arXiv:quant-ph/9609018v2 10 May 1997
Physics Letters A 226, 253-256 (1997)
A Hamiltonian for Quantum Copying
Dima Mozyrsky, Vladimir Privman
Department of Physics, Clarkson University, Potsdam, NY 13699, USA
and
Mark Hillery
Department of Physics and Astronomy
Hunter College of the City University of New York
695 Park Avenue, New York, NY 10021, USA
ABSTRACT
We derive an explicit Hamiltonian for copying the basis up and down
states of a quantum two-state system—a qubit—onto n “copy” qubits
(n ≥ 1) initially all prepared in the down state. In terms of spin compo-
nents, for spin- 1
2 particle spin states, the resulting Hamiltonian involves
n- and (n + 1)-spin interactions. The case n = 1 also corresponds to a
quantum-computing controlled-NOT gate.
PACS numbers: 03.65.Bz, 85.30.St.
– 1 –

Interest in quantum computing [1-27] has boosted studies of quantum
mechanics of two-state systems such as the spin states of spin- 1
2 parti-
cles. We will use “spin” to indicate a two-state system in this context.
The “binary” up and down spin states are of particular signiﬁcance and
the two-state systems are also termed “qubits” in these studies. While
macroscopic “desktop” coherent quantum computational units are still in
the realm of science ﬁction [16,18], miniaturization of computer compo-
nents calls for consideration of quantum-mechanical [14-16,18] aspects of
their operation. Experiments have recently been reported [25,28-29] re-
alizing the simplest quantum gates. Decoherence eﬀects [16,18,26,30-32]
and inherently quantum-mechanical computational algorithms [30-31,33-
36] have been studied.
Here we consider the signal-copying process in two-state systems.
Quantum copying is of interest also in cryptography and signal transmis-
sion [37-52]. The latter applications, in their coherent-quantum-mechanical
version, are on the verge of being experimentally realized [38,39,41-43,47,48].
We assume that n + 1 spins are involved, where spin 1 is the input which
is prepared in the up state, |1〉, or down state, |0〉, at time t. The aim
is to obtain the same state in the n “copy” spin states, i.e., for spins
2, 3, . . . , n + 1, as well as keep the original state of spin 1. Generally,
one cannot copy an arbitrary [53-56] quantum state; however, one can
duplicate a set of basis states such as the qubit states considered here.
– 2 –

One can also discuss an approximate, optimized copying of the linear
combinations of the basis states [55,56].
Another limitation of the copying procedure [53-56] has been that
the initial state of the n copy spins must be ﬁxed. An attempt to allow for
a more general state leads to incomplete copying which is also of interest
[57]. In this work we assume that the initial state, at time t, of all the
copy spins is down, |0〉. Our aim is to derive an explicit Hamiltonian for
the copying process.
We adopt the approach in the quantum-computing literature [1-27]
of assuming that a constant Hamiltonian H acts during the time interval
∆t, i.e., we only consider evolution from t to t + ∆t. The dynamics can
be externally timed, with H being switched on at t and oﬀ at t + ∆t. The
time interval is then related to the strength of couplings in H which are
of order ¯h/∆t. Some time dependence can be allowed [27], of the form
f (t)H, where f (t) averages to 1 over ∆t and vanishes outside this time
interval.
We will denote the qubit states by quantum numbers qj = 0 (down)
and qj = 1 (up), for spin j. The states of the n + 1 spins will then be
expanded in the basis |q1q2 · · · qn+1〉. The actual copying process only
imposes the two conditions
– 3 –

|100 · · · 0〉 → |111 · · · 1〉 , (1)
|000 · · · 0〉 → |000 · · · 0〉 , (2)
up to possible phase factors. Therefore, a unitary transformation that
corresponds to quantum evolution over the time interval ∆t is by no mean
unique (and so the Hamiltonian is not unique). We will choose a particu-
lar transformation that allows analytical calculation and, for n = 1, yields
a controlled-NOT Hamiltonian. The controlled-NOT unitary transfor-
mations have been discussed in the literature [7,13-15,28,58,59]. A recent
preprint [59] also derives an explicit Hamiltonian which is somewhat dif-
ferent from ours; we compare and discuss both results later.
We consider the following unitary transformation,
U =eiβ |111 · · · 1〉〈100 · · · 0| + eiρ|000 · · · 0〉〈000 · · · 0|
+eiα|100 · · · 0〉〈111 · · · 1| + ∑
{qj }′
|q1q2q3 · · · qn+1〉〈q1q2q3 · · · qn+1| .
(3)
Here the ﬁrst two terms accomplish the desired copying transformation.
The third term is needed for unitarity (since the quantum evolution is
reversible). We allowed for general phase factors in these terms. The
sum in the fourth term, {qj }′, is over all the other quantum states of the
– 4 –

system, i.e., excluding the three states |111 · · · 1〉, |100 · · · 0〉, |000 · · · 0〉.
One could maintain analytical tractability while adding phase factors for
each term in this sum; however, the added terms in the Hamiltonian are
not interesting. One can check by explicit calculation that U is unitary,
U †U = 1.
To calculate the Hamiltonian H according to
U = e−iH∆t/¯h , (4)
we diagonalize U . The diagonalization is simple because we only have
to work in the subspace of the three special states identiﬁed in (3), see
the preceding paragraph. Furthermore, the part related to the state
|000 · · · 0〉 is diagonal. In the subspace labeled by |111 · · · 1〉, |100 · · · 0〉,
|000 · · · 0〉, in that order, the operator U is represented by the matrix
U =

 0 eiβ 0
eiα 0 0
0 0 eiρ

 . (5)
The eigenvalues of U are ei(α+β)/2, −ei(α+β)/2, eiρ. Therefore the
eigenenergies of the Hamiltonian in the selected subspace can be of the
form
– 5 –

E1 = − ¯h
2∆t (α + β) + 2π¯h
∆t N1 , (6)
E2 = − ¯h
2∆t (α + β) + 2π¯h
∆t
(
N2 + 1
2
)
, (7)
E3 = − ¯h
∆t ρ + 2π¯h
∆t N3 , (8)
where N1,2,3 are arbitrary integers.
In order to simplify the expressions, we will limit our consideration to
a particular set of parameters. We would like to minimize energy gaps of
the Hamiltonian [27] and generally, keep the energy spectrum symmetric.
The latter condition yields a more elegant answer; actually, analytical
calculation is possible with general parameter values. Thus, we take
ρ = 0, N3 = 0, and also impose the condition E1 + E2 = 0. We then take
the diagonal matrix with E1,2,3 as diagonal elements and apply the inverse
of the unitary transformation that diagonalizes U . All the calculations
are straightforward and require no further explanation or presentation of
details in the matrix notation. We note, however, that one could do all
these calculations directly in the qubit-basis notation such as in (3); the
diagonalization procedure is then the Bogoliubov transformation familiar
from solid-state physics.
– 6 –

The result for the Hamiltonian in the three-state subspace is the
matrix
H = π¯h
∆t
(
N − 1
2
) 
 0 e−iγ 0
eiγ 0 0
0 0 0

 , (9)
which depends on one real parameter
γ = α + β
2 (10)
and on one arbitrary integer
N = N1 − N2 . (11)
The full Hamiltonian H in the 2n+1-dimensional spin space is
H = π¯h
∆t
(
N − 1
2
) (
e−iγ |111 · · · 1〉〈100 · · · 0| + eiγ |100 · · · 0〉〈111 · · · 1|
)
.
(12)
In what follows we make the choice N = 1 to simplify the notation. The
form of the Hamiltonian is misleading in its simplicity. It actually involves
n- and (n + 1)-spin interactions. To see this, we rewrite it in terms of
direct products of the unit matrices and the standard Pauli matrices for
– 7 –

spins 1, . . . , n + 1, where the spins are indicated by superscripts (and
N = 1):
H = π¯h
2n+2∆t
(
1 + σ(1)
z
) (
e−iγ σ(2)
+ σ(3)
+ · · · σ(n+1)
+ + eiγ σ(2)
− σ(3)
− · · · σ(n+1)
−
)
;
(13)
here σ± = σx ± iσy ; σ+ =
( 0 2
0 0
)
, σ− =
( 0 0
2 0
)
.
Multispin interactions are much less familiar and studied in solid-
state and other systems than two-spin interactions. Therefore, the fact
that for n = 1 only single- and two-spin interactions are present is sig-
niﬁcant. In actual quantum-computing and other applications it may
be more practical to make copies in stages, generating only one copy in
each time interval, rather than produce n > 1 copies simultaneously. Let
us explore the n = 1 case further. The Hamiltonian (with N = 1) is,
in terms of spin components (or rather the Pauli matrices to which the
spin-component operators are proportional),
Hn=1 = π¯h
4∆t
(
1 + σ(1)
z
) [
(cos γ)σ(2)
x + (sin γ)σ(2)
y
]
. (14)
This Hamiltonian involves two-spin couplings and also interactions which
are linear in the x and y spin components. The latter may be due to a
magnetic ﬁeld applied in the xy-plane, at an angle γ with the x axis.
– 8 –

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A Hamiltonian for quantum copying

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