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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!

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No. Just take two Bell states. They have identical reduced density matrices yet are orthogonal, that is, as distant from each other as it gets.

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