The most powerful technique known at present for bounding the size of quantum
codes of prescribed minimum distance is the quantum linear programming bound.
Unlike the classical linear programming bound, it is not immediately obvious
that if the quantum linear programming constraints are satisfiable for
dimension K, that the constraints can be satisfied for all lower dimensions. We
show that the quantum linear programming bound is indeed monotonic in this
sense, and give an explicitly monotonic reformulation.Comment: 5 pages, AMSTe

Rains, Eric M.

English

arXiv

The most powerful technique known at present for bounding the size of quantum codes of prescribed minimum distance is the quantum linear programming bound. Unlike the classical linear programming bound, it is not immediately obvious that if the quantum linear programming constraints are satisfiable for dimension K, then the constraints can be satisfied for all lower dimensions. We show that the quantum linear programming bound is monotonic in this sense, and give an explicitly monotonic reformulation

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MONOTONICITY OF THE QUANTUM
LINEAR PROGRAMMING BOUND
Eric M. Rains
AT&T Research
February 17, 1998
Abstract. The most powerful technique known at present for bounding the size of
quantum codes of prescribed minimum distance is the quantum linear programming
bound. Unlike the classical linear programming bound, it is not immediately obvious
that if the quantum linear programming constraints are satisfiable for dimension
K, that the constraints can be satisfied for all lower dimensions. We show that the
quantum linear programming bound is monotonic in this sense, and give an explicitly
monotonic reformulation.
Introduction
The most powerful technique known at present for bounding the size of quantum
codes of prescribed minimum distance is the quantum linear programming bound:
Theorem (Quantum LP bound). If there exists a quantum code encoding K
states in n qubits, with minimum distance d, then there exist homogeneous polyno-
mials A(x, y), B(x, y), and S(x, y) of degree n, satisfying the equations
B(x, y) = A(
x+ 3y
2
,
x− y
2
) (1)
S(x, y) = A(
x+ 3y
2
,
y − x
2
) (2)
A(1, 0) = K2 (3)
B(1, y)−
1
K
A(1, y) = O(yd) (4)
and the inequalities
A(x, y) ≥ 0 (5)
B(x, y)−
1
K
A(x, y) ≥ 0 (6)
S(x, y) ≥ 0, (7)
where P (x, y) ≥ 0 means that the polynomial P has nonnegative coefficients.
Proof. This is theorem 10 of [3]; see also [5]. The polynomials A(x, y), B(x, y), and
S(x, y) are the weight enumerator, dual weight enumerator, and shadow enumera-
tor, respectively, of the quantum code. 
Key words and phrases. quantum codes linear programming.
1
2 ERIC M. RAINS
Remark. In the sequel, we will use the standard notation ((n,K, d)) to denote a
quantum code encoding K states in n qubits, with minimum distance d.
It is clear that the existence of an ((n,K, d)) code implies the existence of an
((n,K ′, d)) code for all K ′ ≤ K, which suggests that the same should be true for
the quantum LP bound, namely that if the quantum LP constraints can be satisfied
for ((n,K, d)), then they can be satisfied for ((n,K ′, d)) for all K ′ ≤ K. At first
glance, this appears to be false; after all, in the inequality (6), decreasingK actually
makes the inequality harder to satisfy. This impression is misleading, however; as
we will see below, the quantum LP bound is indeed monotonic in K.
1. Random subcodes
The reason the quantum LP bound “ought” to be monotonic in K is that if
Q is an ((n,K, d)) code, and Qˆ is a subcode of Q of dimension K ′, then Qˆ is an
((n,K ′, d)) code. Of course, in general, it is impossible to deduce the weight enu-
merator of Qˆ from the weight enumerator of Q, so this is not directly applicable to
the LP bound. However, if instead of picking a specific subcode, we instead aver-
age over all subcodes of a given dimension, the resulting average weight enumerator
turns out to depend only on the original weight enumerators.
Recall that if Q is an ((n,K, d)) code, and PQ is the orthogonal projection onto
Q, then the weight enumerators AQ(x, y) and BQ(x, y) are defined by
AQ(x, y) =
∑
e∈E
Tr(PQe)
2xn−wt(e)ywt(e),
BQ(x, y) =
∑
e∈E
Tr(PQePQe)x
n−wt(e)ywt(e),
where E is the set of all tensor products of matrices from the set {I, σx, σy, σx},
and wt(E) is the number of nonidentity tensor factors in E.
Define
AˆQ(x, y) = EQˆ⊂QAQˆ(x, y),
and similarly for BˆQ(x, y), where the expectation is over subcodes of dimension K
′.
If we write PQ = ΠΠ
† for some 2n ×K matrix Π, then
P
Qˆ
= ΠP ′Π†,
for some K ×K projection operator P ′ with Tr(P ′) = K ′. So
AˆQ(x, y) = EP ′
∑
e∈E
|Tr(ΠP ′Π†e)|2xn−wt(e)ywt(e),
and similarly for BˆQ. But
EP ′ Tr(ΠP
′Π†e)2 = EU∈U(K) Tr(ΠUP
′U †Π†e)2
= EU∈U(K) Tr(Π
†eΠUP ′U †)2.
At this point, we can apply the following lemma:
MONOTONICITY OF THE QUANTUM LINEAR PROGRAMMING BOUND 3
Lemma 1. Define functions
s2(A) =
1
2
(Tr(A)2 +Tr(A2))
s12(A) =
1
2
(Tr(A)2 − Tr(A2)).
For any K ×K matrices A and B,
EU∈U(K)s(AUBU
†) =
s(A)s(B)
s(IK)
,
where s is either s2 or s12 .
Proof. This follows from the theory of zonal polynomials [1]. For A and B unitary,
the relations follow from the fact that s2 and s12 are irreducible characters of the
unitary group. Since they are also polynomial functions of A and B, the relations
must hold for arbitrary matrices. 
In particular,
EU∈U(K) Tr(Π
†eΠUP ′U †)2 = EU∈U(K)s2(Π
†eΠUP ′U †) + s12(Π
†eΠUP ′U †)
=
K ′
2
+K ′
K2 +K
s2(Π
†eΠ) +
K ′
2
−K ′
K2 −K
s12(Π
†eΠ).
It follows that
AˆQ(x, y) =
K ′(K ′K − 1)
K3 −K
AQ(x, y) +
K ′(K −K ′)
K3 −K
BQ(x, y).
Similarly,
BˆQ(x, y) =
K ′(K −K ′)
K3 −K
AQ(x, y) +
K ′(K ′K − 1)
K3 −K
BQ(x, y).
In general, if A(x, y) is a polynomial satisfying the quantum LP constraints for
((n,K, d)), then for any K ′ ≤ K, we can define
Aˆ(x, y) =
K ′(K ′K − 1)
K3 −K
A(x, y) +
K ′(K −K ′)
K3 −K
B(x, y).
The claim is that Aˆ satisfies the quantum LP constraints for K ′. We have:
Aˆ =
K ′
2
K2
A+
K ′(K −K ′)
K3 −K
(B −
1
K
A)
Bˆ −
1
K ′
Aˆ =
K ′
2
− 1
K2 − 1
(B −
1
K
A)
Sˆ =
K ′
2
+K ′
K2 +K
(
S(x, y) + S(−x, y)
2
)
+
K ′
2
−K ′
K2 −K
(
S(x, y)− S(−x, y)
2
)
Since all of the constants appearing above are positive for K ′ ≤ K, and Aˆ(1, 0) =
K ′
2
, the claim follows. So we have proved:
4 ERIC M. RAINS
Theorem 1. The quantum linear programming bound is monotonic in K for fixed
n and d.
Remark. Similarly, the quantum LP bound for pure codes (A(1, y) = 1 +O(yd)) is
monotonic in K, since the random subcode operator preserves purity.
We also obtain the following result of independent interest:
Theorem 2. The average weight enumerator of a random ((n,K)) quantum code
is
A(x, y) =
K(4nK − 2n)
4n − 1
xn +
K(K − 2n)
4n − 1
(x+ 3y)n.
Proof. We have A(x, y) = AˆH, where H is the trivial quantum code consisting of
the entire Hilbert space, with weight enumerator 4nxn. 
A reformulation
Lemma 1 suggests that we should be able to obtain a simpler formulation of the
quantum LP bound by considering the polynomials
C(x, y) =
A(x, y) +B(x, y)
K2 +K
,
D(x, y) =
A(x, y)−B(x, y)
K2 −K
(where D(x, y) is only well-defined for K > 1). In particular, we have the following
result:
Lemma 2. The polynomials C and D are preserved by the average subcode opera-
tor; that is, Cˆ = C and Dˆ = D.
So, if we reformulate the quantum LP bound in terms of C and D, the result
should be explicitly monotonic, in that a feasible solution for K will itself be a
feasible solution for all smaller K.
Theorem 3. If there exists an ((n,K, d)) quantum code (K > 1), then there exist
homogeneous polynomials C(x, y) and D(x, y), satisfying the equations
C(x, y) = C(
x + 3y
2
,
x− y
2
) (8)
D(x, y) = −D(
x+ 3y
2
,
x− y
2
) (9)
C(1, 0) = 1 (10)
C(1, y)−D(1, y) = O(yd) (11)
and satisfying the inequalities
C(x, y)−
K − 1
2K
(C(x, y) −D(x, y)) ≥ 0 (12)
C(x, y)−D(x, y) ≥ 0 (13)
C(
x+ 3y
2
,
y − x
2
) ≥ 0 (14)
D(
x+ 3y
2
,
y − x
2
) ≥ 0. (15)
MONOTONICITY OF THE QUANTUM LINEAR PROGRAMMING BOUND 5
Proof. We have
A(x, y) = K2C(x, y)−
K2 −K
2
(C(x, y)−D(x, y))
B(x, y)−
1
K
A(x, y) =
K2 − 1
2
(C(x, y)−D(x, y))
S(x, y) =
K2 +K
2
C(
x+ 3y
2
,
y − x
2
) +
K2 −K
2
D(
x+ 3y
2
,
y − x
2
).
Equations (8) and (9) are clearly equivalent to (1), while (10) and (11) are together
equivalent to (3) and (4). Similarly, the inequalities (12) and (13) are equivalent to
(5) and (6) respectively.
For (14) and (15), it suffices to note that (8) and (9) imply
C(
x + 3y
2
,
y − x
2
) = C(
−x+ 3y
2
,
y + x
2
)
D(
x + 3y
2
,
y − x
2
) = −D(
−x+ 3y
2
,
y + x
2
)
It follows that the two terms in the expression for S(x, y) have disjoint support. So
(7) becomes (14) and (15). 
Theorem 1 is an obvious corollary; K appears only in (12), and decreasing K
in that equation only makes the constraint easier to satisfy. For pure codes, the
additional constraint C(1, y) = 1+O(yd) holds, and again monotonicity is obvious.
It should also be noted that this theorem carries over readily to nonbinary codes
(from the inequalities in [2]and [4]); in particular, the quantum LP bound is mono-
tonic for larger alphabet codes as well.
References
1. A. T. James, Zonal polynomials of the real positive definite symmetric matrices, Ann. of
Math. 74 (1961), 475–501.
2. E. M. Rains, Polynomial invariants of quantum codes, LANL e-print quant-ph/9704042.
3. E. M. Rains, Quantum shadow enumerators, LANL e-print quant-ph/9611001.
4. E. M. Rains, Quantum weight enumerators, IEEE Trans. Inf. Th. (to appear).
5. P. W. Shor and R. Laflamme, Quantum analog of the MacWilliams identities in classical
coding theory, Phys. Rev. Lett. 78 (1997), 1600–1602.
AT&T Research, Room C290, 180 Park Ave. Florham Park, NJ 07932-0971, USA
E-mail address: rains@research.att.com


Monotonicity of the quantum linear programming bound

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