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Vectorization of quantum operations and its use
Alexei Gilchrist,1 Daniel R. Terno,1 and Christopher Wood2
1Institute for Quantum Science and Technology, Faculty of Science, Macquarie University, NSW 2109, Australia
2Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo ON, N2L 2Y5 Canada
We give a detailed exposition of the “vectorized” notation for dealing with quantum operations.
This notation is used to highlight the relationships between representations of completely-positive
dynamics. Vectorization considerably simplifies the analysis of different methods of quantum process
tomography, and enables us to derive compact representation of the investigated quantum operations
in terms of the resulting data.
I. INTRODUCTION
Quantum process tomography [1] is one of the standard tools of quantum information science, and the efficient
methods of processing tomographic data are of great practical and conceptual interest. Tomographic data manipula-
tions involve massive amounts of the “qubit/qudit algebra” on the finite-dimensional Hilbert spaces. Linear operators
form vector spaces of their own, and the Hilbert-Schmidt inner product
A ·B := tr (A†B) (1)
makes such a vector space into a Hilbert space. As a result, all the formulas that describe state evolution and
measurements can be re-written in this vector form. “Vectorized” formalism is fruitfully used in quantum information
(e.g., [2, 3, 4]), but often on the ad hoc basis and with not very transparent notation.
In this article we consistently apply it to various forms of the dynamics of open systems (Sec. III) and quantum
process tomography (Sec. IV). It allows us to give an easy derivation of known results, and provides a number of new
relationships between the reconstructed data and the investigated processes. A good notation is a clue to successful
derivations. We begin with a pedagogical introduction into the vectorization of finite-dimensional matrices. This is a
well-established part of matrix analysis [5], and part of our presentation is devoted to introducing notation which is
consistent with other areas of areas of theoretical physics and with group representation theory [6].
II. VECTORIZATION OF MATRICES
A. Conventions
We distinguish between upper (contravaraint) and lower (covariant) indices. Components of a vector ψ ∈ H are
labeled as ψk, the basis of H is denoted by {ek}, k = 1, . . . d ≡ dimH. We adapt the Einstein summation convention
where the summation goes over identical upper and lower indices, hence ψ = ψkek.
The inner product allows to identify ek ≡ e †k . When there is a chance of confusion of the vectorial labels with the
components, the former are put inside the brackets: the a-th component of the basis vector k is denoted as
(ek)a = ea(k). (2)
We choose the orthonormal bases, so ea(k) = δak . In the Dirac notation we identify |k〉 ≡ ek, 〈k| ≡ ek. Since
|ψ〉 =
∑
k
ψk|k〉, 〈ψ| =
∑
k
ψk∗〈k|, (3)
the convention for the components of general vectors is
(ψ†)k = ψk∗. (4)
Moreover, when the indices are shown explicitly, the sign of a Hermitian conjugation becomes redundant, since
positioning of the index indicates weather the object is a ket (vector) or a bra (its dual), hence ψk = ψk∗.
Matrices of the size p× q represent the operators between the vector spaces of the corresponding dimensions,
M : H1 → H2, φa = Mabψb, a = 1, . . . p ≡ dimH2, b = 1, . . . , q ≡ dimH1. (5)

2If the bases of the two Hilbert spaces are denoted by el and uk, respectively, then
M = Mklukel ≡
∑
k,l
Mkl|k〉〈l| ∈ H2 ⊗H∗1, (6)
where H∗ is the dual Hilbert space of H. This equation suggest a basis for the space of p× q matrices,
E lk = |k〉〈l|, (7)
which consists of matrices with all but one entry being zero and the unity at the (kl)-th entry. The matrix M reads
then as
M = MklE lk , k = 1, . . . p, l = 1, . . . , q. (8)
Matrices with two lower or two upper indices can be considered as superpositions of direct product states,, as in
Ψ = F klekul ∈ H1 ⊗H2. (9)
With the above conventions the relationships between components of a transposed matrix are given by
(MT )ab = Mba, (MT ) ba = M ba, (10)
the complex conjugation turns the lower into the upper indices, and vice versa,
(M∗) ba = Ma∗b , (11)
and the Hermitian conjugation satisfies
(M †)ab = M b∗a = (MT ) b∗a = (M∗) ab (12)
Kronicker’s delta is real and symmetric, δab = δab = δ ab . For consistency and ease of index manipulations, we set that
δab ∗ = δba. We also keep in mind that (E ba )∗ = E ba and (E ba )T = (E ba )† = E ab , so
M∗ =
∑
a,b
M∗ ba E ba = Ma∗b E ba , M † = (MabE ba )† =
∑
a,b
Ma∗b E ab = M∗ ba E ab . (13)
Consider a tensor product of two matrices, K = M ⊗N . A pair of row indices a1 and a2 is combined into a single
row index of K, and a pair of column indices b1 and b2 is combined into the new column index. The lexicographic
ordering rule is described below. Hence
Kαβ = Ka1a2 ,b1b2 ≡M
a1
b1N
a2
b2 , (14)
and similarly for
Ka1 b2a2,b1 = M
a1
b1N
b2
a2 , (15)
etc.
B. Definition and simple properties
Representation of matrices as vectors on a higher dimensional Hilbert space is called vectorization. It transforms a
p× q matrix M into a (pq)× 1 column vector denoted by vec(M) or −→M [5]. This is done by ordering matrix elements
lexicographically, i.e. by stacking the rows of M to form a vector. Our convention agrees with [2], but is opposite to
[5]. We made this choice as to keep the same concatenation rules both for vectorization and taking tensor products.
For example, a 2 × 2 matrix M is re-arranged to form the four-dimensional vector −→M ,
−→MT = (M11,M12,M21,M12). (16)
To automate matrix manipulations we have to lump a pair of matrix indices ab, a = 1, . . . p, b = 1, . . . q into a single
vector index α. With the stacking convention that we adopted,
α = f(a, b) ≡ q(a− 1) + b. (17)

3The inverse of vectorization restores the matrix form, as mat(−→M) = M . The matrix indices are restored according to
a = bα/qc + 1, b = αmod q ≡ α− qbα/qc, (18)
where bxc is the largest integer less or equal than x.
We now list some useful properties of vectorized matrices [5]. For convenience we consider square matrices that act
on the space H, p = dimH. A matrix
A =
∑
kl
Akl|k〉〈l| ≡ Aklekel ∈ H ⊗H∗, (19)
is transformed to the vector
−→A =
∑
kl
Aklek ⊗ el =
∑
l
(Ael) ⊗ el = (A⊗ Ip)
∑
l
el ⊗ el, (20)
where Ip is the identity matrix on H.
Vectorization is obviously linear: for matrices Aα and scalars aα,
−−−→aαAα = aα
−→Aα. (21)
Vectorization is intrinsically related to the tensor product. Our stacking convention for a matrix Mab is the same
as the convention for the index concatenation for elements of a tensor product space, such as Mab. This will be useful
in the following. It is also easy to discover how the vectorizations of MT , M∗, and M † are related to each other.
Following the definition of vectorization and Eq (4), we see that
−→M∗ = −→M†,
−→
M † =
−−→
MT †, (22)
and we let −→M and
−→
M † to carry contravariant indices, and −→M∗ and
−−→
MT to carry covariant ones. Hence
−→Mα ≡Mα = Mab,
−→M∗α ≡Mα = M∗ ba = Ma∗b =
−→Mα∗, (23)
and
−→
M †α = M †ab = M b∗a ,
−−→
MT α = MT ba = M ba =
−→
M †α∗. (24)
The Hilbert-Schmidt inner product is equivalent to the usual inner product of vectors,
tr (A†B) = A∗ ab Bba =
−→A∗β
−→B β = −→A †β
−→B β = −→A †−→B ≡ 〈−→A,−→B 〉, (25)
where β = f(b, a).
Vectorization relates the tensor and outer product of vectors,
−→A ⊗ −→B = vec(−→A−→B∗†), (26)
as follows from Eq. (22).
To apply the summation convention we use the opposite label positioning for the basis elements vec(E ba ) ≡
−→Eα.
Let α = f(a, b) and β = f(k, l). Then the basic definition
(−→Eα)β = δβα = δkaδbl , (27)
and the conjugation rules for vectors and matrices lead, e.g., to
(−→E∗α)β = (
−→Eα†)β =
−→Eαβ =
−→Eαβ∗, (28)
which agrees with
mat(−→E∗α) lk = (E ba )∗ lk = (E ba )k∗l = (δkaδbl )∗ = δakδlb, (29)
Similarly,
mat(
−→
ETα) lk = (E ba )lk = δlaδbk, (30)

4and
−→M = Mα−→E α,
−→M † = Mα∗−→Eα† = Mα
−→E α = −→M∗. (31)
Consider two matrices A and B, of the sizes p× q and q × r, respectively. Their product C = AB can be written
as a vector −−→AB in two ways. The matrix C results from the left action of A on B, so in a vectorized notation
Cα = MAαβBβ , (32)
where the matrix MA will be now determined. Writing the indices in full and using matrix multiplication rules leads
to
Cab = MAa db,c Bcd = AacBcdIdb = AacI db Bcd, (33)
so
MAa db,c = Aacδ db . (34)
As a result,
−−→AB = (A⊗ Ir)
−→B . (35)
Now we see that Eq. (20) for −→A is a corollary with B 7→ I. The matrix C also results from the right action of B on
A, which is expressed as
−−→AB = (Ip ⊗BT )
−→A. (36)
The following lemma [5] deals with the triple product of matrices:
Lemma 1. Let A,B,C and X be p× q, r × s, p× s and q × r matrices, respectively. Then the matrix equation
AXB = C (37)
for qs unknowns X lm is equivalent to the system of ps equations
(A⊗BT )−→X = −→C . (38)
That is to say,
−−−→AXB = (A⊗BT )−→X. (39)
The proof consists in the repeated application of Eqs. (35) and (36),
−−−→AXB = (A⊗ Is)
−−→XB = (A⊗ Is)(Iq ⊗BT )
−→X = (A⊗BT )−→X. (40)

C. Reshuffling
The SWAP operation changes the order of subsystems in a tensor product:
|ψ〉1 ⊗ |φ〉2 7→ SWAP(|ψ〉1 ⊗ |φ〉2) = |φ〉2 ⊗ |ψ〉1. (41)
For two identical systems this operation swaps their quantum states. Consider the vectors ψ ∈ H1 and φ ∈ H2. In
the component form we write them as
ψ = ψa1ea1 , a1 = 1, . . . , p = dimH1, (42)
and
φ = ψa2ea2 , a2 = 1, . . . , r = dimH2, (43)

5A single index α of the tensor product Ψα = (ψ ⊗ φ)α is built from the indices of its subsystems as in Eq. (18),
α = r(a1 − 1) + a2. (44)
Similarly, a single vectorial index for Φ = φ⊗ ψ is
β = q(a2 − 1)m+ a1. (45)
The bases of the two tensor products are certain permutations of each other. It is easy to see that this permutation
β = σ[r, p](α) is given by the table
(
1 2 . . . s . . . r r + 1 r + 2 . . . (p− 1)r + 1 . . . pr
1 1 + p . . . 1 + (s− 1)p . . . 1 + (r − 1)p 2 2 + p . . . p . . . p+ (r − 1)p
)
, (46)
where the first row contains indices of the components of Ψ that are matched with the corresponding indices of Φ in
the second row. The vertical lines separate the first and the last r components of Ψ from the rest, and s = 1, . . . r.
Its inverse, σ[r, p]−1 associates to each element of Φ an element of Ψ and is obtained by interchanging the rows. As a
result, if we label the SWAP operator as S(r, p), then
(φ⊗ ψ)α = S(r, p)αβ(ψ ⊗ φ)β (47)
where
S(r, p)αβ = δ
σ[r,p](α)
β . (48)
It follows from this construction that S is an orthogonal matrix, and S(r, p)T = S(r, p)−1 = S(p, r). For example, a
qubit SWAP is
S(2, 2) =



1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1



, (49)
and the SWAP of 2 × 3 systems is done with
S(2, 3) =







1 0 0 0 0 0
0 0 0 1 0 0
0 1 0 0 0 0
0 0 0 0 1 0
0 0 1 0 0 0
0 0 0 0 0 1







. (50)
If we restore the original indices we see that SWAP can be treated as a matrix transposition
Φa2a1 = Ψa1a2 = Sa2a1,b1b2Ψ
b1b2 , (51)
and
Sa2a1,b1b2 = δ
a2
b2 δ
a1
b1 . (52)
For two square matrices M and N that act on the spaces H1 and H2, respectively, the SWAP operation results in
N ⊗M = S(r, p)(M ⊗N)S(r, p)T . (53)
As we will see in the next sections, it is important to relate vec(M) ⊗ vec(N) and vec(M ⊗N). It is done by the
operation of reshuffling [2]. Let the matrices M and N be of the sizes p× q and r× s, respectively. A matrix element
Ma1b1N
a2
b2 (54)
can be interpreted either according to “ first vectorize, then tensor”, as
Ma1b1N
a2
b2 =: C
a1 a2
b1, b2
vec7→ Cαβ ⊗7→ CA = (vec(M) ⊗ vec(N))A , (55)

6or as
Ma1b1N
a2
b2 =: C
a1a2
,b1b2
⊗7→ Cγδ
vec7→ CB = vec(M ⊗N)B, (56)
according to “first tensor, then vectorize” precept.
A repeated application of the index concatenation definition gives
A = rsq(a1 − 1) + rs(b1 − 1) + s(a2 − 1) + b2, B = rsq(a1 − 1) + qs(a2 − 1) + s(b1 − 1) + b2. (57)
Hence swapping the indices a2 and b1 brings
−→M ⊗−→N to −−−−→M ⊗N . This can be formalized by the following
Definition 1 (Reshuffling Matrix). For matrices M and N of the size p × q and r × s, respectively, the reshuffling
matrix R(M,N) is defined by
R(M,N) ≡ R(p, q, r, s) = Ip ⊗ S(r, q) ⊗ Is, (58)
Our discussion established the
Proposition 2 (Reshuffling). Let the matrices M and N be of the size p× q and r × s, respectively. Then
vec(M ⊗N) = R(p, q, r, s)(−→M ⊗−→N ). (59)
III. APPLICATIONS OF VECTORIZATION TO OPEN QUANTUM SYSTEMS
The simplest use of the vectorization is in analyzing the convex probability domains of quantum measurements
[3]. Vectorization makes it nearly obvious that for a positive operator valued measure (POVM) with N outcomes
the domain in the probability space that is formed by probabilities of all outcomes with all possible states of a
n-dimensional system has at most n2 − 1 dimensions.
A. Completely Positive Maps
Evolution of an open quantum system whose initial state is uncorrelated with the environment is described by a
completely positive map. Label the initial state of the system (H1) as ρ, and the state of its environment (H2) by ω.
If the joint state is given by τ12 = ρ ⊗ ω, and the overall unitary evolution is U12, then the final state of the system
is given by
ρ 7→ T(ρ) = tr H2(U12ρ⊗ ωU †12), (60)
where the operation of partial tracing is defined by
(tr 2τ)a1b1 = τ
a1a2
,b1a2 . (61)
Such a map T is trace-preserving, convex-linear, completely positive map [2]. This latter property means that T
is positive (maps positive matrices to positive matrices), and also if we introduce an auxiliary system of arbitrary
dimension then the map T ⊗ I on the joint system is positive, where I is the identity map on the auxiliary system.
A map satisfying these three axioms is referred to as a completely positive, trace preserving (CPTP) map. It is
possible to relax the trace condition to tr [T(ρ)] ≤ tr [ρ], allowing for completely positive trace decreasing maps. We
will only be concerned with CPTP maps and we shall refer to maps satisfying all three of these axioms simply as
completely-positive maps (CP maps).
Any CPTP map has a convenient operator sum representation:
Theorem 3 (Kraus representation). A map T acting on density operators of H is CPTP if and only if there exists
a set of bounded operators {Kn} acting on H such that
T(ρ) =
∑
n
KnρK†n where
∑
n
K†nKn = I. (62)

7The operators Kn are called Kraus matrices and they satisfy
∑
nK†nKn = I, which is known as the completeness
relation. A Kraus representation of a given process T is not unique. This can be useful as different system-environment
interactions may still give rise to the same reduced dynamics on the system.
In general, a linear hermiticity-preserving transformation T (completely positive or not) acting on the space of
density matrices may be represented by the dynamical (Choi) matrix D(T) [2, 7, 8],
ρ′ab = Da dc,b ρcd, (63)
and a vectorized version of this relation
−→ρ ′α = D¯αβ−→ρ
β , (64)
uses the reshuffled matrix D¯,
D¯a db,c = Da dc,b , (65)
vec(D¯) = R−→D .
Dynamical matrix has a number of useful properties [2]. The trace preserving condition is equivalent to the
constraint on the partial trace of the dynamical matrix,
Da dc,a = δdc , (66)
which implies that the eigenvalues sum up to the system’s dimension, ∑a λa = d1. Moreover, if the map is unital,
i.e., it maps the maximally mixed state into the maximally mixed state, then
Da ss,b = δab . (67)
The dynamical matrix is Hermitian, D†αβ = Dαβ , and due to a theorem of Choi [8] its positivity is equivalent to the
complete positivity of T:
Theorem 4 (Dynamical Matrix). A quantum operation T on a d-dimensional system S is CP if and only if its
dynamical matrix D(T) is positive-semidefinite (D(T) ≥ 0).
The proof is based on the eigendecomposition of the Choi matrix,
Da dc,b = Dαβ =
∑
n
λn
−→Mnα
−→Mn†β =
∑
n
λn
−→Mnα
−→M∗nβ =
∑
n
λnMnacM∗n db , (68)
where α = f(a, c), β = f(b, d), and if all the eigenvalues λn are non–negative it is possible to define Kraus matrices
by absorbing the eigenvalues,
Kn =
√
λnMn. (69)
We also get a compact expression for the reshuffled matrix,
D¯ =
∑
n
Kn ⊗K∗n. (70)
B. Linear Superoperator
Vectorization allows a different perspective on Choi matrix and its reshuffled version. We consider a combination
of basis matrices E ba and T(E ba ) ≡ E′ ba , with suitably arranged component indices.
A linear superoperator ΦT acts on vectorized density matrices and is defined through
ΦT−→ρ ≡
−−→
T(ρ). (71)
On the one hand, since
〈Eα,Eβ〉 =
−→Eα−→Eβ = tr (E ba †E dc ) = δac δdb = δαβ , (72)

8we have
ρβ
−→
E′α
−→Eα−→Eβ = ρβ
−→
E′β = ΦT−→ρ . (73)
As a result,
ΦT =
∑
α
−→
E′α
−→Eα† =
−→
E′α
−→Eα. (74)
On the other hand, a comparison with Eq. (64) identifies the linear superoperator with the reshuffled dynamical
matrix, ΦT = D¯(T). The matrix elements of ΦT are obtained by
ΦTκµ =
−→
E′ακ
−→Eαλ =
∑
a,b
(E′ ba )kl(E ba ∗) nm = (E′ ba )klδamδnb = (E′ nm )kl = (E′µ)κ. (75)
Indeed, ρ′κ = ΦTκµρµ, so
D¯k nl,m = (E′ nm )kl =
∑
x
(Kx)km(K∗x) nl . (76)
We note that when expressed in terms of the Kraus matrices Kn the reshuffled matrix D¯ is built according to “first
tensor, then vectorize” prescription. However, the opposite approach “first vectorize, then tensor” is taken when −→ΦT
is expressed in terms of the basis matrices and their transforms.
C. Process matrix
We define the process matrix ΛT for a map T as a matrix on a fictitious space H1 ⊗H2, where the two spaces are
copies of H. Without going through the vectorization of Eqs. (72), (73), the transformation ρ′ = T(ρ) can be written
as
ρ′ = ρcdE′ ba tr (E ab E dc ). (77)
Thus it is given by
T(ρ) = tr 2
[
ΛT2T (I ⊗ ρ)
]
, ΛT2T ≡ E′ ba ⊗ E ab . (78)
For our purposes it is more convenient to use the matrix arrangement of ΦT, and define a new four-indexed object
according to “first tensor, then vectorize” prescription,
ΛT ≡ E′ ba ⊗ E ab † =
∑
a,b
E′ ba ⊗ E ba ∗, (79)
where we keep a complex conjugation as a reminder for the correct index placement. Then
T(ρ) = tr 2
[
ΛT(I ⊗ ρT )
]
, (80)
and
ΛTk nl,m = (
∑
a,b
E′ ba ⊗ E ba ∗)k nl,m = (E′ ba )kmδal δnb = (E′ nl )km. (81)
A comparison with Eq (63) shows that D(T) = ΛT.
D. Summary of relationships
Jamio lkowski isomorphism [2, 9] identifies dynamical matrices of CP maps with certain entangled states. Consider
a generalization of the Bell state |Φ+〉 = (|+ +〉+ | −−〉)/2 on the space H1 ⊗H2 of the previous section. Its density
matrix is given by
τ+ =
∑
a,b
E ba ⊗ E ba /d, (82)

9where d = dimH. It is now easy to write the matrix elements of ρ′ = (T ⊗ I)ρ,
τ ′+kl,mn = (E′ nl )km/d = ΛTk nl,m /d, (83)
which establishes
Theorem 5 (Jamio lkowski isomorphism). Any linear map T acting on the space of mixed states on the Hilbert space
H can be associated, via its dynamical matrix D(T), with an operator on the enlarged Hilbert spacee H⊗H,
D(T) ∼= (T ⊗ I)τ+ = τ ′+, (84)
where a completely positive map T the matrix τ ′+ is a valid quantum state.
As a useful computational aid we present a compact summary of relationships between different representations of
a completely positive map.
To\From D(T) ΦT Kn
T(ρ) = tr 2
ˆ
D(I1 ⊗ ρT )
˜
mat(ΦT−→ρ )
P
n KnρK
†
n
D = D mat(R−1−→ΦT)
P
n
−→Kn
−→Kn†
ΦT = mat(R
−→
D ) ΦT
P
n Kn ⊗K∗n
Kn =
√
λnmat(
−→Mn) ←− Kn
TABLE I: Relationships between mathematical representations of CP maps.
Two possible viewpoints on both D and ΦT as obtained either according to “first tensor, then vectorize” or “first
vectorize, then tensor” approaches (with a counterpart being obtained by the other procedure) are consistent, because
the reshuffling matrix involved in the process satisfies R = R−1. The latter result follows from Eq. (46), with all the
dimensions being equal to dimH.
IV. PROCESS TOMOGRAPHY USING THE SUPEROPERATOR REPRESENTATION
A. Dual bases
To understand the working of a quantum device in the standard process tomography [1] we need a tomographically
complete set of input states that are sent through the investigated device. Then its action is reconstructed by analyzing
the outputs. In the mathematics of reconstruction a set of dual states D = {Dν} for a tomographically complete set
ρin = {ρµ}, µ = 1, . . . , n = d2, where d = dimH, plays an important role. It is defined by the orthogonality relation
tr (D†νρµ) = δνµ, ∀µ, ν.
If we rewrite this in vectorized notation we have obtain
〈−→Dµ,−→ρν
〉
= δµν (85)
hence
−→ρµ
−→ρµ−→Dν = −→ρµδµν = −→ρν (86)
Introducing
P =
∑
µ
−→ρµ −→ρµ† ≡ −→ρµ
−→ρµ, (87)
we see that P−→Dν = −→ρν , hence
−→Dν = P−1−→ρν . (88)
It is possible to write this relationship between a tomographic set and its dual in a simpler and computationally
more efficient way, if we write vectors of the set ρin columns of a matrix, as
[ρin] = (−→ρ1, . . . ,−→ρn). (89)

10
Since its columns are linearly independent, this d2 × d2 matrix is invertible. Its Hermitian conjugate can be written
as
[ρin]† =



−→ρ1†
...
−→ρn†



. (90)
With this notation
R = [ρin][ρin]†. (91)
As a result,
−→Dµ = [ρin]−1†[ρin]−1−→ρµ. (92)
If we introduce [D] = (−→D1, . . . ,
−→Dn), then an even simpler expression is obtained:
[D] = [ρin]−1†[ρin]−1[ρin] = [ρin]−1†. (93)
Now we apply this to the standard process tomography.
B. Process tomography with ΦT
Proposition 6. Let ρin = {ρµ} be a tomographically complete set of input states with dual basis D = {Dµ}. Then
the linear superoperator ΦT for a CP map T : ρ 7→ T(ρ) is given by
ΦT =
∑
µ
−−−→
T(ρµ)
−→Dµ† ≡
−→ρ′µ
−→Dµ (94)
Proof. Consider an arbitrary state ρ = pµρµ. We have to show ΦT−→ρ = vecT(ρ). Applying the above expression to −→ρ
we see that
ΦT(−→ρ ) = pν
−→ρ′µ 〈Dµ, ρν〉 = pν
−→ρ′µδµν = pµ
−→ρ′µ = T(ρ). (95)
We simplify this expression by using the matrix of vectorized input states, Eq. (89). This leads us to the following
Proposition 7. For the set of output states ρout = {T(ρµ)} where ρin = {ρµ} is a tomographically complete set of
input states, the linear superoperator ΦT is given by
ΦT = [ρout][ρin]−1 (96)
Proof. We use Eq. (94).
[ρout][ρin]−1 = [ρout][D]† =
−→ρ′µ
−→Dµ = ΦT. (97)
We rewrite this result in terms of the probabilities of various experimental outcomes.
Proposition 8. Let {Mµ} be a tomographically complete measurement set (Mµ ≥ 0,
∑Mµ = I) with a dual basis
{Eν}, tr (E†νMµ) = δνµ. Then
ΦT = [E] [m] [D]† , (98)
where [E] = (−→E1, . . . ,
−→En) is a matrix of the vectorized dual elements, and [m] = (mµν) is a matrix of probabilities,
mµν = tr (M †µρ′ν).
Proof. Since any ρ is reconstructed according to ρ = Eµmµ, where mµ = tr (Mµρ),
[ρout] = [E][m], (99)
and [D]† = [ρin]−1.

11
C. Ancilla-assisted process tomography with linear superoperator
Presentation of Jamio lkowski isomorphism and the manipulation of data in the ancilla-assisted quantum tomography
(AAPT) [4, 10] also benifit from the vectorized notation.
We introduce an auxiliary system (ancilla) H2 to our principal system H1, so that the state space of the joint
system is given by H1 ⊗ H2. AAPT aims to reconstruct the CP map T on the states of H the action of T ⊗ I on a
single state τ12 of this combined system. Evolution on H1 results in an operation T described by a superoperator ΦT,
and the ancilla does not evolve. Any initial state τ12 of the joint system can be represented as
τ12 =
∑
µ
wµρµ ⊗ ωµ,
∑
µ
wµ = 1 (100)
which is in general entangled (it is separable if and only if all wµ ≥ 0).
Lemma 9. The joint system dynamical matrix has the form
ΦT⊗I = R(ΦT ⊗ I¯)R−1, (101)
where R is the reshuffling matrix, which in this case again satisfies R = R−1, and the linear superoperator of the
identity map is I¯ ≡ ΦI.
Proof. Linearity of the evolution is expressed as
τ ′ = (T ⊗ I)τ =
∑
µ
wµT(ρµ) ⊗ ωµ. (102)
Hence
−→τ ′ =
∑
µ
wµ
−−−−−→
ρ′µ ⊗ ωµ =
∑
µ
wµR(−→ρµ′ ⊗−→ωµ) = R(ΦT ⊗ I¯)
∑
µ
wµ−→ρµ ⊗−→ωµ. (103)
Matrix elements of I¯ satisfy I¯k nl,m = δkmδnl = Iαβ . Another reshuffling leads to the desired result,
−→τ ′ = R(ΦT ⊗ I¯)R−1−→τ . (104)
Both vectors
−→τinA = −→τinαβ ≡ (R−1−→τ )αβ = (R−1−→τ )k ml, n =
∑
µ
wµρµkl ωµmn, (105)
and −−→τoutA ≡ (R−1−→τ ′)αβ correspond to the right hand side of Eq. (59). Their relationship is of the forms of Eq. (35):
−−→τoutαβ = (ΦTαγIβδ)−→τin
γδ. (106)
Introducing Φτ ≡ mat(−→τin) we have
−−→τout = (ΦT ⊗ I)−→τin = (I ⊗ ΦTτ )
−→ΦT. (107)
Hence we can recover the dynamical matrix ΦT from the output state τAS when the matrix Φτ is invertible:
Proposition 10. A linear superoperator ΦT is recovered from the output of AAPT with the initial state τ according
to
−→ΦTαβ = (I ⊗ (Φ−1τ )T )αβ,γδ−−→τout
γδ (108)
An important special case is the entanglement-assisted process tomography, where the input state is a maximally
entangled
τ = τ+ =
∑
ij
|i〉〈j| ⊗ |i〉〈j|/d. (109)
It corresponds to Φτ+ = mat(−→τ+) = I. Hence we established a useful expression for the dynamical matrix and a dual
form of the Jamio lkowski isomorphism
Corollary 11 (entanglement-assisted process tomography). In the entanglement assisted process tomography (AAPT
with the maximally entangled initial state τ+) the dynamical matrix the linear superoperator ΨT is determined by the
output state according to
−→ΦT = −−→τout. (110)

12
V. CONCLUSIONS AND OUTLOOK
Adopting vectorized notation allows transparent and consistent representation of various forms of open system
dynamics, isomorphism between states and operations and representation of process tomography. Neat expressions
for process tomography (98), (108) and (110) use already reconstructed output states. Processing of actual state
tomographic data is much more involved. In particular, relative frequencies cannot be directly taken as probabilities
[3, 10], assumption of completely positive dynamics should be justified or may not be true [11, 12], matrices may have
only generalized inverses [13]. A transparent and versatile notation is a great asset in dealing with these issues, and
we expect that it will simplify some of the existing formulas and bring to light new useful relationship, similarly to
presented in this work.
Acknowledgments
We thank Karol Z˙yczkowski for many useful discussions. CW is supported by Perimeter Scholars International
program.
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