplitude-damping scenario is the local action on one subsystem of a Werner state,
[
\rho'=\sum_i (I\otimes E_i)\rho(I\otimes E_i^\dagger),
]
with
[
E_0^{{(\mathrm{AD})}=} \begin{pmatrix} 1 & 0\ 0 & \sqrt{1-p} \end{pmatrix}, \qquad
E_1^{{(\mathrm{AD})}=} \begin{pmatrix} 0 & \sqrt{p}\ 0 & 0 \end{pmatrix}.
]
For the Werner input
[
\rho_W=z|\Psi^{-\rangle\langle\Psi^{-|+\frac{1-z}{4}I,}}
\qquad
|\Psi^{-\rangle=\frac{1}{\sqrt2}(|01\rangle-|10\rangle),}
]
the transformed Bloch parameters become
[
y_3=p,\qquad T_1=T_2=-\sqrt{1-p}\,z,\qquad T_3=-(1-p)z,
]
and
[
C(\rho_W^{{(\mathrm{AD})})} = 1+\frac{1-\sqrt{1-p}\,z}{2}\log_2(1-\sqrt{1-p}\,z) +\frac{1+\sqrt{1-p}\,z}{2}\log_2(1+\sqrt{1-p}\,z).
]
The non-symmetric (X)-state study emphasizes a sharper conclusion: local amplitude damping can create quantum discord in some cases, but it cannot create LAQC in the examples analyzed. That observation is presented as evidence suggesting, but not proving, monotonicity under LOCC [2204.07552].

The non-Markovian extension uses a common-bath phase-flip channel with
[
K_0=\sqrt{\frac{1+A(t)}{2}}\; \mathbb I,\qquad
K_1=\sqrt{\frac{1-A(t)}{2}}\;\sigma_z,
]
for which (T_{11}) and (T_{22}) acquire the factor (A^2(t)) while the remaining Bloch parameters remain unchanged. Under Random Telegraph Noise,
[
A(t)=e^{-\gamma t}\left[\cos(\omega \gamma t)+\frac{1}{\omega}\sin(\omega \gamma t)\right],
\qquad
\omega=\sqrt{\left(\frac{2a}{\gamma}\right)^2-1},
]
finite-time zeros of (A(t)) produce sudden death and revival of RQC and LAQC. Under Modified Ornstein-Uhlenbeck noise, (A(t)\to 0) only asymptotically, so the decay is asymptotic and lacks revival. This gives a dynamical distinction absent from the Markovian studies [2603.13648].

5. Relation to discord, entanglement, measurement-induced correlations, and coherence

LAQC emerged in a conceptual environment shaped by several adjacent critiques of discord-centered language. One influential line of argument holds that nonzero quantum discord is necessary but not sufficient for correlations above the classically achievable limit. In that view, discord characterizes local quantumness, whereas the rank (L) of the correlation matrix captures the actual correlation content. The same work shows that local operations cannot increase (L), that some nonzero-discord states can be created from classical states by a single local operation, and that the set of such locally generable states has Lebesgue measure zero [1202.1959].

A second adjacent distinction is between local and nonlocal quantumness. In the generalized Werner-state analysis of local superposition, discord is argued to probe both local quantumness and nonlocal quantumness; nonzero discord in separable states can then arise from local superposition rather than entanglement. This suggests a conceptual neighborhood for LAQC, but not an identity of formalisms [1502.00857].

A third neighboring framework studies quantum correlations induced by local von Neumann measurement. There the semiquantum correlations
[
Q_1(\rho)= \min_{\Phi_1}|\rho-\Phi_1(\rho)|^2,\qquad
Q_2(\rho)= \min_{\Phi_2}|\rho-\Phi_2(\rho)|^2,
]
the total quantum correlation
[
Q_{12}(\rho)= \min_{\Phi_{12}}|\rho-\Phi_{12}(\rho)|^2,
]
and the joint term
[
\delta(\rho)=Q_1(\rho)+Q_2(\rho)-Q_{12}(\rho)
]
are defined via local measurement maps. For pure states with Schmidt coefficients (\lambda_i),
[
Q_{12}=Q_1=Q_2=\delta=1-\sum_i\lambda_i^{4=\frac{C^2}{2}.}
]
This is a measurement-induced framework closely related to local accessibility, but it is not the LAQC construction [1306.0674].

A more direct overlap appears in the coherence-theoretic approach based on net global coherence,
[
\mathfrak{C}^{\rm net}(\hat{\varrho}{AB}) = \mathfrak{C}(\hat{\varrho}{AB}) - \mathfrak{C}(\hat{\varrho}A) - \mathfrak{C}(\hat{\varrho}_B),
]
and, for relative entropy of coherence,
[
\mathfrak{C}_r^{\rm net}(\hat{\varrho}{AB}) = I(\hat{\varrho}{AB}) - I(\Delta{AB}[\hat{\varrho}_{AB}]).
]
That quantity is nonnegative and vanishes iff the state is either a product state or a classical-classical state in the chosen basis. The framework gives an operational interpretation in terms of coherence localization under LICC and is explicitly presented as distinct from, but strongly aligned with, discord and entanglement [1706.00478].

Within the LAQC literature itself, a recurring misconception concerns direct identification with discord. The Bell-diagonal, symmetric (X)-state, and non-symmetric (X)-state studies consistently treat LAQC as a distinct quantifier: it is often below discord, it can remain finite when concurrence vanishes, and in the amplitude-damping examples it is not locally generated even when discord is [1803.02426, 2107.00158, 2204.07552].

6. Swapping, distribution, and network-oriented developments

A network-oriented discord protocol from 2024 shows how quantum correlations can be distributed to memories from an initially classically correlated carrier state,
[
\rho_{C_1C_2} = \frac{1}{2} \left( \ketbra{++}{++} + \ketbra{--}{--} \right){C_1C_2},
]
using only bilocal controlled-(Z) operations and projective measurements. The final memory state
[
\rho'{M_1M_2} = \mathcal{N}\, \mathrm{Tr}{C_1C_2}!\left[ \Pi\psi\,U_{CM} (\rho_{C_1C_2}\otimes \rho_{M_1M_2}) U_{CM}^\dagger \Pi_\psi \right]
]
can exhibit discord and global quantum discord even though the carriers are separable. The work is explicitly discord-based rather than LAQC-based, but it exemplifies the broader theme that local operations plus classical resources can distribute nonclassical correlations [2408.05490].

A direct LAQC generalization of repeater-type ideas appears in the study of LAQC swapping for one-parameter two-qubit (X) states. Two independent states (\rho_{AB}) and (\rho_{CD}) are combined, and a projective measurement is performed on (BC) with
[
|\nu\rangle=\cos!\left(\frac{\eta}{2}\right)|00\rangle_{BC} +\sin!\left(\frac{\eta}{2}\right)|11\rangle_{BC},
\qquad -\frac{\pi}{2}<\eta<\frac{\pi}{2}.
]
The output state on (AD) remains of (X) form, with transformed Bloch parameters such as
[
T_1^{{AD}=T_1^{{AB}T_1^{{CD}\sin\eta,\qquad}}}
T_2^{{AD}=T_2^{{AB}T_2^{{CD}\sin\eta.}}}
]
Across the five families analyzed—Werner states, (a)-states, (B)-states, (\rho_v), and MEMS—the principal qualitative rule is that if the initial state and the one used for the projective measurement are correlated, the final state has non-zero LAQC [2507.23142].

The family-specific outcomes show that swapped LAQC is not reducible to entanglement. For (a)-states, the swapped state has
[
L(\rho_{AD})= 1+\frac{1+a_{AB}a_{CD}\sin\eta}{2}\log_2(1+a_{AB}a_{CD}\sin\eta) +\frac{1-a_{AB}a_{CD}\sin\eta}{2}\log_2(1-a_{AB}a_{CD}\sin\eta),
]
while
[
C(\rho_{AD})=0.
]
For MEMS, the same paper states
[
C(\rho_{AD})=0 \qquad \text{for all }0\le y_{AB},y_{CD}\le 1,
]
yet the swapped LAQC remains finite. This establishes, within the studied one-parameter families, that LAQC can be redistributed by a swapping protocol even when the final state is separable [2507.23142].

Taken together, the available results position LAQC as a complementary-basis correlation measure with exact analytics for all two-qubit (X) states, nontrivial and channel-dependent dynamical behavior, a presently unresolved but motivated LOCC-monotonicity question, and an operational profile distinct from both concurrence and standard discord [2204.07552, 2603.13648].