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We introduce a new class of entangled subspaces -- completely entangledsubspaces of entanglement depth $k$ ($k$-CESs). These are subspaces ofmultipartite Hilbert spaces containing only pure states with the entanglementdepth at least $k$. We present an efficient construction of $k$-CESs of anyachievable dimensionality in any multipartite scenario. Further, we discuss therelation between these subspaces and unextendible product bases (UPBs). Inparticular, we establish that there is a non-trivial bound on the cardinalityof a UPB whose orthocomplement is a $k$-CES. Further, we discuss existence ofsuch UPBs for qubit systems.

Completely entangled subspaces of entanglement depth $k$