Probabilistic cloning and deleting of quantum states
Yuan Feng, Shengyu Zhang, and Mingsheng Ying*
State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology,
Tsinghua University, Beijing, China, 100084
Received 17 August 2001; published 10 April 2002
We construct a probabilistic cloning and deleting machine which, taking several copies of an input quantum
state, can output a linear superposition of multiple cloning and deleting states. Since the machine can perform
cloning and deleting in a single unitary evolution, the probabilistic cloning and other cloning machines pro-
posed in the previous literature can be thought of as special cases of our machine. A sufficient and necessary
condition for successful cloning and deleting is presented, and it requires that the copies of an arbitrarily
presumed number of the input states are linearly independent. This simply generalizes some results for cloning.
We also derive an upper bound for the success probability of the cloning and deleting machine.
DOI: 10.1103/PhysRevA.65.042324 PACS numbers: 03.67.-a, 03.65.Ta
In quantum mechanics, one well-known fact is the no-
cloning theorem 1,2, which asserts that, unlike in classical
world, an arbitrary unknown quantum state cannot be cloned
perfectly because of the linearity of quantum operations.
However, inaccurate copying is possible 3. On the other
hand, states chosen from a linearly independent set can be
probabilistically cloned by a unitary-reduction process 4,5.
This is an impressive result, and, more interestingly, using
the cloning machine introduced in 6, nonorthogonal states
from a linearly independent set can evolve into a linear su-
perposition of multiple cloning states.
Recently, deleting unknown quantum states was also
found to be impossible, where ‘‘deleting’’ means ‘‘uncopy-
ing,’’ that is, deleting one or more copies of the input state by
a linear trace preserving operation 7. At first glance it
seems that copying and deleting should be treated separately
since, as pointed out in Ref. 7, the deleting process is in-
dependent of cloning. In general, deleting is not the inverse
of copying; only if copying and deleting are performed by
unitary operation is it so. By a careful analysis, however, it
may be seen that the mechanisms of copying and deleting are
quite similar. This suggests that we look for a unified way of
dealing with copying and deleting of quantum states.
In this short note, we construct a quantum machine which,
taking several copies of an input quantum state, can output a
linear superposition of multiple cloning states and multiple
deleting states. The probabilistic cloning machine of Duan
and Guo 4,5 and the cloning machine of Pati 6 can both
be thought of as special cases of our cloning and deleting
machine. What we would like to emphasize is that in our
construction both the copying and deleting procedures occur
in a single machine. We show that if |
i
is chosen from S
= |
i
: i =1,2, . . . , m , then |
i
k
can be probabilistically
cloned and deleted if and only |
1
k
, |
2
k
,..., |
m
k
are linearly independent. We also give an upper bound for
the success probability of the quantum machine.
Consider a quantum state set S = |
1
, |
2
,..., |
m
whose elements belong to an N
A
-dimensional Hilbert space
with N
A
m the subscript A is used to indicate that this is
the original system. A quantum cloning process 4,5 is de-
fined by an evolution as
|
i
| | P
0
→|
i
|
i
| P
i
,
where | is the input state of an ancillary system, and P
0
and P
i
are the initial state and the final state after cloning
|
i
of the cloning apparatus, respectively. Considering the
ability of multiple cloning, the cloning process of 6 is given
by
|
i
| | P
0
→
n =1
M
p
n
( i )
|
i
( n +1)
| 0
( M-n )
| P
n
,
where M is the total number of states of the ancilla whose
initial state is denoted by | and p
n
( i )
is the success prob-
ability of producing n exact copies of |
i
. Obviously, this
cloning machine is a generalization of that considered in
4,5, and the major distinction between them is that in the
former cloning procedures of different copies are embedded
in a single machine.
On the other hand, a quantum deleting machine 7 which
can delete one of two copies and replace it with some stan-
dard state | 0 is defined as
|
i
|
i
| P
0
→|
i
| 0 | P
i
.
In general, deleting cannot be seen as the inverse process of
copying and they are independent of each other; but if they
are performed by unitary operation then deleting is the in-
verse of copying. By combining ideas from 4,5,7, further-
more, the quantum deleting machine depicted above may
easily be extended to a probabilistic multiple deleting pro-
cess, which can be expressed by the following transforma-
tion:
|
i
k
| P
0
→
n =1
k -1
p
n
( i )
|
i
n
| 0
( k -n )
| P
n
.
The purpose of this short note is to extend all the concepts
mentioned above in a unified way to answer the following
question: if we have several identical copies of |
i
, is it
possible to have a quantum superposition of the multiple
cloning and deleting states described as follows: *Corresponding author. Email address: yingmsh@tsinghua.edu.cn
PHYSICAL REVIEW A, VOLUME 65, 042324
1050-2947/2002/654/0423244/$20.00 ©2002 The American Physical Society 65 042324-1