  The preparation of initial superposition states of discrete-time quantum
walks (DTQWs) are necessary for the study and applications of DTQWs. In linear
optics, it is easy to prepare initial superposition states of the coin, which
are always encoded by polarization states; while the preparation of
superposition states of the walker is challenging. Based on a novel encoding
method, we here propose a DTQW protocol in linear optics which enables the
preparation of arbitrary initial superposition states of the walker and the
coin. With this protocol, we report an experimental demonstration of DTQW with
the walker initially in superposition states, by using only passive
linear-optical elements. The effects of the walker's different initial
superposition states on the spread speed of the DTQW and on the entanglement
between the coin and the walker are also experimentally investigated, which
have not been reported before. When the walker starts with superposition
states, we show that the properties of DTQW are very different from those of
DTQW starting with a single position. Our findings reveal different properties
of DTQW and paves an avenue to study DTQW with arbitrary initial states.
Moreover, the encoding method enables one to encode an arbitrary
high-dimensional quantum state using a single physical qubit and may be adopted
to implement other quantum information tasks.
