Let
M
n
\mathcal {M}_n
be the collection of
n
×
n
n\times n
complex matrices equipped with operator norm. Suppose
U
,
V
∈
M
n
U, V \in \mathcal {M}_n
are two unitary matrices, each possessing a gap larger than
Δ
\Delta
in their spectrum, which satisfy
‖
U
V
−
V
U
‖
≤
ϵ
\|UV-VU\| \le \epsilon
. Then it is shown that there are two unitary operators
X
X
and
Y
Y
satisfying
X
Y
−
Y
X
=
0
XY-YX = 0
and
‖
U
−
X
‖
+
‖
V
−
Y
‖
≤
E
(
Δ
2
/
ϵ
)
(
ϵ
Δ
2
)
1
6
\|U-X\| + \|V-Y\| \le E(\Delta ^2/\epsilon ) \left (\frac {\epsilon }{\Delta ^2}\right )^{\frac 16}
, where
E
(
x
)
E(x)
is a function growing slower than
x
1
k
x^{\frac {1}{k}}
for any positive integer
k
k
.