The Inverse Eigenvalue Problem
for Quantum Channels

Michael M. Wolf1, David Perez-Garcia2

1 Niels Bohr Institute, 2100 Copenhagen, Denmark
2 Dpt. Análisis Matemático and IMI, Universidad Complutense de Madrid, 28040 Madrid, Spain

(Dated: May 26, 2010)

Given a list of complex numbers λ1, . . . , λn, when can it be the spectrum of a quantum channel, i.e., a
completely positive trace preserving map? We provide an explicit solution for the n = 4 case and show
that in general the characterization of the non-zero part of the spectrum can essentially be given in terms of
its classical counterpart—the non-zero spectrum of a stochastic matrix. A detailed comparison between the
classical and quantum case is given. We discuss applications of our findings in the analysis of time-series
and correlation functions and provide a general characterization of the peripheral spectrum, i.e., the set of
eigenvalues of modulus one. We show that while the peripheral eigen-system has the same structure for all
Schwarz maps, the constraints imposed on the rest of the spectrum change immediately if one departs from
complete positivity.

Contents

I. Introduction and outline 1

II. Basic spectral properties of quantum channels 2

III. The n = 4 case – qubit channels 3

IV. The non-zero part of the spectrum 6
A. The spectral set 6
B. Moments and the non-zero part of the spectrum 7

V. Cycles and the peripheral spectrum 10

VI. Classical vs. quantum inverse eigenvalue problems 12
A. The classical inverse eigenvalue problem 13
B. Comparing classical and quantum 13

VII. Applications for time series and correlation functions 14

References 15

I. INTRODUCTION AND OUTLINE

Quantum channels provide general input-output relations for quantum mechanical evolutions.
If the input space equals the output space, we can assign a spectrum to each quantum channel.
The spectral properties play a role wherever successive powers of the channel are applied. These
may describe discrete time-evolution or they may appear in particular representations of quantum
spin chains. Apart from the most basic facts which coincide with their ‘classical’ counterparts for
stochastic matrices, not much is known about the spectral properties of completely positive maps.

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2

The present work aims at a deeper analysis, beyond Perron-Frobenius theory, and provides nec-
essary, and in some cases also sufficient, conditions for a list of numbers to be the spectrum of
a quantum channel. That is, we address the ‘inverse eigenvalue problem’ for completely positive
maps—occasionally being content with plain (not complete) positivity.

After having collected some basic results in Sec.II, Sec.III provides a rather exhaustive analysis of
qubit-maps—a complete characterization of the spectra of (completely) positive qubit maps is given.
In Sec.IV the non-zero part of the spectrum of general (completely) positive maps is investigated and
close analogies to the classical case of stochastic matrices are found. Sec.V provides a detailed study
of the peripheral spectrum (i.e., eigenvalues whose modulus equals the spectral radius) and of the
corresponding eigenvectors. Before outlining some applications in the analysis of time-series and
correlation functions in Sec.VII we compare the quantum and the classical world, i.e., the spectral
properties of quantum channels and those of stochastic matrices in Sec.VI.

II. BASIC SPECTRAL PROPERTIES OF QUANTUM CHANNELS

We begin with fixing some notation and recalling basic results. Let T : Md(C) → Md(C) be
a linear map on the space of complex valued d × d matrices. T is said to be positive if it maps the
cone of positive semidefinite matrices into itself. If T is positive, then it is in particular Hermiticity
preserving, i.e., T (A)† = T (A†) for all A ∈ Md(C). The adjoint map T ∗ is defined by imposing
tr [AT ∗(B)] = tr [T (A)B] for all A,B ∈ Md(C). T is trace-preserving iff T ∗ is unital, meaning
that T ∗(1) = 1. A linear map onMd(C) is completely positive iff T ⊗ idd is positive. A completely
positive and trace-preserving linear map is referred to as quantum channel.

As a vector spaceMd(C) is isomorphic to Cd2 so that T can be represented as a d2 × d2 matrix
for which we will write T̂ . Fixing a Hilbert-Schmidt orthonormal basis inMd(C), i.e., a collection
of d2 operators {Ei ∈ Md(C)} with tr

[
E†iEj

]
= δij , we get T̂ij = tr

[
E†i T (Ej)

]
. If T is

Hermiticity preserving, then T ∗ is represented by the adjoint matrix T̂ †.
When analyzing the spectrum of a map we distinguish between the spectrum as a set, where each

element only appears once, and as a multiset (unordered n-tuple), where elements appear according
to their algebraic multiplicity. More precisely, the spectral set of T is the set of complex numbers λ
for which T−λid is not invertible. The spectrum (as a multiset) is the list of roots of the characteristic
polynomial det[T̂ − λ1] where each root appears with a multiplicity according to its degeneracy.
We will write spec(T) if we refer to the multiset and {spec(T)} if the spectral set is meant.

From Hermitian conjugation of the eigenvalue equation T (X) = λX we see that the eigenvalues
of Hermiticity preserving maps are either real or come in complex conjugate pairs. This holds
even if we take multiplicities into account. Since spec(T) = spec(T∗), every trace-preserving
map has an eigenvalue one. If the map is positive in addition, then its spectral radius is one, i.e.,
max{|λ|

∣∣λ ∈ spec(T)} = 1.
A positive map is irreducible [EHK78] iff the spectral radius is a non-degenerate eigenvalue and

the corresponding eigen-vector is positive definite. If in addition there is no other eigenvalue whose
modulus matches the spectral radius then the map is called primitive [SPGWC09].

The following shows that a characterization of the spectra of (completely) positive maps is essen-
tially independent of whether or not the maps are required to be trace-preserving:

Proposition 1 (Similarity with trace-preserving maps) Let T : Md(C) → Md(C) be a (com-
pletely) positive map with spectral radius % > 0. There is a trace-preserving (completely) positive
map T ′ onMd(C) such that spec(T) = % spec(T′).

Proof. Define a map Tε : X 7→ T (X) + εtr [X]1 and denote its spectral radius by %ε. Following
[EHK78] Tε is irreducible for all ε > 0 and for ε → 0 we have that spec(Tε) → spec(T) continu-
ously. Irreducibility implies the existence of a positive definite P = Tε(P )%−1ε > 0. Hence the map



3

T ′ε(X) := %−1ε P−1/2Tε(P
1/2XP 1/2)P−1/2 satisfies spec(Tε) = spec(T′ε)%ε and is (completely)

positive and unital. Since these maps form a compact set, the limit T ′ = limε→0(T ′∗ε ) exists and has
all desired properties.

In the following we will consider trace-preserving maps only.

III. THE n = 4 CASE – QUBIT CHANNELS

We continue the discussion with completely positive qubit maps T : M2(C) → M2(C) which
allow for an exhaustive analysis. Some of the results derived in this section will be the basis for
the general characterization of the non-zero part of the spectrum in Sec.IV A. We will first recall
and derive some preparatory results and then provide an explicit solution of the inverse eigenvalue
problem in Thm.1.

Choosing normalized Pauli matrices as operator basis (with σ0 = 1) we can represent T as a
4 × 4 matrix T̂ij := tr [σiT (σj)] /2. If T is Hermiticity preserving, then T̂ is real, and if T is
trace preserving, then (T1,j) = (1, 0, 0, 0). Since the maps of our interest satisfy both, they can be
represented as

T̂ =

(
1 0
v ∆

)
, (1)

where ∆ is a real 3× 3 matrix and v ∈ R3. The spectrum of T is thus the union

spec(T) = {1} ∪ spec(∆). (2)

The Jamiolkowski state corresponding to T̂ is given by τ = 1
4

∑
ij T̂ijσi ⊗ σTj so that v describes

its reduced density operator and ∆ its correlations. The (generally cumbersome) conditions for
T̂ to correspond to a completely positive map have to be read off from τ ≥ 0 (cf. [RSW02]).
The parametrization allows, however, for a nice geometric interpretation: parameterizing a density
operator via ρ = (1 +

∑3
k=1 xkσk)/2, i.e., in terms of a vector x ∈ R3 within the Bloch ball

||x||2 ≤ 1, the action of T as parametrized in Eq.(1) is a simple affine transformation

x 7→ v + ∆x. (3)

From here conditions for T to be positive are readily derived as the vector has to stay within the
Bloch ball. As the center of the Bloch ball corresponds to the maximally mixed state, v = 0 holds iff
T is unital. In this case positivity is equivalent to ||∆||∞ ≤ 1. The following is a useful proposition
which allows us to restrict ourselves to the unital case:

Proposition 2 (Reduction to unital maps) Let T : M2(C) → M2(C) be a trace-preserving lin-
ear map with ∆ij := tr [σiT (σj)] /2 the lower-right submatrix (i.e., i, j = 1, 2, 3) of the matrix
representation T̂ . Then the map defined by T̂ ′ := 1⊕∆ is unital, trace-preserving and has the same
spectrum as T . Moreover, it is (completely) positive if T is.

Proof. By construction T ′ is trace-preserving and unital and has the same spectrum as T , so it
remains to show that (complete) positivity is preserved when going from T to T ′. To this end
we use that D := diag(1,−1,−1,−1) is the matrix representation of time-reversal, i.e., matrix
transposition in some basis. A look at the Choi matrix reveals that the map DT̂D is completely
positive if T̂ is. Similarly, if T is merely positive, then DT̂D is positive as well, since it is a



4

concatenation of positive maps. Consequently, the convex combination (T̂+DT̂D)/2 = T̂ ′ inherits
the property of being (completely) positive from T .1

For a qubit map of the form in Eq.(1) with v = 0 complete positivity can be expressed in
terms of the singular values of ∆ and its determinant. In order to understand this, suppose that
one acts with a unitary before and another one after applying the map T so that the overall action
is ρ 7→ U2T (U1ρU

†
1 )U†2 . The matrix representing this concatenation is then given by the prod-

uct (1 ⊕ O2)T̂ (1 ⊕ O1), where Oi ∈ SO(3) are real rotations of the Bloch sphere. This reflects
the two-to-one group homomorphism SU(2) → SO(3). Hence, we can use this to diagonalize
∆ → diag(s1, s2, s3) without changing the complete positivity property. Moreover, we can choose
the Oi’s such that the components of s follow a particular order and satisfy s1 ≥ s2 ≥ |s3|. Then
a generally necessary and for unital maps also sufficient condition for complete positivity is that
[VV02]

s1 ≤ 1 and s1 + s2 ≤ 1 + s3. (4)

These conditions can, alternatively, be summarized by “s ∈ T ” [RSW02] where T ⊂ R
3 is the

tetrahedron whose corners have coordinates with components si = ±1 satisfying s1s2s3 = 1. Note
that s ∈ T is independent of the ordering and sign pattern attained from a particular choice of the
Oi ∈ SO(3).

Since our interest lies, following Eq.(2), in the eigenvalues of ∆ but the constraints are essentially
given in terms of its singular values we need to relate singular values and eigenvalues. This is the
content of a classic result of Weyl [Wey49] and Horn [Hor54] of which we need a real version
provided in [LM01]. Applied to our context this gives:

Lemma 1 Given λ ∈ C3 and s ∈ R3, there is a ∆ ∈ M3(R) with spec(∆) = λ and O1∆O2 =
diag(s1, s2, s3) for some Oi ∈ SO(3) iff the following three requirements are fulfilled:

1. λ is, as a multiset, invariant under complex conjugation, i.e., either λ ∈ R3 or there is a single
real λi and a complex conjugate pair.

2.
∏3
i=1 si =

∏3
i=1 λi.

3. |s↓1s
↓
2| ≥ |λ

↓
1λ
↓
2| and |s↓1| ≥ |λ

↓
1| where ↓ refers to a (re-)ordering with decreasingly ordered

absolute values, i.e., |s↓i | ≥ |s
↓
i+1| and similarly |λ↓i | ≥ |λ

↓
i+1|.

Proof. Necessity: 1. follows from ∆ being real, 2. follows from both sides being equal to det(∆),
and 3. follows from Weyl’s theorem [Wey49] since the |si| are the singular values of ∆.

Sufficiency: Following [LM01] the conditions 1.-3. guarantee the existence of a matrix M ∈
M3(R) with spec(M) = λ and singular values |si|. The latter implies that there are Õi ∈ SO(3)

s.t. Õ1MÕ2 = diag(s̃1, s̃2, s̃3) where |s̃i| = |si|. Moreover, since
∏
i si =

∏
i s̃i (exploiting

condition 2.) we have that s and s̃ differ either by no, or by two signs. In both cases we can set
∆ := M . If two signs are different we simply multiply one of the Õi’s by a diagonal matrix like
diag(1,−1,−1) or a permutation thereof in order to obtain the Oi’s.

Now we are prepared for solving the inverse eigenvalue problem for qubit channels:

Theorem 1 (Spectra of qubit channels) Given a multiset Λ ∈ C4 the following statements are
equivalent:

1 Note that in higher dimensions the strategy of the proof fails: the higher-dimensional analogue of the above D maps
ρ 7→ 2tr [ρ]1/d− ρ, which is no longer a positive map if d > 2. Similarly, DT̂D then can fail to be completely positive
even if T is so.



5

1. There is a trace-preserving completely positive linear map T : M2(C) →M2(C) such that
spec(T) = Λ.

2. There is a unital and trace-preserving completely positive linear map T :M2(C)→M2(C)
such that spec(T) = Λ.

3. Λ = {1}∪λ where λ ∈ C3 is, as a multiset, closed under complex conjugation. Furthermore,
if we define s ∈ R

3 by si := λi if λi ∈ R and si := |λi| otherwise, then (with T the
tetrahedron defined below Eq.(4))

s ∈ T . (5)

Proof. The equivalence 1.⇔2. is a direct consequence of Prop.2.
3.⇒2.: by Lemma 1 there exists a ∆ ∈ M3(R) with spec(∆) = λ and O1∆O2 =

diag(s1, s2, s3) for some Oi ∈ SO(3). The unital and trace-preserving map defined via T̂ = 1⊕∆
has thus spec(T) = {1} ∪ λ = Λ and is completely positive due to Eq.(5).

2.⇒3.: as discussed above, the matrix representation of the channel has the form T̂ = 1⊕∆ and
spec(T) = {1} ∪ λ = Λ with λ = spec(∆). Our aim is to argue that diag(1, s1, s2, s3) represents
a valid quantum channels as well so that Eq.(5) has to be fulfilled. Making use of the real Schur
decomposition we can w.l.o.g. assume that ∆ has one of the following forms λ1 ∗ ∗

λ2 ∗
λ3

 ,

 λ1 ∗ ∗
a b
−b a

 , (6)

where ∗ denotes arbitrary entries and the ‘real form’ on the left is such that λi ∈ R and the ‘complex
form’ on the right is such that λ1 ∈ R and λ2,3 = a± ib with a, b ∈ R.

Let us consider the real form first. Define a new matrix ∆′ ∈M3(R) via the convex combination

∆′ :=
1

|S|
∑
Di∈S

DiΛDi, (7)

where S is the set of four diagonal matrices Di which have diagonal entries ±1 and satisfy
det(Di) = 1 so that ∆′ = diag(λ1, λ2, λ3). Note that T̂ ′ := 1 ⊕ ∆′ represents a valid quan-
tum channel since Eq.(7) corresponds to a convex combination of channels which merely differ by
unitary conjugations (represented by 1 ⊕ Di). Therefore λ ∈ T and since in the real case s = λ
Eq.(5) follows.

Consider now the complex form. Again we can get rid of the ∗-entries by convex combination, if
we choose S = {1,diag(1,−1,−1)} this time. Since the two-by-two block matrix in the lower right
corner has singular values |λ2|, |λ3| (which equal s2 and s3) there are special orthogonal matrices
O1, O2 such that O1∆′O2 = diag(s1, s2, s3). Consequently, s ∈ T because diag(1, s1, s2, s3)
represents a valid quantum channel since it can be obtained from T by convex combinations and
concatenating with unitary conjugations.

A counterpart of this theorem for positive maps can easily be derived:

Theorem 2 (Spectra of positive qubit maps) . Let T :M2(C)→M2(C) be a positive and trace-
preserving linear map. Then the multiset Λ := spec(T) satisfies

1. 1 ∈ Λ and Λ is invariant under complex conjugation,

2. |λ| ≤ 1 for all λ ∈ Λ.

Conversely, if a multiset Λ of four complex numbers satisfies these two conditions, then there is a
unital and trace-preserving positive linear map T onM2(C) such that spec(T) = Λ.



6

Proof. As discussed before, conditions 1. and 2. are necessary for positive trace-preserving linear
maps in any dimension. For the converse we construct a map via T̂ := 1 ⊕ ∆ with ∆ of the form
in Eq.(6), depending on whether or not Λ = {1, λ1, λ2, λ3} contains a complex conjugate pair, and
∗ = 0 in both cases. By construction we have spec(T) = Λ and T is trace-preserving and unital.
Moreover, it is positive since ||∆||∞ ≤ 1 guarantees that T maps the Bloch sphere into itself.

IV. THE NON-ZERO PART OF THE SPECTRUM

A. The spectral set

In this subsection we will regard the spectrum as a set. That is, {spec(T)} is a set of complex
numbers which contains each element only once so that degeneracies are not taken into account.
This considerably simplifies the inverse eigenvalue problem if we allow for an additional null space:

Theorem 3 (Spectral sets without zero) Let Λ = {λ1, . . . , λN} be a set of non-zero complex num-
bers such that

1. it is closed under complex conjugation: Λ = Λ̄,

2. it contains one: 1 ∈ Λ, and

3. all elements are contained in the unit disc: |λ| ≤ 1, ∀λ ∈ Λ.

Then there exists a completely positive and trace-preserving linear map T : Md(C) → Md(C)
with d ≤ 2 max{N − 1, 1} such that

{spec(T)}\{0} = Λ. (8)

Conversely, if Λ is the spectral set of a positive and trace-preserving linear map, then conditions
1.-3. have to be fulfilled.

Proof. For each λj ∈ Λ consider the multiset Λj := {1, 1, λj , λ̄j}. According to Thm.1 there is
a quantum channel Tj : B(Hj) → B(Hj), where Hj ' C2, such that spec(Tj) = Λj. Defining a
Hilbert space H :=

⊕
j Hj and isometries Vj : H → Hj we can construct a quantum channel on

B(H) via

T (ρ) :=
∑
j

V †j Tj

(
VjρV

†
j

)
Vj . (9)

By construction λj ∈ spec(T) for all λj ∈ Λ. Moreover, since T
(
|ψ〉〈φ|

)
= 0 for all ψ ∈ Hk,

φ ∈ Hl with k 6= l, the kernel completes the spectrum of T such that indeed

{spec(T)} = Λ ∪ {0}. (10)

What is d = dimH? Assume that Nr and Nc are the numbers of real elements different from 1 and
complex conjugate pairs in Λ respectively, i.e., N = Nr + 2Nc + 1. Then our construction requires
d = 2(Nr +Nc) ≤ 2N − 2 degrees of freedom if N ≥ 2 and d = 2 if Λ = {1}.

The converse statement in the theorem is just a restatement of the general properties discussed in
Sec.II.



7

B. Moments and the non-zero part of the spectrum

From now on we consider the spectrum again as a multiset. That is, every eigenvalue appears
according to its algebraic multipliticity. A central notion in the following discussion of the non-zero
part of the spectrum is that of moments. Given a multiset of complex numbers Λ = {λ1, . . . , λn}
the k’th moment of Λ is defined via

µk(Λ) :=

n∑
j=1

λkj , k ∈ N. (11)

The collection of moments determines the non-zero part of Λ and viceversa. Obviously, all moments
are real if Λ = Λ̄. Conversely, positivity of all moments µk(Λ) ≥ 0, k ∈ N implies Λ = Λ̄. Positivity
also guarantees that Λ contains a positive element % ∈ Λ for which ∀λ ∈ Λ : |λ| ≤ % holds [Fri09].

Note that if Λ is the spectrum of some linear map with matrix representation T̂ , then µk(Λ) =

tr
[
T̂ k
]
. If T is a linear map on matrices, this can be rewritten as follows:

Lemma 2 (Moments of maps on matrices) Let T :Md(C)→Md(C) be a linear map, then

tr
[
T̂ k
]

= 〈Ω
∣∣(T k ⊗ id

)(
|Ω〉〈Ω|

)∣∣Ω〉, (12)

where |Ω〉 :=
∑d
i=1 |ii〉. Moreover, if T is completely positive and has Kraus operators {Kj} then

tr
[
T̂ k
]

=
∑

j1,...,jk

∣∣∣tr [Kj1 · · ·Kjk ]
∣∣∣2. (13)

Proof. Both statements are elementary observations when using matrix units |i〉〈j| as basis elements
for computing the trace in Hilbert-Schmidt space as

tr
[
T̂ k
]

=

d∑
i,j=1

〈i|T k
(
|i〉〈j|

)
|j〉.

This leads us to the main theorem of this section:

Theorem 4 (Non-zero part of the spectrum) Let T : Md(C) → Md(C) be a trace-preserving
and completely positive linear map. Then µk

(
spec(T)

)
≥ 0 for all k ∈ N.

Conversely, consider a multiset Λ = {λ1, . . . , λn} of non-zero complex numbers such that
max{|λ|

∣∣ λ ∈ Λ} = 1 is attained for a unique element λp ∈ Λ. If for all k,m ∈ N

µk
(
Λ
)
≥ 0, and (14)

µm
(
Λ
)
> 0 implies µkm

(
Λ
)
> 0, (15)

then there exists a primitive trace-preserving completely positive linear map T such that Λ is the
non-zero part of the spectrum of T , i.e., spec(T)\{0} = Λ.

Proof. Positivity of the moments follows from Lemma 2 and the fact that T is assumed to be
completely positive. For the converse statement we exploit a highly non-trivial result of [BH91].
This states that Eqs.(14,15) together with the uniqueness of λp are necessary and sufficient for the
existence of someN ∈ N and a primitive non-negative matrixM ∈MN (R+) with spec(M)\{0} =
Λ. From this one can easily construct a stochastic matrix with the same spectrum: denote by 〈L| =



8

〈L|M the left Perron eigenvector and define a diagonal matrix X with entries Xii = 〈L|i〉 for a set
of orthonormal basis vectors {|i〉}. Then S := XMX−1 is a primitive stochastic matrix which has
the same spectrum as M . Now define a map T :MN (C)→MN (C) via

T (ρ) :=

N∑
i,j=1

Sij 〈j|ρ|j〉 |i〉〈i|. (16)

If S|p〉 = λ|p〉, then T (ρ) = λρ for ρ =
∑
i pi|i〉〈i| and thus spec(T) ⊇ Λ, i.e., theN eigenvalues of

S (taking algebraic multiplicities into account) appear in the spectrum of T . Moreover, T
(
|i〉〈j|

)
=

0 for all i 6= j gives rise to a kernel of dimension N(N − 1) which thus completes the spectrum so
that indeed Λ is the entire non-zero part of the spectrum of T . By construction T is trace-preserving
and completely positive. In fact, it is an entanglement breaking quantum channel. Primitivity of T
follows from primitivity of S: since S has an entrywise positive fixed point, T has a positive definite
fixed point. Together with the fact that T has a unique eigenvalue of magnitude one this implies
primitivity [SPGWC09].

Some remarks on the above theorem are in order:

1. Positivity does not imply positive moments. Take any positive map T on Md(C) which is
not completely positive. Then there is a ψ ∈ Cd2 such that 〈ψ|(T ⊗ id)(|Ω〉〈Ω|)|ψ〉 < 0.
We can always write |ψ〉 = (Y ⊗ 1)|Ω〉 for some Y ∈ Md(C). Using Eq.(12) the map
T ′(ρ) := Y †T (ρ)Y will then be such that tr

[
T̂ ′
]
< 0 although T ′ is positive. The same

holds for n-positive maps with n < d.

2. The required number of zeros. It was shown in [BH91] that even for the case n = 4 there
cannot be an upper bound on the number N −n of zeros which one has to add in order realize
a given Λ (which satisfies all necessary requirements) as the non-zero part of the spectrum
of some N × N non-negative matrix M . A priori, this does not imply that such a bound
does not exist when the realization is in terms of a quantum channel. In fact, the inequality in
Eq.(27) from which the necessity of a large number of zeros is usually derived in the classical
case, fails to hold in the quantum case. And indeed, the additional freedom provided by
quantum mechanics often allows for much more efficient realizations — see [WPG09] and
the discussion in Sec. VI for more details. We will, however, see below Thm.6 that also in the
quantum context ancillary zeros can be necessary in order to realize a certain spectrum.
In the classical context of non-negative matrices, it has been shown in [JLL96] that if Λ is
the non-zero part of the spectrum of some M ∈ MN (R+) with rank(M) = r, then there is
an M ′ ∈ Mr2(R+) for which spec(M′)\{0} = Λ. That is, the unboundedness of N in the
classical context is related to large or numerous nilpotent Jordan blocks.

3. Eq.(15) is not necessary. Note that in the converse part of Thm.4 all the imposed conditions
on Λ but Eq.(15) are necessary. While for the analogous classical statement for non-negative
matrices in [BH91] Eq.(15) is evidently necessary, too, it fails to be so in the quantum context.
Let us construct an example of a primitive quantum channel where Eq.(15) does not hold:
Define the following d+ 1 Kraus operators Kj ∈Md(C):

Kj :=
1√
2

{
|j〉〈j + 1 mod d| for j = 0, . . . d− 1,∑d−1
k=0 exp (πik/d) |k〉〈k| for j = d.

(17)

The map T (ρ) :=
∑d
j=0KjρK

†
j is then completely positive, trace-preserving and unital.

Moreover, following Eq.(13) we get tr
[
T̂
]
> 0 although tr

[
T̂ 2
]

= 0 if d > 2. Primitivity of
the map T is guaranteed since for n ≥ d we have

span
{
Kj1 · · ·Kjn

}
= Md(C) (18)



9

which implies primitivity by virtue of [SPGWC09].

Although tr
[
T̂ 2
]

= 0 in the above example, we note that primitivity generally implies that

tr
[
T̂ k
]
> 0 for all k ≥ p where p depends on the dimension only. This follows from

Wielandt’s inequality which in the classical case of stochastic matrices in Md(R+) yields
p = d2−2d+2 and in the context of quantum channels onMd(C) it gives p = (d2−2)(d2−1)
[SPGWC09].

4. Classical vs. quantum channels. Consider a matrix representation T̂ of a map T :Md(C)→
Md(C) expressed in terms of matrix units, i.e., 〈k, l|T̂ |i, j〉 := 〈l|T

(
|i〉〈j|

)
|k〉 for i, j, k, l =

1, . . . , d. Then the d × d matrix S defined via Sij := 〈ii|T̂ |jj〉 is entry-wise non-negative
if T is a positive map and it is stochastic if in addition T is trace-preserving. That is, every
quantum channel T̂ contains a stochastic matrix S as a principal submatrix. Eq.(16) reverses
this: it defines a quantum channel from this stochastic submatrix by setting all other matrix
elements of T̂ to zero.

The close relation between the non-zero spectra of quantum channels and of stochastic matrices
implied by Eq.(16) allows to exploit other results from the classical world. In the following we
derive in this way two sufficient conditions for realizable spectra for which a bound on the number
of ancillary zeros can be given:

Theorem 5 (Non-zero spectrum with non-positive real parts) Let Λ = {λ1, . . . , λn} be a multi-
set of non-zero complex numbers with λ1 = 1 and Re(λj) ≤ 0 for all j ≥ 2, and let d ∈ N be the
smallest number such that µ1(Λ)2 ≤ dµ2(Λ). If Λ = Λ̄ and

µ1(Λ) ≥ 0 and µ2(Λ) > 0, (19)

then there exists a primitive, completely positive and trace-preserving linear map T : Md(C) →
Md(C) such that Λ = spec(T)\{0}.

Proof. The theorem is a direct consequence of its classical counterpart derived in [LS06] inserted
into Eq.(16). As discussed already in the proof of Thm.4 primitivity is carried over from the classical
case.

Theorem 6 (Real non-zero spectrum for n = 4) Let Λ = {λ1, . . . , λ4} be a multiset of non-zero
real numbers. If (i) Λ = Λ̄, (ii) for all j : |λj | ≤ 1 ∈ Λ and (iii)

∑4
j=1 λj ≥ 0, then there

exists a completely positive and trace-preserving linear map T : M4(C) → M4(C) such that
Λ = spec(T)\{0}.

Proof. Following [LL78] the stated conditions are sufficient for the existence of a non-negative (and
by similarity transformation stochastic) matrix with spectrum Λ. Using Eq.(16) again this gives the
desired result.

Thm.6 together with Thm.1 shows that ancillary zeros can be necessary to make a certain spectrum
realizable. Consider for instance Λ = {1, 1, 1,−1}. Then Thm.1 implies that this cannot be the
spectrum of a quantum channel onM2(C) whereas Thm.6 tells us that it can be the non-zero part
of the spectrum of a quantum channel onM4(C).

Finally, we use the similarity between the classical and the quantum case to characterize the non-
zero part of the spectrum of quantum channels with ‘full Kraus rank’. The latter means that the
complex linear span of the Kraus operators of a quantum channel on Md(C) is the entire matrix
spaceMd(C). Equivalently, the Choi matrix (or Jamiolkowski state) has full rank. This condition
can be seen as the quantum counterpart to entry-wise positivity of stochastic matrices. Note that, in
particular, every quantum channel with full Kraus rank is primitive.



10

Theorem 7 (Non-zero spectrum under full Kraus rank) Let Λ = {1, λ2 . . . λn} be a multiset of
non-zero complex numbers. There is a completely positive and trace-preserving linear map T with
full Kraus rank such that Λ = spec(T)\{0} iff |λj | < 1 for all j ≥ 2 and µk(Λ) > 0 for all k ∈ N.

Proof. Necessity of the conditions is implied by Eq.(12) and the general discussion in Sec.II.
Following [BH91] the stated conditions are sufficient for the existence of an entry-wise positive
matrix M with Λ = spec(M)\{0}. As mentioned in the proof of Thm.4 this can, by similarity
transformation, be made a stochastic matrix S which is again entry-wise positive. Inserting this
into Eq.(16) the latter guarantees full Kraus rank since the Choi matrix of the constructed quantum
channel reads (

T ⊗ id
)(
|Ω〉〈Ω|

)
=
∑
ij

Sij|i〉〈i| ⊗ |j〉〈j| ≥ s1, (20)

where s = min{Sij} > 0.

V. CYCLES AND THE PERIPHERAL SPECTRUM

We will now have a closer look at the space corresponding to the peripheral spectrum (i.e., eigen-
values which are phases) of a positive linear map T : Md(C) → Md(C) which is assumed to be
either trace-preserving or unital. Denote the complex linear span of all the respective eigenspaces
by

XT := span
{
X ∈Md(C)

∣∣∃ϕ ∈ R : T (X) = eiϕX
}
, (21)

for which we will simply write X if the dependence on T is clear from the context. The questions
which we address first are: what’s the structure of X ? what’s the structure of T restricted to X ? and
what does this imply for the peripheral spectrum?

Recall that if T is trace-preserving and positive, there cannot be any non-trivial Jordan-block
associated to an eigenvalue of modulus one.

The following shows that X is a ∗-algebra (w.r.t. a modified product) and that T acts on it either
by permuting blocks or by unitary conjugation on subsystems. The property which we use in order
to prove the assertion lies in between positivity and complete positivity: we call a positive trace-
preserving map T on Md(C) a Schwarz map if T ∗(A†A) ≥ T ∗(A†)T ∗(A) holds for all A ∈
Md(C). This is in particular true if T is completely positive.2

Theorem 8 (Structure of cycles) Let T :Md(C)→Md(C) be a trace-preserving Schwarz map.

1. There exists a decomposition of the Hilbert space Cd = H0 ⊕
⊕K

k=1Hk into a direct sum of
tensor products3 Hk = Hk,1 ⊗Hk,2 and positive definite density matrices ρk acting on Hk,2
such that

XT = 0⊕
K⊕
k=1

Mdk ⊗ ρk, (22)

2 In fact, 2-positivity, i.e., positivity of T ⊗ id2, is sufficient but not necessary for T to be a Schwarz map.
3 Note that we may have dimH0 = 0 as well as dimHk,i = 1. Moreover, the direct sum in the decomposition does not

necessarily correspond to a block structure in computational basis – just in some basis.



11

whereMdk is a full complex matrix algebra onHk,1 of dimension dk := dim(Hk,1). That is,
for every X ∈ XT there are xk ∈ B(Hk,1) such that

X = 0⊕
⊕
k

xk ⊗ ρk. (23)

2. There exist unitaries Uk ∈ B(Hk,1) and a permutation π, which permutes within subsets
of {1, . . . ,K} for which the corresponding Hk,1’s have equal dimension, so that for every
X ∈ XT represented as in Eq.(23)

T (X) = 0⊕
K⊕
k=1

Ukxπ(k)U
†
k ⊗ ρk. (24)

Proof. The basic ingredients of the proof are those which already appeared in [WC08] where pos-
itive maps with unit determinant were characterized. Exploiting Dirichlet’s lemma on Diophantine
approximations4 we can find an ascending subsequence ni ∈ N so that limi→∞ Tni converges to
a map I with eigenvalues which are either one or zero. Clearly, I is again a trace-preserving, posi-
tive linear map which satisfies the Schwarz inequality since all these properties are preserved under
concatenation. Moreover, XT = XI is the fixed-point space of I . So, statement 1. of the theorem
follows from the structure of the space of fixed points provided in [Lin99].

2. The map T−1 := limi→∞ Tni−1 is the inverse of T on X since by construction T−1T = I .
Hence, both T and T−1 map X → X in a bijective way. The crucial point here is that T−1 is a again
a positive, trace-preserving map since it is constructed as a limit of such maps. To understand the
consequences, consider any pure state in X , i.e., a density operator σ ∈ X which has no non-trivial
convex decomposition within X . Then the image of σ under T has to be a pure state as well: assume
this is not the case, i.e., T (σ) =

∑
i λiσi is a non-trivial convex decomposition into states σi ∈ X .

Then applying T−1 to this equation leads to a contradiction since σ =
∑
i λiT

−1(σi) is not pure.
Consequently, both T and T−1 map pure states in X onto pure states. Note that a pure state can
only have support in one of the K blocks, for instance σ = x ⊗ ρk where x ∈ B(Hk,1) is a rank
one projection. Now we know that T (σ) = x′ ⊗ ρk′ for some k′ and some rank one projection
x′ ∈ B(Hk′,1). By continuity and the fact that T is a bijective linear map on X , we have that within
X : (i) every element of block k is mapped to an element of the block k′ (i.e., k′ does not depend on
x), and (ii) the spacesHk,1 andHk′,1 must have equal dimension. Therefore, there is a permutation
π which permutes blocks with equal dk so that X ∈ X is mapped to

T (X) = 0⊕
⊕
k

Tk(xπ(k))⊕ ρk,

with some linear maps Tk : Mdk → Mdk . Since the latter have, together with their inverses,
to be positive and trace-preserving they must be either matrix transpositions or unitary conjuga-
tions [WC08]. Matrix transpositions are, however, ruled out by the requirement that T and thus each
Tk is a Schwarz-map.

The action of T on XT derived in Eq.(24) now determines the possible peripheral spectra:

Theorem 9 (Peripheral spectrum) Let T : Md(C) → Md(C) be a trace-preserving Schwarz
map. There are integers nc, dc ∈ N satisfying

∑
c ncdc ≤ d, integers mc ∈ Znc and vectors

4 Dirichlet’s lemma states the following [Sch80]: let x ∈ R
m and q > 1 any integer. Then there exist integers

n, p1, . . . , pm such that 1 ≤ n ≤ qm and |xkn− pk| ≤ 1/q for all k.



12

µ(c) ∈ Cdc whose components are phases (i.e., ∀k : |µ(c)
k | = 1) so that the peripheral spectrum of

T (including algebraic multiplicity) is given by the
∑
c ncd

2
c numbers

µ
(c)
k µ̄

(c)
l e

2πimc
nc . (25)

Conversely, given such a set of numbers there is a completely positive, trace-preserving and unital
linear map T onMd(C) with d =

∑
c ncdc so that its peripheral spectrum coincides with the given

numbers and it has no other non-zero eigenvalue.

Proof. First note that the eigenvalue equation T (X) = λX for X ∈ XT is, via Thm.8, equivalent
to

∀k : xπ(k) = λU†kxkUk. (26)

Assume for the moment that π is a cycle of length nc, i.e., after relabeling we have k ∈ Znc and
π(k) = k + 1 mod nc – the general case will later reduce to this one by decomposing a general
permutation into cycles. All Uk’s and xk’s now act on spaces of equal dimension, dc say. For given
Uk’s and λ the eigenvalue equation determines each xk from x0 and leads to the additional constraint
x0 = λncŨ†x0Ũ with Ũ := U0U1 · · ·Unc−1. Now define any unitary U on Cdc for which Unc = Ũ
and consider any of the d2c solutions of the eigenvalue equation y = µU†yU , y ∈ B(Cdc). Then
x0 = y and λ = µ exp [2πim/nc] leads to a solution of Eq.(26) for everym ∈ Znc . By construction
the eigenvectors X of T which correspond to different m’s will be orthogonal, so that in total the
construction leads ncd2c and thus all solutions. If µ1, . . . , µdc are the eigenvalues of U , then the
multiset of eigenvalues λ is given by

{
µkµ̄j exp [2πim/nc]

}
with k, j = 1, . . . , dc and m ∈ Znc .

The case of a general permutation now follows by decomposing it into cycles (labeled by c and
of possibly different lengths nc, which implies that

∑
c nc = K) and observing that the peripheral

spectrum of T is just the union of the spectra of all the ’cycle-building-blocks’.
In order to prove the converse we construct a map T guided by the structure appearing in Thm.8

(but now with dimHk,2 = 1). The desired T is obtained as a concatenation of three types of maps:
a pinching onto the

∑
c ncdc blocks of Cd =

⊕
c Cdc⊗nc , a permutation with cycles of length nc

which permute blocks of dimension dc and, finally, unitary conjugations on each block. In this way
we achieve that T is (i) completely positive, unital and trace-preserving, (ii) it projectsMd(C) onto
a direct sum of matrix algebras of the form in Eq.(22) (albeit with ρk = 1 and without kernel)
and (iii) it acts on this sub-algebra as in Eq.(24). It follows then from the previous analysis that an
appropriate choice of the unitary conjugations will lead to the predetermined spectrum.

Note that a different converse can be proven along the same lines as in Thm.9: for every trace-
preserving Schwarz-map T there is a trace-preserving completely positive map T̃ such that T̃ = T
on XT .5 That is, complete positivity is not only irrelevant for the peripheral spectrum but for the
entire peripheral eigen-system. As shown below Thm.4 this does no longer hold for the rest of the
spectrum, let alone the corresponding eigen-system.

VI. CLASSICAL VS. QUANTUM INVERSE EIGENVALUE PROBLEMS

We will briefly review the classical counterpart of our treaties—the inverse eigenvalue problem
for stochastic matrices to which we resorted already in Sec.IV B. For a more detailed overview see
[CG05]. Recall that the spectrum of any stochastic matrix is contained in the unit disc, it contains

5 This is closely related to the well known fact that any conditional expectation, i.e., positive projection onto a *-subalgebra
ofMd(C), is automatically completely positive.



13

one, and it is, as a multiset, invariant under complex conjugation. A non-negative matrix6 is ir-
reducible iff the spectral radius is a non-degenerate eigenvalue and the corresponding eigenvector
is entrywise positive definite. An irreducible matrix in turn is primitive iff there is only a single
eigenvalue whose modulus matches the spectral radius.

A. The classical inverse eigenvalue problem

Every stochastic matrix S is, after a permutation, upper block-triangular with the diagonal blocks
being irreducible. That is, the spectrum of S is the union of spectra of irreducible substochastic
matrices. As used in the proof of Thm.4 already, every irreducible matrix is, up to a rescaling,
similar to a stochastic matrix. For every irreducible matrix M ∈ Md(R+) there is an index of
cyclicity k ∈ {1, . . . , d} so that the spectrum of M is invariant under 2π/k-rotations in the complex
plane. That is, the eigenvalues come in k rotated d/k-tuples. Correspondingly,Mk is block-diagonal
with primitive matrices in the diagonal blocks. So finally, the spectrum of any non-negative matrix,
stochastic or not, is essentially determined by the spectra of primitive matrices.

The non-zero part of the spectrum of primitive matrices is completely characterized by Eqs.(14,15)
together with the fact that the spectral radius is attained only for a single eigenvalue [BH91]. The
number of additionally required zero eigenvalues is, however, generally unbounded. One way to
show this is to exploit the dimension dependent condition

µk(Λ)m ≤ dm−1µkm(Λ), k,m ∈ N (27)

which is valid for any non-negative matrix in Md(R+) [LL78, Joh81]. For a multiset Λ of three
complex numbers Eqs.(14,27) are in fact necessary and sufficient for Λ being the spectrum of a
matrix inM3(R+) [LL78]. For d = 4, however, these conditions are no longer sufficient.

B. Comparing classical and quantum

Maybe the most basic difference between the spectra of classical and quantum channels concerns
the location of single eigenvalues [WPG09]. While for quantum channels on Md(C) the entire
unit disc is accessible for any d ≥ 2, stochastic matrices inMd(R+) obey a dimension dependent
constraint on the location of every single eigenvalue: a necessary condition for λ to be a valid
eigenvalue is that it is contained in the convex hull of all roots of unity of order up to d. A complete
characterization of the accessible region within the unit disc can be found in [Kar51, DD46].

As a consequence, for certain sequences of four non-zero eigenvalues which obey Eq.(14) there
is a quantum channel onM2(C) but no classical channel for any bounded dimension. That is, while
on the classical side an unbounded number of zeros have to be added, no extra kernel is required on
the quantum side.

As discussed below Thm.4 already, it is, however, open whether or not there is a general bound
on the number of required zeros in the quantum case. Eq.(27), which is classically used to bring
the dimension into play, fails to hold for spectra of quantum channels: the channel constructed in
Eq.(17) leads to µ1 > 0 with µ2 = 0 and thus violates Eq.(27) for k = 1, m = 2.

Another crucial difference between the classical and quantum case is the breakdown of the prob-
lem into irreducible and finally into primitive blocks. The decomposition valid for stochastic ma-
trices fails to hold for quantum channels and no counterpart of this is known for the quantum case.
Maybe the structure of the peripheral eigen-system proven in Thm.8 hints at a more general decom-
position.

6 A matrix is said to be ’non-negative’ if all its entries are non-negative.



14

VII. APPLICATIONS FOR TIME SERIES AND CORRELATION FUNCTIONS

Suppose we are given a bounded sequence a ∈ l∞ which is generated via

at = 〈A
∣∣T̂ t∣∣ρ〉, t ∈ N0, (28)

from powers of a linear operator T̂ ∈MD(C). This may for instance be the discrete time evolution
of some expectation value or, as explained later, a two-point correlation function. Then a contains
information about the spectrum of T̂ . A simple relation between spec(T) and a is provided by the
Cayley-Hamilton theorem, i.e., the fact that T̂ is a solution of its characteristic equation. So let
p(x) =

∏D
k=1(λk−x) =

∑D
k=0 ckx

k be the characteristic polynomial of T̂ , then
∑D
k=0 ak+lck = 0

for all l ∈ N0. Depending on degeneracies of the eigenvalues and their Jordan-block structure there
can be annihilating polynomials of degree smaller than D [Gan74a].

Conversely, eigenvalues of T̂ can be determined as roots of a polynomial which is constructed
directly from the sequence a. To this end define for each τ ∈ N0 a vector vτ ∈ l∞ via vτ :=
(aτ , aτ+1, . . .). If the real linear space V := span{vτ}τ∈N0 has dimension r < ∞, then there are
coefficients ck such that for all l ∈ N0: vl+r =

∑r−1
k=0 ckvl+k. These coefficients, which can be

determined from a, define a polynomial p(x) := xr −
∑r−1
i=0 ckx

k. The roots of p then turn out
to be eigenvalues of T (since they coincide with the poles of the z-transform discussed below, see
[Gan74b] p. 205 ff).

A related connection between spec(T̂) and a is provided by the z-transform L : C→ C:

L(z) :=
1

z

∑
t∈N0

at
zt

= 〈A
∣∣(z1− T̂ )−1∣∣ρ〉. (29)

Note that the series which defines L converges outside a disc with radius equal to the spectral radius
of T̂ and L is defined inside by analytic continuation.

The poles of L correspond to eigenvalues of T̂ . Depending on the interplay between T̂ , A and ρ,
not all eigenvalues may, however, appear as poles. Consider as an example the sequence

at =
(

Im
[
(1− i)txt

]
+ yt

)
/
√

2, x, y ∈ [−1, 1]. (30)

In this case L has poles at z1,2 = (1 ± i)x and at z3 = y. So according to Thm.1 the sequence can
be generated by a quantum channel acting onM2(C) only if (|z1|, |z2|, z3) ∈ T . In fact, it can be
generated by a map onM2(C) with matrix representation

T̂ =

(
1 0
0 y

)
⊕
(

x x
−x x

)
,

with ρ and A given by (1, 1/
√

2, 1/
√

2, 0) and (0, 1, 0, 1) respectively. So the example can be
generated by a qubit channel iff the poles are valid eigenvalues. Also note that T is a positive map
iff |y| ≤ 1 and |x| ≤ 1/

√
2.

Using the above relations, knowledge about the spectra of quantum channels can be used to decide
whether or not a given sequence a has a representation of the form in Eq.(28) with T being a quantum
channel of a certain dimension. Such sequences arise naturally for instance as discrete, homogeneous
time-evolutions. They also appear as two-point correlation functions in quantum spin chains whose
state is finitely correlated [FNW92]. The Kraus operators of the T then correspond to the matrices of
the matrix product state [PGVWC07] and t+ 1 is the distance between the two considered sites on
the chain. So a detailed spectral analysis might help in this context if one wants to find an accurate
finitely correlated approximation to given two-point correlations.



15

Out[250]=

5 10 15 20 25 30

-0.2

0.2

0.4

0.6

FIG. 1: Sequences given by Eq.(30) with y = 2/3. While the solid curve (x = 0.58) can be generated by an
evolution of a qubit quantum channel, the dashed curve (x = 0.59) cannot since the poles of its z-transform do
not correspond to valid eigenvalues of a qubit quantum channel. The sequence can, however, be generated by a
positive trace-preserving map.

Acknowledgments We acknowledge financial support by the Danish research council (FNU),
the Spanish grants I-MATH, MTM2008-01366, S2009/ESP-1594 and the European projects QUE-
VADIS and COQUIT.

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http://arxiv.org/abs/0909.5347
http://arxiv.org/abs/quant-ph/0202124

	 Contents
	I Introduction and outline
	II Basic spectral properties of quantum channels
	III The n=4 case – qubit channels
	IV The non-zero part of the spectrum
	A The spectral set
	B Moments and the non-zero part of the spectrum

	V Cycles and the peripheral spectrum
	VI Classical vs. quantum inverse eigenvalue problems
	A The classical inverse eigenvalue problem
	B Comparing classical and quantum

	VII Applications for time series and correlation functions
	 References