Higher-Rank Numerical Ranges of Unitary and Normal Matrices

1.1 MB·English

by  Man-Duen Choi

HIGHER-RANK NUMERICAL RANGES OF UNITARY
AND NORMAL MATRICES
MAN-DUEN CHOI1, JOHN A. HOLBROOK2, DAVID W. KRIBS2,3
7 AND KAROL Z˙YCZKOWSKI4,5
0
0
2
Abstract. Weverifyaconjectureonthestructureofhigher-rank
n
numerical ranges for a wide class of unitary and normal matrices.
a
J Using analytic and geometric techniques, we show precisely how
the higher-rank numerical ranges for a generic unitary matrix are
1
3 given by complex polygons determined by the spectral structure
of the matrix. We discuss applications of the results to quantum

2 error correction, specifically to the problem of identification and
v
construction of codes for binary unitary noise models.
4
4
2
8
0
1. Introduction
6
0
h/ The study of higher-rank numerical ranges of matrices was initiated
p in [6], with a basic problem in quantum error correction [5] giving
-
t the primary motivation. Higher-rank numerical ranges generalize the
n
classical numerical range of a matrix, and arise as a special case of the
a
u matricialrangefora matrix [8, 15]. In[6], three ofus conjectured that
q
the higher-rank numerical ranges of normal matrices depend, in a very
:
v
precise way, on the spectral structure of the matrix. The conjecture
i
X reduces in the rank-1 case to a well-known property of the classical nu-
r merical range for a normal matrix, and it has opened the door to some
a
interesting new mathematical problems. Its verification (or refutation)
would also yield information for quantum error correction.
In this paper we verify the higher-rank numerical range conjecture
forawide variety ofunitaryandnormalmatrices. This isaccomplished
by introducing a number of new geometric techniques into the analysis.
We show in the case of a generic N N unitary with non-degenerate
×
spectrum and positive integer k 1 with N 3k, that the kth numeri-
≥ ≥
calrangeis given by acertain polygoninthe complex planedetermined
by the eigenvalues of the unitary, and thus verify the conjecture. In
2000 Mathematics Subject Classification. 15A60, 15A90, 47A12, 81P68.
key words and phrases. higher-rank numerical range, unitary matrix, quantum
error correction.
1
2 M.–D. CHOI,J.A. HOLBROOK,D.W.KRIBS,K. Z˙YCZKOWSKI
other cases, such as N = 5m and k = 2m, we also verify the conjec-
ture. But the analysis in these cases is more delicate, and our proof is
non-constructive in nature. Figure 1 below provides a chart indicating
the cases we verify, together with the various constraints.
Our results may be applied to construct error correcting codes for a
special class of quantum channels. A “binary unitary channel” [5] is
E
anoisemodeldescribed by two unitaryerrorsthatcanoccur during the
time evolution of a given quantum system. The construction of codes
for such a channel relies on the structure of the higher-rank numerical
rangesforasingleunitaryU. SupposeU actsonN-dimensionalHilbert
space. Then an “[N,k]-code” for is a k-dimensional subspace code
E
that is correctable for . The results described above for N 3k yield
E ≥
a simple algorithm to determine the existence of codes, and an explicit
construction of codes when they exist.
The paper is organized as follows. Section 2 contains a discussion of
somebasicsofquantumerrorcorrectionthatgiveimportantmotivation
for consideration of higher-rank numerical ranges. In Section 3 we
discuss the conjecture, and show in particular how the general normal
case relies on the unitary case. Section 4 deals with the structure
of the aforementioned polygon and a derivation of conditions under
which it is nonempty. Section 5 includes the proof for N 3k and
≥
the construction of codes for binary unitary channels. In Section 6 we
derive a number of other cases non-constructively based on the case
N = 5 and k = 2. We finish with a brief discussion on possible further
avenues of research and limitations in Section 7.
2. Error Correction in Quantum Computing and Binary
Unitary Channels
For a more complete discussion on the material of this section see [5]
and the references therein. We start with a quantum system in contact
with an external environment having finitely many degrees of freedom
represented on a Hilbert space such that dim = N < . Consider
H H ∞
a unitary time evolution of the combined system and environment in-
duced bya given Hamiltonianassociated withsome quantum computa-
tionimplemented on . The actionofthe evolution maponthesystem
H
is obtained by tracing out the environment, and the resulting map is
called a quantum channel or operation. Such a map is described by a
completely positive, trace preserving map : ( ) ( ), and can
E L H → L H
always be represented in the operator-sum form as (ρ) = E ρE†
E Pa a a
for a set of operators E ( ) satisfying E†E = I. As a
{ a} ⊆ L H Pa a a
HIGHER-RANK NUMERICAL RANGES 3
convenience we shall write = E when the operators E determine
a a
E { }
in this way. The E are interpreted as the noise or errors induced by
a
E
.
E
In the standard approach to quantum error correction, a code on
H
is given by a subspace of dimension at least two. Denote the
C ⊆ H
projection of onto by P . A code is (ideally) correctable for an
C
H C C
operation if there is a quantum operation : ( ) ( ) that
E R L H → L H
acts as a left inverse of on ;
E C
(1) (ρ) = ρ ρ P ( )P .
(cid:0) (cid:1) C C
R◦E ∀ ∈ L H
Given a representation for = E , a code is correctable for if
a
E { } C E
and only if there are complex numbers (λ ) such that
ab
(2) P E†E P = λ P a,b.
C a b C ab C ∀
Thus, the problem of finding correctable codes for is equivalent to
E
simultaneously solvingthefamilyofequationsinEqs.(2)forthescalars
λ and projections P , for all pairs a,b.
ab C
Of central importance in quantum computing, communication and
cryptography is the class of randomized unitary channels [1, 3]. Such
a channel has a representation of the form = √p U where each
a a
E { }
U is a unitary operator and the p form a classical probability dis-
a a
{ }
tribution; p > 0, p = 1. Hence,
a Pa a
(3) E(ρ) = XpaUaρUa† ∀ρ ∈ L(H).
a
The associated quantum operation is given by the scenario in which
the error U occurs with probability p . By Eqs. (2), finding ideal
a a
correctable codes for = √p U is equivalent to solving the (un-
a a
E { }
normalized) equations
(4) PU†U P = λ P a,b,
a b ab ∀
for λ and P. Note that each operator U†U is unitary.
ab a b
The following observation illustrates the importance of randomized
unitary channels in error correction.
Proposition 2.1. Let be a quantum operation. Then is a cor-
E C
rectable code for if and only if there is a randomized unitary channel
E
such that is correctable for and
F C F
(5) (ρ) = (ρ) ρ P ( )P .
C C
E F ∀ ∈ L H
Proof. In fact, = √p U can be chosen so that the partial isome-
a a
F { }
tries U P have mutually orthogonal ranges for distinct a. This follows
a C
4 M.–D. CHOI,J.A. HOLBROOK,D.W.KRIBS,K. Z˙YCZKOWSKI
directly from the usual construction of a correction operation for
R
on [10]. The matrix (λ ) from Eqs.(2) is a density matrix, and
ab
E C
the unitary which diagonalizes it can be used to find a set of error
operators F that implement , where ( ) = P ( )P , such
a C C C C
{ } E ◦ P P · ·
that P F†F P = δ d P . The polar decomposition yields a partial
C a b C ab aa C
isometry U , which can be extended to a unitary on the entire space,
a
such that FaPC = UapPCFa†FaPC = √daaUaPC. The √daa form a
{ } (cid:4)
probability distribution by the trace preservation of .
E
Of course, the randomized unitary channel given by the restriction
of to the code can only be obtained when the code itself is known.
E C
Thus, this result in itself is of little practical use for the general prob-
lem of finding error correcting codes. Nevertheless, it indicates that
the general problem is equivalent to finding codes for the class of ran-
domized unitary channels.
Observe that a code is correctable for a binary unitary noise model
= √pV,√1 pW , where V and W are unitary, if and only if it is
E { − }
correctable for = √pI,√1 pV†W .
E { − }
Definition 2.2. A binary unitary channel on a Hilbert space is
H
a channel of the form
(6) = √pI,p1 pU
E { − }
for some unitary U ( ) and fixed probability 0 < p < 1. Thus, the
∈ L H
action of is given by
E
(7) (ρ) = pρ+(1 p)UρU† ρ ( ).
E − ∀ ∈ L H
Remark 2.3. Binary unitary channels form a rather restrictive class
of physical noise maps, but they provide a useful set of “toy” examples
for testing the “compression” approach [5] to build quantum error cor-
recting codes enabled by consideration of higher-rank numerical ranges.
Observe that from Eqs. (2), the problem of finding ideal correctable
codes for a given binary unitary channel is equivalent to solving four
equations, but that the entire problem reduces to solving the single (un-
normalized) equation for λ and P given by:
(8) PUP = λP.
An immediate consequence of what follows is an algorithm to construct
codes for a wide class of binary unitary channels. This is encapsulated
in the discussion of Section 5.
HIGHER-RANK NUMERICAL RANGES 5
3. Higher-Rank Numerical Range Conjecture
Givenafixedpositiveintegerk 1andT ( ),thekth numerical
≥ ∈ L H
range of T is the set of complex numbers
Λ (T) = λ C : PTP = λP for some rank k projection P .
k (cid:8) (cid:9)
∈ −
The classical numerical range W(T) = Λ (T) is obtained when k = 1.
1
Definition 3.1. Let T ( ) and let k 1 be a fixed positive inte-
∈ L H ≥
ger. Then we define Ω (T) to be the intersection of the convex hulls
k
conv(Γ), where Γ runs through all (N k+1)-point subsets (counting
−
multiplicities) of the set of eigenvalues spec(T) for T. That is,
Ωk(T) = \ conv(Γ).
Γ⊆spec(T);|Γ|=N−k+1
Thus, Ω (T) is a convex subset of the complex plane that can be
k
computed directly from the spectrum of T. Below we will show how
this set can typically be computed as an intersection of much fewer
than N sets.
(cid:0) (cid:1)
N−k+1
It is easy to see that Ω (T) contains Λ (T) for normal T. We include
k k
a short proof for completeness and notational purposes.
Proposition3.2. Let T ( ) be a normaloperatorand fix a positive
∈ L H
integer k 1. Then Λ (T) Ω (T).
k k
≥ ⊆
Proof. Let ψ ,..., ψ be a complete set of orthonormal eigen-
1 N
{| i | i}
vectors for T with eigenvalues T ψ = λ ψ . Let λ Λ (T) and
j j j k
| i | i ∈
let P = k φ φ be a rank-k projection such that PTP = λP.
Then TφPφi=1|=iδih λi|for all i,j. Let A be a subset of 1,...,N with
i j ij
cardinhality| Ai = k 1. Choose a unit vector φ in th{e k-dimen}sional
| | − | i
subspace P = span φ ,..., φ that is perpendicular to all the
1 k
ψ for whiHch j A;{|andi so, |φ i=} z ψ with z 2 = 1.
| ji ∈ | i Pj∈/A j| ji Pj| j|
Then we have
(9) λ = Tφ φ = X zj 2λj conv λj : j / A ,
h | i | | ∈ { ∈ }
j∈/A
and it follows that λ belongs to Ω (T). (cid:4)
k
It is well-known and easy to verify that the numerical range of a
normal operator T coincides with the convex hull of its eigenvalues
(that is, Λ (T) = Ω (T)). In [6] the following conjecture was asserted
1 1
as a generalization of this fact.
6 M.–D. CHOI,J.A. HOLBROOK,D.W.KRIBS,K. Z˙YCZKOWSKI
Conjecture A. Let be an N-dimensional Hilbert space and let
H
k 1 be a positive integer. Then for every normal operator T ( ),
≥ ∈ L H
(10) Λ (T) = Ω (T).
k k
The Hermitian case [6] and the normal N 4 case [5] of the conjec-
≤
ture have been verified previously. We show that the general normal
case of Conjecture A can be reduced to the unitary case.
Proposition3.3. Conjecture A holds if and onlyif the conjecture holds
for all unitary matrices.
Proof. First note that for a fixed k, a standard translation argument
shows the statement Λ (T) = Ω (T) for all normal T (CN) is
k k
∈ L
equivalent to the statement 0 Λ (T) if and only if 0 Ω (T) for all
k k
normal T (CN). We focus ∈on the latter formulation∈.
∈ L
EverynormaloperatorT decomposesasT = T 0 ,whereT isnor-
1 m 1
⊕
malandinvertible. Thecasem k iseasily handled, soassume m < k.
≥
One can check that 0 Λ (T) if and only if 0 Λ (T ), and 0
k k−m 1
∈ ∈ ∈
Ω (T) if and only if 0 Ω (T ).Let λ ,...,λ bethe(non-
k k−m 1 1 N−m
zero) eigenvalues for T , a∈nd let U be the{unitary on CN}−m obtained
1
from the polar decomposition of T with eigenvalues λ1 ,..., λN−m .
1 {|λ1| |λN−m|}
By assumption we have Λ (U) = Ω (U), and hence zero belongs
k−m k−m
to both sets or neither set. Thus, we complete the proof by show-
ing that: (i) 0 Λ (T ) if and only if 0 Λ (U), and (ii)
k−m 1 k−m
∈ ∈
0 Ω (U) if and only if 0 Ω (T ).
k−m k−m 1
∈ ∈
Note that by invertibility we have T = UR = RU = √RU√R
1
where R = √T†T 0 is invertible with eigenvalues λ ,..., λ .
1 k−m
≥ {| | | |}
Thus, (i) follows from the more general principle that if T = X†SX
where X is invertible, then 0 Λ (T) if and only if 0 Λ (S). Indeed,
k k
∈ ∈
if P is a rank-k projection such that PTP = 0, then the rank-k range
projection Q of XP satisfies QSQ = 0.
For (ii), note that by definition 0 / Ω (T ) precisely when 0 does
k−m 1
∈
not belong to the convex hull of N k + m + 1 of the eigenvalues
−
λ ,...,λ . This is equivalent to the existence of a line passing
1 N−m
{ }
through the origin that does not meet this convex hull. By the same
argument, this geometric condition is equivalent to 0 / Ω (U). (cid:4)
k−m
∈
This result, combined with the motivation from quantum computing
discussed above, naturally leads to a focus on the unitary case of the
conjecture. We introduce the following nomenclature to delineate the
generic unitary subcases.
HIGHER-RANK NUMERICAL RANGES 7
Definition 3.4. We will use the notation Conj(N,k) to denote the
sub-conjecture of Conjecture A given by the statement Eq. (10) for
a given pair [N,k] and all unitary operators on N-dimensional Hilbert
space with non-degenerate spectrum. Further, we will say Conj(N,k) is
constructively verified for a given pair [N,k] if it is shown that Eq. (10)
holds for every unitary operator U on = CN, and if, whenever Λ (U)
k
H
is nonempty, for every λ Λ (U) a rank-k projection P can be explic-
k
∈
itly constructed such that PUP = λP.
k
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7 ?
8
9 ?
10
11 ?
12 ?
N 13 ? ?
14 ?
15 ?
16 ? ?
17 ? ?
18 ? ?
19 ? ?
20
21 ?
22 ?
23
24
25
means C(N,k) true constructively
means C(N,k) true vacuously (Ω empty)
k
means C(N,k) true nonconstructively
? means C(N,k) is unsettled
means C(N,k) true constructively but Ω may be empty
k
etc
Figure 1. Conj(N,k) for nondegenerate unitary U.
8 M.–D. CHOI,J.A. HOLBROOK,D.W.KRIBS,K. Z˙YCZKOWSKI
4. The Structure of Ω
k
In this section we analyse the geometric structure of the set Ω (U)
k
for a generic unitary U on N-dimensional Hilbert space with non-
degenerate spectrum. Let us first establish notation we will use for
the rest of the paper.
We shall consider the case of a unitary U with eigenvalues λ =
j
exp(iθ ), j = 1,...,N, such that 0 θ < θ < ... < θ < 2π.
j 1 2 N
≤
Thus, the eigenvalues λ are ordered counterclockwise around the unit
j
circle ∂D in C (we use D to denote the closed unit disc). For multiple
eigenvalues the numbering is arbitrary, but we choose an orthonormal
system of eigenvectors ψ such that
j
| i ∈ H
(11) U ψ = λ ψ .
j j j
| i | i
When appropriate we extend the numbering of the λ and ψ cycli-
j j
| i
cally: for example, λ means λ . Given integers i,j with i < j
N+1 1
i+N, let D(i,j,U) denote the compact convex subset of C bounde≤d
by the line segment from λ to λ and the counterclockwise circular arc
i j
from λ to λ ; recall our conventions about cyclical numbering of the
j i
λ . We interpret D(i,i+N,U) as λ .
j i
{ }
We first show that the form of Ω (U) is simpler than what Defini-
k
tion 3.1 suggests. In particular, Ω (U) is a filled convex polygon with
k
at most N sides.
Lemma 4.1. For all k 1 and every unitary U (CN), the set
≥ ∈ L
Ω (U) is the convex polygon given by
k
N
(12) Ωk(U) = \D(i,i+k,U).
i=1
Proof. Let Ω denote the intersection of Eq. (12). The cardinality of
the set
S = 1,2,...,N i+1,i+2,...,i+k 1
{ }\{ − }
(where integers are interpreted modulo N) is S = N k +1. Thus
| | −
Ω (U) conv( λ : j S ) D(i,i+k,U).
k j
⊆ { ∈ } ⊆
Since this holds for all i, we have Ω (U) Ω.
k
⊆
On the other hand, if S = N k + 1 = s we may write S =
| | −
i ,i ,...,i with 1 i < i < < i N and
1 2 s 1 2 s
{ } ≤ ··· ≤
s
conv( λj : j S ) = \D(ij,ij+1,U)
{ ∈ }
j=1
HIGHER-RANK NUMERICAL RANGES 9
(it is understood here that i = i ). Since between i and i there
s+1 1 j j+1
arei i 1integersthatareomittedfromS,wemusthavei i
j+1 j j+1 j
− − − ≤
k, so thatD(i ,i ,U) D(i ,i +k,U). It followsthatΩiscontained
j j+1 j j
⊇
in conv( λ : j S ) for each such S. Hence Ω (U) Ω, and equality
j k
{ ∈ } ⊇ (cid:4)
is verified.
The following containments are direct consequences of this result.
Corollary 4.2. For all U and all k, we have Ω (U) Ω (U).
k+1 k
⊆
Proof. In view of Lemma 4.1, we need only observe that, for any i,
D(i,i+k +1,U) D(i,i+k,U). (cid:4)
⊆
Corollary 4.3. If V is unitary and spec(U) spec(V), then Ω (U)
k
⊆ ⊆
Ω (V) for all k.
k
Proof. It is enough to treat the case where spec(V) = spec(U)
∪
λ and to arrange the geometric ordering so that spec(U) =
N+1
{ }
λ ,λ ,...,λ and spec(V) = λ ,λ ,...,λ . For 1 i N k
1 2 N 1 2 N+1
{ } { } ≤ ≤ −
we have D(i,i+ k,V) = D(i,i+ k,U); for N k < i N, we have
− ≤
D(i,i+k,V) D(i,i+k,U) (notethat theextended numbering within
⊇
spec(V) is done modulo N + 1); finally, D(N + 1,N + 1 + k,V)
D(N,N +k,U). Using Lemma 4.1, we see that Ω (V) Ω (U). (cid:4)⊇
k k
⊇
The following corollary points out that Conj(N,k) is easy to verify
when k divides N.
Corollary 4.4. Suppose that k divides N, with m = N/k. Then Ω (U)
k
is the intersection of k m–gons, and Conj(N,k) follows.
Proof. Let S = i,i+k,i+2k,...,i+(m 1)k (i = 1,2,...,k); these
i
{ − }
index sets partition 1,2,...,N . In view of Lemma 4.1, the m–gons
{ }
Gi = \ D(i,i+k,U)
j∈Si
intersect to form Ω (U). Now consider λ Ω (U). Since λ G for
k k i
∈ ∈
each i we may write
λ = Xtijλj
j∈Si
as a convex combination (t 0, t = 1). For i = 1,2,...,k,
ij ≥ Pj∈Si ij
let
φi = Xptij ψj ;
| i | i
j∈Si
10 M.–D. CHOI,J.A. HOLBROOK,D.W.KRIBS,K. Z˙YCZKOWSKI
1
0.5
0 (a)
−0.5
−1
−1 −0.5 0 0.5 1
1
0.5
(b)
0
−0.5
−1
−1 −0.5 0 0.5 1
Figure 2. Corollary 4.4 in action: (a) Ω (U) as inter-
3
section of 3 quadrilaterals when N = 12; (b) Ω (U) as
4
intersection of 4 triangles when N = 12.
clearly the φ are unit vectors, and they are orthogonal because the
i
| i
Si are disjoint. We see that (U λI) φi φi′ for all i,i′, so that =
− | i ⊥ | i V
span φ : i = 1,2,...,k satisfies (U λI) . We can thus define
i
{| i } − V ⊥ V
the rank-k projection P = k φ φ onto . Then P(U λI)P = 0
Pi=1| iih i| V −
and it follows that λ Λ (U). Together with Proposition 3.2, we have
k
verified Conj(N,k) in∈these cases. (cid:4)