# Does the trace distance between marginals bound the distance between the overall states?

If the quantum states of the subsystems of two systems are close (for example: in terms of trace distance), are the states of the larger systems also close, i.e., if $$||\rho_A - \rho_{A^\prime}||\leq \epsilon$$ and $$||\rho_B - \rho_{B^\prime}||\leq \epsilon,$$ can we claim that $$||\rho_{AB} - \rho_{A^\prime B^\prime}||\leq \delta(\epsilon)$$? Thanks in advance!