  In recent years, tensor network states have emerged as a very useful
conceptual and simulation framework to study quantum many-body systems at low
energies. In this paper, we describe a particular way in which any given tensor
network can be viewed as a representation of two different quantum many-body
states. The two quantum many-body states are said to correspond to each other
by means of the tensor network. We apply this "tensor network state
correspondence"---a correspondence between quantum many-body states mediated by
tensor networks as we describe---to the multi-scale entanglement
renormalization ansatz (MERA) representation of ground states of one
dimensional (1D) quantum many-body systems. Since the MERA is a 2D hyperbolic
tensor network (the extra dimension is identified as the length scale of the 1D
system), the two quantum many-body states obtained from the MERA, via tensor
network state correspondence, are seen to live in the bulk and on the boundary
of a discrete hyperbolic geometry. The bulk state so obtained from a MERA
exhibits interesting features, some of which caricature known features of the
holographic correspondence of String theory. We show how (i) the bulk state
admits a description in terms of "holographic screens", (ii) the conformal
field theory data associated with a critical ground state can be obtained from
the corresponding bulk state, in particular, how pointlike boundary operators
are identified with extended bulk operators. (iii) We also present numerical
results to illustrate that bulk states, dual to ground states of several
critical spin chains, have exponentially decaying correlations, and that the
bulk correlation length generally decreases with increase in central charge for
these spin chains.
