Kinematic budget of quantum correlations

Abstract

The diversity of quantum correlations -- discord, entanglement, steering, and Bell nonlocality -- disappears at the observable second-moment kinematic level. By treating state purity as a finite resource, we introduce a local-unitary-invariant budget split of symmetrised second moments into local and nonlocal sectors that maps quantum systems onto compact, two-dimensional, hole-free manifolds. The topology of these manifolds is governed by state purity and time-reversal symmetry. This dimensional reduction reveals a deep structural link: exceeding classical capacity limits forces the activation of time-asymmetric generators, guaranteeing non-positive partial transpose entanglement. For two qubits, the geometry is analytically solvable. A single boundary elegantly isolates classical correlations, while nested regions physically dictate entanglement, steering, Bell nonlocality, and bounds on non-stabiliser magic. Beyond two qubits, dimensional capacity bottlenecks enforce these universal kinematic limits on correlation structures. Because this macroscopic representation is completely determined by global and marginal purities, it bypasses the exponential scaling of full-state tomography. By coarse-graining over gauge-like first moments, the budget geometry acts as a thermodynamic phase diagram, exposing both the static hierarchy of quantum resources and their dynamic redistribution under decoherence.

Figures (10)

• Figure 1: Local basis-independent purity-budget geometry for two-qubit states. ( a) The entire two-qubit state space, projected onto a single, compact geometric manifold where the kinematic bounds are exposed. The coordinates $X$ and $Y$ track the fundamental tug-of-war between local polarisation and nonlocal correlations. The grey region is unphysical, forbidden by the time-symmetric nature of quantum mechanics ($Q < 0$). Within the physical regime, exact analytic boundaries organise the space into the familiar correlation hierarchy. The classical envelope (purple curve) sets a hard capacity limit for all classically structured states; crossing this threshold guarantees nonclassicality, ensuring the state is both entangled and steerable. Additional curves delineate the threshold for guaranteed CHSH-Bell violations (blue dot-dashed) and the absolute separability limit (red dashed). ( b) Monte Carlo sampling of the state space. Distinct families of quantum states perfectly populate their analytically defined regions, confirming that this hole-free projection captures the entire physical space.
• Figure 2: Purity budget-controlled maxima of entanglement and computational magic.a--b, ( a) Maximum attainable entanglement negativity and ( b) stabiliser Rényi-2 magic for two-qubit states at fixed purity. Both are plotted against the budget-allocation angle $\theta$, which tracks the internal trade-off between local and nonlocal budgets. Even when nonclassical resources are permitted, both exhibit a sharp, purity-dependent ceiling; entanglement exclusively emerges beyond the absolute separability threshold ($R=1/3$). c--d, Extension of the magic bounds to ( c) 5-qubit and ( d) 7-qubit systems. The expanded capacity of the Pauli spectrum in higher dimensions flattens the maximum magic profile, rendering the upper bounds largely insensitive to specific angular redistributions of the budget. Collectively, these envelopes establish the purity budget as a universal kinematic constraint -- indicating the onset of quantum resources and capping their maximum possible magnitude.
• Figure 3: Dimensional bottlenecks and guaranteed NPT entanglement.a-- c, The purity-budget projection for ( a) qubit-qutrit, ( b) qutrit-qutrit, and ( c) three-qubit systems (the explicit construction of these geometries is given in Supplementary Information). The grey region remains unphysical ($Q < 0$). Dimensional asymmetry introduces a strict capacity bottleneck: dictated by the Schmidt decomposition, the marginals of a pure state must share the same spectrum. Consequently, the smaller qubit permanently caps the qutrit's entanglement capacity, forcing the correlation limit strictly below the $Y=1$ apex. For multipartite systems (e.g., c), sequential local depolarisation traces a piecewise, multi-segmented positivity wall. d-- g, Zones of guaranteed NPT entanglement. The classical-quantum envelopes set the absolute correlation capacity of the time-symmetric generator sector. Traversing above these boundaries mathematically forces the activation of time-odd, asymmetric generators to maintain global state positivity. The mandatory presence of these generators structurally guarantees negative eigenvalues under partial transposition. Any geometric excursion into these shaded regions, therefore, acts as a universal, basis-independent witness for NPT entanglement. For three qubits, this structural hierarchy precisely isolates entanglement classes: exceeding the $QC^2$ envelope guarantees biseparable NPT entanglement ( f), while crossing the higher $Q^2C$ boundary precludes any single-party classical description, structurally forcing genuine multipartite NPT entanglement ( g) (e.g., the W state Dur:2000zz).
• Figure 4: Kinematic flows of decoherence.a-- d, Parametric decoherence trajectories in the $(X,Y)$ plane for ( a) a maximally entangled Bell state, ( b) a pure-product state, ( c) a mixed-entangled state, and ( d) a mixed-separable state. The kinematic constraints and flow patterns generalise to higher dimensions beyond two-qubit systems. As noise strength increases, the resulting kinematic flows map how specific quantum channels redistribute the purity budget. Local noise degrades nonclassicality: depolarisation contracts states radially inward toward the origin, while dephasing drives them vertically downward to selectively suppress nonlocal correlations. Conversely, correlated noise redistributes these resources. By dissipating local polarisation faster than nonlocal correlations, correlated amplitude damping enables states to cross into steerable or entangled regimes despite an overall loss of purity. These trajectories reveals an empirical arrow of decoherence: under natural open-system dynamics, global purity ($P$) and time-reflection overlap ($Q$) cannot increase simultaneously (also see Fig. \ref{['fig:arrow']}).
• Figure 5: The empirical arrow of decoherence in the macroscopic $(P,Q)$ plane.a--f, Open-system dynamics manifest as constrained flows, illustrated here by plots in the global purity ($P$) vs the time-reflection overlap ($Q$) plane for six distinct initial states: (a) a Bell state, (b) a CHSH state, (c) a mixed-entangled state, (d) a pure-product state, (e) a mixed-separable state, and (f) a classical state. Under natural, memoryless open-system dynamics, $P$ and $Q$ cannot increase simultaneously. Physically, environments that extract entropy to cool the system (increasing $P$) inevitably polarise the state, destroying time-reversal symmetry and lowering $Q$. Conversely, scrambling environments that restore this symmetry inherently inject entropy, thereby lowering $P$. This no-go rule acts as a macroscopic, geometric arrow of decoherence, permanently restricting the accessible regions of the second-moment space during natural evolution in all two-qubit systems and beyond.