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Quantum Information and Computation, Vol. 9, No. 1&2 (2009) 0103–0130
c© Rinton Press
SUB– AND SUPER–FIDELITY AS BOUNDS FOR QUANTUM FIDELITY
JAROS LAW ADAM MISZCZAK ZBIGNIEW PUCHA LA
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences,
Ba ltycka 5, 44-100 Gliwice, Poland
PAWE L HORODECKI
Faculty of Applied Physics and Mathematics, Gdan´sk University of Technology,
Narutowicza 11/12, 80-952 Gdan´sk, Poland
and
National Quantum Information Centre of Gdan´sk,
Andersa 27, 81-824 Sopot, Poland
ARMIN UHLMANN
Institute of Theoretical Physics, University of Leipzig,
Vor dem Hospitaltore 1, D-04103 Leipzig, Germany
KAROL Z˙YCZKOWSKI
Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagiellon´ski,
Reymonta 4, 30-059 Krako´w, Poland
and
Centrum Fizyki Teoretycznej, Polska Akademia Nauk,
Aleja Lotniko´w 32/44, 02-668 Warszawa, Poland
Received May 19, 2008
Revised September 30, 2008
We derive several bounds on fidelity between quantum states. In particular we show that
fidelity is bounded from above by a simple to compute quantity we call super–fidelity.
It is analogous to another quantity called sub–fidelity. For any two states of a two–
dimensional quantum system (N = 2) all three quantities coincide. We demonstrate
that sub– and super–fidelity are concave functions. We also show that super–fidelity is
super–multiplicative while sub–fidelity is sub–multiplicative and design feasible schemes
to measure these quantities in an experiment. Super–fidelity can be used to define a
distance between quantum states. With respect to this metric the set of quantum states
forms a part of a N2 − 1 dimensional hypersphere.
Keywords: quantum fidelity, quantum states, Bures distance, distances in state space
Communicated by : R Jozsa & M Mosca
1 Introduction
By processing quantum information we wish to transform a quantum state in a controlled way.
Taking into account inevitable interaction with an environment and possible imperfection of
real dynamics it is then crucial to characterize quantitatively, to what extend a given quantum
state gets close to its target. For this purpose one often uses fidelity [1], here denoted by F .
That quantity has also been called transition probability [2]: Operationally it is the maximal
success probability of changing a state to another one by a measurement in a larger quantum
103

104 Sub– and super–fidelity as bounds for quantum fidelity
system. If both quantum states are pure, fidelity is the squared overlap between them.
In the general case fidelity between any two mixed states is the function of the trace
norm of the product of their square roots. Thus analytical evaluation of fidelity, or its direct
experimental measurement becomes a cumbersome task. Hence there is a need for other
quantities, which bound fidelity and are easier to compute and measure.
The aim of this work is to present some bounds for fidelity and to develop experimental
schemes to estimate it for an arbitrary pair of mixed quantum states. In particular we find an
upper bound for fidelity by a simple quantity which is the function of purity of both states and
the trace of their product. Since it possesses some nice algebraic properties we believe it may
become useful in future research and propose to call it super–fidelity. In a sense it is a quantity
complementary to the one forming the lower bound proved in [3], and we tend to call sub–
fidelity. For any two one–qubit states all three quantities coincide. Fidelity is well known to
be multiplicative with respect to the tensor product. In this work we prove that super–fidelity
is concave and super–multiplicative, while sub–fidelity is concave and sub–multiplicative.
Fidelity can be used to define the Bures distance between quantum states and the Bures
angle. As shown by Uhlmann in [4] the Bures geometry of the set of one–qubit states (N = 2),
is equivalent to a three-dimensional hemisphere 12S3. The set of density operators, ΩN ,
becomes the space of non-constant curvature by the Bures metric for N ≥ 3, [5].
We construct distance and angle analogous to the Bures distance out of super–fidelity in a
similar way. With respect to this metric the set ΩN forms a fragment of a N2−1 dimensional
hypersphere with the maximally mixed state ρ∗ := I/N at the pole. A linear function of
super–fidelity was earlier used by Chen et al. [6] to analyze the set of mixed quantum states
and demonstrate its hyperbolic interpretation.
This paper is organized as follows. In Section II the definition and basic properties of
fidelity are reviewed. Sections III and IV are devoted to bounds on fidelity. In Section V
we define sup– and super–fidelity and investigate their properties. Experimental schemes
designed to measure these quantities are presented in section VI. In Section VII we analyze
the geometry of the set of quantum states induced by the distance derived by super–fidelity.
Concluding remarks are followed by appendices, in which we prove necessary lemmas and
present the collection of useful algebraic facts.
2 Fidelity between quantum states
Consider an N– dimensional Hilbert space HN . A linear operator ρ : HN → HN represents
quantum state if it is Hermitian, semipositive, ρ = ρ† ≥ 0, and normalized, trρ = 1. Let ΩN
denote the set of all mixed quantum states of size N .
Fidelity between quantum states ρ1 and ρ2 is defined as [2, 1],
F (ρ1, ρ2) = (tr|
√ρ1
√ρ2|)2 = ||ρ1/21 ρ
1/2
2 ||21, (1)
where || · ||1 is Schatten 1-norm (trace norm),
||A||1 = tr|A| := tr
√
AA†. (2)
Alternatively, the trace norm of an operator can be expressed as the sum of its singular values,
||A||1 =
∑n
i=1 σi(A). Here σi(A) is equal to the square root of the corresponding eigenvalue
of the positive matrix AA† – see e.g. [7].

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 105
There are different uses of the name fidelity. In [1] the older notion transition probability
has been renamed fidelity by Jozsa. In [8]
√
F has been called fidelity, while [9] uses Jozsa’s
notion, and to the latter convention we shall stick in calling fidelity the expression in Eq. (1).
For pure states the definition (1) is reduced to the transition probability. If one state is
pure, ρ1 = |ψ〉〈ψ|, then F (ρ1, ρ2) = 〈ψ|ρ2|ψ〉. Hence for any two pure states their fidelity is
equal to their squared overlap, F (ψ, φ) = |〈ψ|φ〉|2 =: κ.
Fidelity enjoys several important properties [2, 10, 11, 12, 1], which can also be proved on
state spaces of unital C∗-algebras. Some of them are:
i) Bounds: 0 ≤ F (ρ1, ρ2) ≤ 1. Furthermore F (ρ1, ρ2) = 1 iff ρ1 = ρ2, while F (ρ1, ρ2) = 0
iff supp(ρ1) ⊥ supp(ρ2).
ii) Symmetry: F (ρ1, ρ2) = F (ρ2, ρ1).
iii) Unitary invariance: F (ρ1, ρ2) = F (Uρ1U †, Uρ2U †), for any unitary operator U .
iv) Concavity: F (ρ, aρ1 + (1− a)ρ2) ≥ aF (ρ, ρ1) + (1 − a)F (ρ, ρ2), for a ∈ [0, 1].
v) Multiplicativity: F (ρ1 ⊗ ρ2, ρ3 ⊗ ρ4) = F (ρ1, ρ3)F (ρ2, ρ4).
vi) Joint concavity:
√
F (aρ1+(1−a)ρ2, aρ′1+(1−a)ρ′2) ≥ a
√
F (ρ1, ρ′1)+(1−a)
√
F (ρ2, ρ′2),
for a ∈ [0, 1].
For further analysis of fidelity properties it is instructive to work with eigenvalues of a ma-
trix
√
ρ1/21 ρ2ρ
1/2
1 . Let us denote them by λi, i = 1, . . . , N . This matrix is positive so its
eigenvalues and singular values coincide. Unless otherwise stated, we tacitly assume that
λ1 ≥ λ2 ≥ · · · ≥ λN . The root fidelity reads
√
F (ρ1, ρ2) = tr
√√ρ1ρ2
√ρ1 =
N
∑
i=1
λi. (3)
Squaring this equation one obtains a compact expression for fidelity,
F (ρ1, ρ2) =
( N
∑
i=1
λi
)2
= trρ1ρ2 + 2
∑
i<j
λiλj , (4)
where we have taken into account that trρ1ρ2 = tr
√ρ1ρ2
√ρ1 =
∑N
i=1 λ2i . The matrix√ρ1ρ2
√ρ1 is similar to ρ1ρ2 and they share the same set of N eigenvalues.
3 Bounds for fidelity
We shall need some further algebraic definitions. For any matrix X of size N with a set of
eigenvalues {λ1, . . . , λN} we define elementary symmetric functions sm(X) as the elementary
symmetric function of its eigenvalues [17, Def. 1.2.9]. For instance, the second and third
elementary symmetric functions read
s2(X) =
∑
i<j
λiλj , (5)
s3(X) =
∑
i<j<k
λiλjλk. (6)

106 Sub– and super–fidelity as bounds for quantum fidelity
For any matrix of rank r the highest non-vanishing symmetric function reads sr(X) =
∏r
i=1 λi.
In the generic case r = N we have sN (X) = det(X).
In this section we shall list several bounds for fidelity, some of which are well known in
the literature. Let us start by stating a simple result,
F (ρ1, ρ2) ≤ trρ1trρ2, (7)
which follows directly from Fact 1 (see Appendix A) if we set ν = 1/2. This fact implies the
property F (ρ1, ρ2) ≤ 1.
Expression (4) implies the following lower bound
trρ1ρ2 ≤ F (ρ1, ρ2) ≤ Ntr|ρ1ρ2|. (8)
To get the upper bound we use Fact 2 (see Appendix A) and set ν = 1/2 to obtain ||√ρ1
√ρ2||21 ≤
N ||ρ1ρ2||1.
Let us now denote the spectra of the states ρ1 and ρ2, by vectors ~p and ~q, respectively.
The fidelity between them is then bounded by the classical fidelity between diagonal density
matrices [13]
F (p↑, q↓) ≤ F (ρ1, ρ2) ≤ F (p↑, q↑), (9)
where the arrows up (down) indicate that the eigenvalues are put in the nondecreasing (non-
increasing) order.
The lower bound in (8) can be improved, since the following result is true [3]
F (ρ1, ρ2) ≥ trρ1ρ2 +
√
2
√
(trρ1ρ2)2 − trρ1ρ2ρ1ρ2. (10)
The above inequality is saturated for any pair of one–qubit states. Furthermore, the above
inequality is an equality if the rank of ρ1ρ2 does not exceed two. On the other hand, the
inequality is strict if that rank is larger than two — see Appendix E. For completeness we
present the simple proof of inequality (10) in Appendix B.
Another lower bound is obtained if the rank of ρ1ρ2 is exactly r. If sr denotes the rth
elementary symmetric function then
F (ρ1, ρ2) ≥ trρ1ρ2 + r(r − 1) r
√
sr(ρ1ρ2). (11)
This bound is proved in Appendix C. If both states are generic, i.e. if they are of the maximal
rank the above formula reads
F (ρ1, ρ2) ≥ trρ1ρ2 +N(N − 1) N
√
det ρ1 det ρ2. (12)
The key result of this paper consist in the following upper bound, in a sense complementary
to (10).
Theorem 1 For any density matrices ρ1 and ρ2 we have
F (ρ1, ρ2) ≤ trρ1ρ2 +
√
(1− trρ21)(1 − trρ22). (13)

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 107
Before presenting the proof in the subsequent section let us first note that the bound is
saturated if at least one of the states is pure. Furthermore, an equality holds for any two
mixed states of size N = 2. To show this property observe that in this case the sum in
(4) consists of a single term 2λ1λ2 = 2
√
det(ρ1ρ2) =
√
2det(ρ1)
√
2det(ρ2). Since for any
one-qubit state one has 2det(ρ) = 1− trρ2 an equality in (13) follows. This fact was already
known to Hu¨bner [14]. In a similar way we treat the more general case of N = 3 in Appendix
E, for which some other equations for fidelity are derived.
4 Proof of the main upper bound
The notion of the second symmetric function (5) allows us to write the expression
[(trX)2 − (trX2)] = 2s2(X). (14)
Note that if X has nonnegative eigenvalues then s2(X) ≥ 0.
Using (14) we can rewrite fidelity
F (ρ1, ρ2) = trρ1ρ2 + 2s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
, (15)
and
√
(1− trρ21)(1 − trρ22) = 2
√
s2(ρ1)s2(ρ2). (16)
Thus the Theorem 1 can be equally expressed as an inequality
s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
≤
√
s2(ρ1)s2(ρ2). (17)
The proof of (17) is decomposed into two Lemmas, the proof of which can be found in
Appendix D.
Lemma 1 For given density matrices ρ1, ρ2 with eigenvalues p1, . . . , pn and q1, . . . , qn respec-
tively
s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
≤ s2
(
√
diag(p)diag(q)
)
, (18)
where diag(p) and diag(q) denote diagonal matrices with entries on diagonal p1, . . . , pn and
q1, . . . , qn respectively.
Lemma 2 With notation as in Lemma 1, we have
s2
(
√
diag(p)diag(q)
)
≤
√
s2(diag(p))s2(diag(q)) =
√
s2(ρ1)s2(ρ2). (19)
Proof of Theorem 1. For given density matrices ρ1 and ρ2 with eigenvalues p1, . . . , pn
and q1, . . . , qn respectively. We denote diagonal matrices with entries on diagonal p1, . . . , pn
and q1, . . . , qn as diag(p) and diag(q) respectively.
F (ρ1, ρ2) = trρ1ρ2 + 2s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
≤ trρ1ρ2 + 2s2
(
√
diag(p)diag(q)
)
≤ trρ1ρ2 + 2
√
s2(ρ1)s2(ρ2) = trρ1ρ2 +
√
(1− trρ21)(1 − trρ22).
Making use of Lemma 1 for the first inequality and of Lemma 2 for the second one we arrive
at the inequality (13). 2.

108 Sub– and super–fidelity as bounds for quantum fidelity
5 Sub– and super–fidelity and their properties
5.1 Definition and basic facts
We shall start this section with a general definition. For any two hermitian operators A and
B let us define two quantities
E(A,B) = trAB +
√
2[(trAB)2 − trABAB], (20)
G(A,B) = trAB +
√
(trA)2 − trA2
√
(trB)2 − trB2. (21)
For any two density operators their traces are equal to unity, so E(ρ1, ρ2) and G(ρ1, ρ2)
have lower bound (10) and upper bound (13), respectively. Thus both universal bounds for
fidelity can be rewritten as
E(ρ1, ρ2) ≤ F (ρ1, ρ2) ≤ G(ρ1, ρ2). (22)
Note that both bounds require the evaluation of three traces only, so they are easier to
compute than the original fidelity. As shown in Section 3 for N = 2 all three quantities are
equal, so we propose to call E(ρ1, ρ2) and G(ρ1, ρ2) as sub– and super–fidelity. These names
are additionally motivated by the following appealing properties:
i’) Bounds: 0 ≤ E(ρ1, ρ2) ≤ 1 and 0 ≤ G(ρ1, ρ2) ≤ 1.
ii’) Symmetry: E(ρ1, ρ2) = E(ρ2, ρ1) and G(ρ1, ρ2) = G(ρ2, ρ1).
iii’) Unitary invariance: E(ρ1, ρ2) = E(Uρ1U †, Uρ2U †) andG(ρ1, ρ2) = G(Uρ1U †, Uρ2U †),
for any unitary operator U .
iv’) Concavity:
Proposition 1 Sub– and super–fidelity are concave, that is for A,B,C,D ∈ ΩN and
α ∈ [0, 1] we have
E(A,αB + (1 − α)C) ≥ αE(A,B) + (1 − α)E(A,C), (23)
G(A,αB + (1 − α)C) ≥ αG(A,B) + (1− α)G(A,C). (24)
v’) Properties of the tensor product:
Proposition 2 Super–fidelity is super–multiplicative, that is for A,B,C,D ∈ ΩN
G(A ⊗B,C ⊗D) ≥ G(A,C)G(B,D), (25)
while
Proposition 3 Sub–fidelity is sub–multiplicative, that is for A,B,C,D ∈ ΩN
E(A⊗B,C ⊗D) ≤ E(A,C)E(B,D). (26)

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 109
Properties i’), ii’) and iii’) follow from the properties of trAB and definitions (20) and (21).
In this section we prove properties iv’) and v’).
Proof of Proposition 1. The definitions (20) and (21) can be rewritten in terms of the
aforementioned elementary symmetric functions (5) using relation (14),
E(A,B) = trAB + 2
√
s2(AB), (27)
G(A,B) = trAB + 2
√
s2(A)s2(B). (28)
In general rth root of sr is concave on the cone of positive operators [41]. This implies
concavity of G directly. To get concavity of E we can replace matrix AB by the similar
matrix A1/2BA1/2 which is positive. Using the concavity of
√
s2(A1/2BA1/2) we obtain the
result. 2.
Proof of Proposition 2. First we note that super–fidelity is not multiplicative. As an
example we can take
A =
(
1 0
0 0
)
, B =
( 1
2 0
0 12
)
, C =
(
0 0
0 1
)
, D =
( 1
2 0
0 12
)
, (29)
in which case we have
1
2
= G(A⊗B,C ⊗D) > G(A,C)G(B,D) = 0. (30)
To prove the proposition we write
G(A⊗B,C ⊗D) = trACtrBD +
√
(1− trA2trB2)(1− trC2trD2), (31)
and
G(A,C)G(B,D) =
(
trAC +
√
(1 − trA2)(1 − trC2)
)(
trBD +
√
(1− trB2)(1 − trD2)
)
.
(32)
Denoting trA2 = α, trB2 = β, trC2 = γ and trD2 = δ we have to show that
√
(1− αβ)(1 − γδ) ≥ trAC
√
(1− β)(1 − δ) + trBD
√
(1− α)(1 − γ)
+
√
(1 − α)(1− γ)(1− β)(1 − δ).
Now from Fact 6 with a = 2 (see Appendix B) one has
trAC ≤
√
trA2trC2 = √αγ (33)
and
trBD ≤
√
trB2trD2 =
√
βδ. (34)
Thus it is enough to show that
√
(1− αβ)(1 − γδ) ≥ √αγ
√
(1 − β)(1− δ) +
√
βδ
√
(1− α)(1 − γ)
+
√
(1− α)(1 − γ)(1− β)(1 − δ). (35)
We define two vectors
X =


√α√1− β√
β
√
1− α√
1− α
√
1− β

 and Y =


√γ
√
1− δ√
δ√1− γ√
1− γ
√
1− δ

 . (36)

110 Sub– and super–fidelity as bounds for quantum fidelity
Note that
〈X |Y 〉 = √αγ
√
(1− β)(1 − δ)+
√
βδ
√
(1− α)(1 − γ)+
√
(1− α)(1 − γ)(1− β)(1− δ) (37)
and
〈X |X〉 = (1− αβ) and 〈Y |Y 〉 = (1− γδ). (38)
Now by combining (38) with (37) and using Cauchy–Schwarz inequality
√
〈X |X〉〈Y |Y 〉 ≥ 〈X |Y 〉, (39)
we obtain (35). 2.
Proof of Proposition 3. To show sub–multiplicativity of sub–fidelity we write the defini-
tion (20) for a tensor product,
E(A⊗B,C ⊗D) = tr[(A⊗B)(C ⊗D)]
+
√
2[
(
tr(A⊗B)(C ⊗D)
)2 − tr(A⊗B)(C ⊗D)(A⊗B)(C ⊗D)]
= trACtrBD +
√
2[(trACtrBD)2 − trACACtrBDBD].
The product of two sub–fidelities reads
E(A,C)E(B,D) = (trAC +
√
2[(trAC)2 − trACAC])(trBD +
√
2[(trBD)2 − trBDBD])
= trACtrBD + trAC
√
2[(trBD)2 − trBDBD] + trBD
√
2[(trAC)2 − trACAC]
+
√
2[(trAC)2 − trACAC]
√
2[(trBD)2 − trBDBD].
For short we denote
α = trAC, a = trACAC,
β = trBD, b = trBDBD.
We have α2 ≥ a and β2 ≥ b. By rewriting above expressions in the new notation we
obtain
E(A⊗B,C ⊗D) = αβ +
√
2[α2β2 − ab]
and
E(A,C)E(B,D) = αβ + α
√
2[β2 − b] + β
√
2[α2 − a] +
√
2[α2 − a]
√
2[β2 − b].
Now we write
√
2[α2β2 − ab] =
√
2[α2β2 − ab+ ab− ab− α2b+ α2b− β2a+ β2a]
=
√
2[(α2 − a)(β2 − b) + b(α2 − a) + a(β2 − b)].
Making use of subadditivity of square root we obtain
√
2[α2β2 − ab] ≤
√
2(α2 − a)(β2 − b) +
√
2b(α2 − a) +
√
2a(β2 − b)].
Because 2 < 4, a ≤ α2 and b ≤ β2 we get
√
2[α2β2 − ab] ≤
√
4(α2 − a)(β2 − b) + β
√
2(α2 − a) + α
√
2(β2 − b)].

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 111
And as a result we obtain the desired inequality
E(A ⊗B,C ⊗D) ≤ E(A,C)E(B,D).
2.
For any pair of Hermitian operatorsX1 and X2 let us now define a quadratic Lorentz form
(X1, X2)L := [(trX1)(trX2)− trX1X2]. (40)
To find out the motivation standing behind this name let us expand a Hermitian operator
X in an operator basis, X =∑N
2−1
j=0 ajHj . We assume that the basis is orthogonal, trHjHk =
δjk, the first operator is proportional to identity, H0 = I/
√
N , and all other operators Hj are
traceless. Then the form (40) gives
(X,X)L = (trX)2 − trX2 =
(N − 1)
N a
2
0 −
N2−1
∑
j=1
a2j , (41)
which is of Minkowski–Lorentz type. By the help of this notion, super–fidelity can be written
as
G(A,B) = trAB + [(A,A)L(B,B)L]1/2, (42)
while sub–fidelity reads
E(A,B) = trAB + [2(AB,AB)L]1/2. (43)
The forward cone with respect to the form (40) is given by operators X satisfying
(X,X)L ≥ 0 and trX ≥ 0. (44)
Since the density matrices are normalized, trρ = 1, the form (ρ, ρ)L is non–negative.
For a Lorentz form any two forward directed Hermitian matrices A and B satisfy
(A,A)L(B,B)L ≤ [(A,B)L]2. (45)
Substituting this bound into expression (42) we arrive at an upper bound for super–fidelity
G(A,B) ≤ trAB + (trA)(trB)− (trAB) = (trA)(trB). (46)
For the case of normalized density matrices, trρ = 1 we get G(ρ1, ρ2) ≤ 1.
Using the bound (11) for density operators we introduce a third quantity
E′(A,B) = trAB + r(r − 1) r
√
sr(AB), (47)
where r is the rank of matrix AB. Note that for r = 2 this formula is reduced to an expression
(27) for sub–fidelity, hence in this case E′ = E.
Since (sr(X))1/r is concave for density operators we infer that the quantity E′(A,B),
defined in equation (47), is separately concave in A and in B.

112 Sub– and super–fidelity as bounds for quantum fidelity
5.2 Examples and classical analogues
To observe sub– and super–fidelity in action consider a family of mixed states
ρa = a|ψ〉〈ψ|+ (1 − a)I/N, (48)
which interpolates between arbitrary pure state |ψ〉 and the maximally mixed state. It is
straightforward to compute the fidelity between the state ρa and the maximally mixed state
ρ∗ := I/N ,
F (ρa, ρ∗) =
1
N2
(
√
(N − 1)a+ 1 + (N − 1)
√
1− a
)2
, (49)
as well as other bounds
E(ρa, ρ∗) =
1
N +
√
2
1
N
√
1− 1N
√
1− a2, (50)
E′(ρa, ρ∗) =
1
N +
(
1− 1N
)
N
√
((N − 1)a+ 1)(1− a)N−1, (51)
G(ρa, ρ∗) =
1
N +
(
1− 1N
)
√
1− a2. (52)
These results are plotted in Fig. 1 for N = 2, 3, 4, 5.
0
0.25
0.5
0.75
1
0 0.2 0.4 0.6 0.8 1
E E′ F G
0
0.25
0.5
0.75
1
0 0.2 0.4 0.6 0.8 1
E E′ F G
0
0.25
0.5
0.75
1
0 0.2 0.4 0.6 0.8 1
E E′ F G
0
0.25
0.5
0.75
1
0 0.2 0.4 0.6 0.8 1
E E′ F G
a) N = 2
F (a)
a
E = E′ = F = G
b) N = 3
F (a)
a
c) N = 4
F (a)
a
d) N = 5
F (a)
a
Fig. 1. The comparison of sub–fidelity E, bound E′, fidelity F (solid line) and super–fidelity
G. Each plot shows these quantities calculated for the maximally mixed state and a state (48)
depending on the parameter a. For a one–qubit system, case a) N = 2, one has E = E′ = F = G.
Note the difference between these quantities shown for N = 3, 4, 5. In this case E > E′ for a close
to unity.
For N = 2 all these quantities coincide, and the quality of the approximation goes down
with the system size N , as expected. Looking at the graph one could imagine that relation

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 113
E ≤ E′ is fulfilled. However, such an equality does not hold as we found a counter example:
the pair of states analyzed in the figure with parameter a very close to unity.
One may work out several other examples, for which sub– and super–fidelity are easy to
find. Explicit formulas are simple in the case of two commuting density matrices ρp and ρq
with spectra given by vectors ~p and ~q, respectively. In such a classical case these quantities
read
E(ρp, ρq) =
N
∑
i=1
piqi +
√
√
√
√
√2


( N
∑
i=1
piqi
)2
−
N
∑
i=1
p2i q2i

, (53)
F (ρp, ρq) =
( N
∑
i=1
√piqi
)2
, (54)
G(ρp, ρq) =
N
∑
i=1
piqi +
√
√
√
√
(
1−
N
∑
i=1
p2i
)(
1−
N
∑
i=1
q2i
)
. (55)
5.3 The difference G− F
In view of the inequality (22) it is natural to ask how big the difference G − F might be.
Since both quantities coincide if one of the states is pure, let us analyze the case of two mixed
states living in orthogonal subspaces.
More precisely, let us fix an even dimensionality of the Hilbert space N = 2M , and define
two diagonal states, each supported in M dimensional space, ρ1 = 2N diag(1, . . . , 1, 0, . . . , 0)
and ρ2 = 2N diag(0, . . . , 0, 1, . . . , 1). Since they are supported by orthogonal subspaces their
fidelity vanishes, F (ρ1, ρ2) = 0. On the other hand the definition (21) gives their super–fidelity
G(ρ1, ρ2) =
N − 2
N , (56)
equal in this case to the difference G−F . As expected for N = 2 we get G = F = 0. However,
for N large enough the difference G− F may become arbitrarily close to unity.
Thus working with super–fidelity G in place of fidelity F one needs to remember that this
approximation works fine for small systems or where at least one of the states is pure enough.
6 On measurement methods
6.1 Associated physical observables
Here we shall shortly discuss possibilities of measurement of both sub– and super–fidelities
in physical experiments. The approach below follows the techniques used in state spectrum
estimation [20] and nonlinear entanglement detection and/or estimation which has been devel-
oped significantly last years (see [21, 22] and references therein). Those approaches exploited
the properties of SWAP operator and other permutation unitary operations to get the prop-
erties of single state rather than the relation of different states. There were little exceptions:
one was a quantum network measurement of an overlap of the two states [20]. Here we shall
follow the latter idea since we want to estimate the distance of two different quantum states.
In particular we shall see that it is possible to measure these quantities with help of not
more than two collective observables. This fact may be helpful in experimental comparison of

114 Sub– and super–fidelity as bounds for quantum fidelity
two different stationary sources of quantum states. Quite remarkably, as we shall see below,
with help of similar techniques, super–fidelity can be represented by only three experimental
probabilities which makes it very friendly from an experimental point of view.
We start by providing a simple example. First one can see that to calculate sub– and
super–fidelity it is necessary to calculate the values of the terms of the form trAB. Let
A,B ∈M2(C). In this case
A =
(
a11 a12
a21 a22
)
, B =
(
b11 b12
b21 b22
)
(57)
and
trAB = tr
[(
a11b11 + a12b21 a11b12 + a12b22
a21b11 + a22b21 a21b12 + a22b22
)]
= a11b11 + a21b12 + a12b21 + a22b22. (58)
On the other hand this value can be calculated using SWAP gate as
tr [SWAP(A⊗B)] = tr








1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1








a11b11 a11b12 a12b11 a12b12
a11b21 a11b22 a12b21 a12b22
a21b11 a21b12 a22b11 a22b12
a21b21 a21b22 a22b21 a22b22








(59)
= tr








a11b11 a11b12 a12b11 a12b12
a21b11 a21b12 a22b11 a22b12
a11b21 a11b22 a12b21 a12b22
a21b21 a21b22 a22b21 a22b22








= trAB. (60)
To address the question of measurability of the quantities (20), (21) let us first recall the
corresponding permutation operators which we shall need subsequently. The first one will be
just SWAP operator (example of which is the SWAP gate presented above) V12 : HN ⊗HN →
HN ⊗HN which is defined by the action
V12|φ1〉 ⊗ |ψ2〉 = |ψ2〉 ⊗ |φ1〉. (61)
This is a Hermitian operator and as a such it represents an observable. It has a simple
eigendecomposition in the form
V12 = P (+)12 − P
(−)
12 , (62)
where projections P (±)12 onto symmetric and antisymmetric subspaces of HN ⊗HN are
P±12 =
1
2
(I12 ± V12). (63)
Below we shall omit the indices and use the notation V and P (±), if it does not lead to
confusion. An important property usually exploited in case of entanglement detection is that
the formula (60) holds for the SWAP operator V of any dimension [23]. Apart form that
operation we will also need a family of unitary permutation matrices V π1234 : H⊗4N → H⊗4N ,
V π1234|ψ1〉 ⊗ |ψ2〉 ⊗ |ψ3〉 ⊗ |ψ4〉 = |ψπ(1)〉 ⊗ |ψπ(2)〉 ⊗ |ψπ(3)〉 ⊗ |ψπ(4)〉, (64)
where π represents any chosen permutation of the indices (1, 2, 3, 4). For simplicity we shall
drop the indices using the notation V π.

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 115
Let us define the set S of all eight permutations that do not map the sequence (1, 2, 3, 4)
into a one having odd or even elements one after another. For instance, the permutations
defined by the ranges (2341) or (3214) belong to S, while (2314) or (1423) do not. For a fixed
set S′ ⊂ S and some π0 ∈ S we define the following observable:
WS′,π0 = 1
2|S′|
(
∑
π∈S′
V πV π0V π +
∑
π∈S′
V π−1V π−10 V π−1
)
. (65)
A special case is the observable W {π0},π0 = (V π0 + V π−10 )/2 with π0 being just some cyclic
permutation (cf. [21] and references therein). The choice of the permutation π0 and/or the
subset S′ may be motivated by a specific physical situation.
In the case of single qubit sources (N = 2) the observables (65) have highly degener-
ated spectra and the corresponding eigenvectors have very symmetric forms. In particular
the observable W {π0},π0 has spectrum {1,−1, 0} which means that its mean value requires
probabilities of only two outcomes of incomplete von Neumann measurement. The observable
has the spectral decomposition W {π0},π0 = Q(+) −Q(−) where support of the projector Q(+)
is spanned by eigenvectors {|φ1〉 = |0000〉, |φ2〉 = |1111〉, |φ3〉 = (|0111〉 + |1011〉 + |1101〉+
|1110〉)/2, |φ4〉 = (|0011〉 + |0110〉 + |1001〉 + |1100〉)/2, |φ5〉 = (|0101〉 + |1010〉)/
√
2, |φ6〉 =
σ⊗4x |ψ4〉} while the support of the second projector Q(−) (orthogonal to Q(+)) corresponds to
{I ⊗ σ⊗2z ⊗ I|φ3〉, I⊗2 ⊗ σ⊗2z |φ4〉, I⊗3 ⊗ σz|φ5〉, I ⊗ σ⊗2z ⊗ I|φ6〉}.
To illustrate how to measure the quantities E and G suppose now we can perform collective
measurements on two and four copies of both quantum states. We plan measurements that
allow two or four copies of analyzed states to interact. Then (cf. [20, 21, 22] and references
therein) the sub– and super–fidelities can be represented in terms of averages of following
observables,
E(ρ1, ρ2) = trV ρ1 ⊗ ρ2 +
√
2[(trV ρ1 ⊗ ρ2)2 − trWS,π0ρ1 ⊗ ρ2 ⊗ ρ1 ⊗ ρ2], (66)
G(ρ1, ρ2) = trV ρ1 ⊗ ρ2 +
√
1− trV ρ1 ⊗ ρ1
√
1− trV ρ2 ⊗ ρ2. (67)
There are two simple but important observations to be made. The sub–fidelity E can be
measured with help of two setups: (i) the one measuring the observable V and (ii) the second
one measuring observable WS,π0 . Each setup requires one source: setup (i) needs the source
that creates, say, pairs ρ1 ⊗ ρ2, while setup (ii) requires a source producing quadruples of the
form, say, ρ1 ⊗ ρ2 ⊗ ρ1 ⊗ ρ2.
Our scheme will work also for a worse source that produces one of the pairs (quadruples)
{ρ1 ⊗ ρ2, ρ2 ⊗ ρ1} ({ρ1 ⊗ ρ2 ⊗ ρ1 ⊗ ρ2, ρ2 ⊗ ρ1 ⊗ ρ2 ⊗ ρ1}) at random according to an unknown
biased probability distribution, which will not affect the results of the corresponding estimate
for sub–fidelity.
The second observation is that the super–fidelity G can be measured with help of single
setup, namely the one that measures observable V , but requires its application to three types
of sources i.e. the ones creating pairs ρ1 ⊗ ρ1, ρ2 ⊗ ρ2, and, say, ρ1 ⊗ ρ2. Again, the last
source may produce at random one of the pairs {ρ1 ⊗ ρ2, ρ2 ⊗ ρ1} and this will not affect the
estimate for super–fidelity.
It is very interesting to study the form of super–fidelity in terms of directly measurable
quantities, i.e. probabilities, since it has a simple optical implementation. Let us introduce

116 Sub– and super–fidelity as bounds for quantum fidelity
the probabilities of the projection onto the antisymmetric subspace of HN ⊗HN :
p(−)ij = trP (−)ρi ⊗ ρj , i, j = 1, 2. (68)
Then super–fidelity has a particularly nice form,
G(ρ1, ρ2) = 1− 2
(
p(−)12 −
√
p(−)11 p
(−)
22
)
, (69)
which is crucial for further discussion. Note that the super–fidelity can be represented in
terms of only three probabilities that can be measured in a single set-up. One can perform
a simple consistency test by checking, whether the combination of experimental probabilities
satisfy (up to error bars) the condition p(−)12 −
√
p(−)11 p
(−)
22 ≤ 0.5 – otherwise one had an
unphysical result, since super–fidelity can not be negative. Note that the probability p(−)11
has been already measured experimentally for two copies of composite systems in context of
entanglement detection [25] or estimation [26] under some assumptions about the nature of
the sources. In subsection below we shall refer to the scheme analogous to the one utilized in
Ref. [25].
It is interesting to note that if the state ̺ is of d-dimensional type, then reproduction
of sub– and super–fidelity via quantum tomography requires 2d2 − 2 independent quantities
to be estimated since each of the two states is described by d2 − 1 real parameters. On the
other hand, to find the quantities E and G in the way described above one requires only two
or three independent real quantities (probabilities) to be estimated independly on how large
the dimension d is. The price to be payed is, of course, that one must perform collective
experiments. Preparation of reliabe setups of such experiments might be a good test for
quantum engeneering.
6.2 Measuring super–fidelity of states representing photons polarizations
Consider now physical setup that would compare two states of polarization of single photon
in terms of super–fidelity G. In this case the density matrix is defined on Hilbert space
isomorphic to C2 where the horizontal (vertical) polarization, usually denoted by |H〉 (|V 〉)
corresponds to the standard basis element |0〉 (|1〉). Suppose one has memoryless sources of
two types Si (i = 1, 2) sending photons in polarization states ρ1, ρ2 respectively.
The experimental setup is elementary. We have sources Si, Sj , where we put either
i = j = 1, 2 (sources of the same type) or, say i = 1, j = 2 (different sources) then we have
a beamsplitter (in equal distance to the source) and two detectors behind it (see Fig. 2). If
two photons form sources Si, Sj meet on the beamsplitter and the two detectors click, we
have so–called anticoalescence event, which happens with probability p(−)ij [25]. Otherwise
we deal with a coalescence result which occurs with probability p(+)ij = 1 − p
(−)
ij . Putting
all three probabilities of anti-coalescence into formula (69) we reproduce the expression for
super–fidelity.
This seems to be the most easy experiment with two sources to perform. Such an experi-
ment can be realized for two sources of photons engineered with help of controlled decoherence
(in a way similar to Ref. [27]) corresponding to two different mixed states of a qubit.
The above scheme immediately extends to the case of states ρ1, ρ2 are defined on N = 2n-
dimensional Hilbert space representing polarization degrees of freedom of n photons. In this

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 117
BS
D1 D’1
SjSi
Fig. 2. Elementary scheme with one beam-splitter BS. The sources Si, Sj are chosen to be, in turn,
ot the same (i = j = 1, 2) and different (i = 1, j = 2) type. Single click in either of the detectors
D, D′ corresponds to projection into symmetric two-qubit subspace of photon polarization, while
two clicks represent projection onto one-dimensional antisymmetric (singlet) subspace.
case the total Hilbert space is HN = (C2)⊗n and the scheme reads as in Fig. 3 (compare
[28, 22]). If the probability p(sk),kij with sk = −1 (sk = +1) corresponds to anticoalescence
(coalescence) on k-th beamsplitter, i.e. it represents the probability of two clicks (one click)
in the pair of detectors Dk, D′k, then the total probabilities:
p(−)ij =
∑
s1,s2...,sn: s1s2...sn=−1
p(s1),1ij p
(s2),2
ij ...p
(sn),n
ij , i, j = 1, 2 (70)
are these we put into (69). In the formula above we count all the cases when an odd number of
anti-coalescence events occurs, provided that there is no photon losses during the experiment.
Si
BS1
D1 D’1
BS2
D2 D’2
BSn
Dn D’n
Sj
....
....
….
Fig. 3. The scheme for measurement of super–fidelity of the states of n-photon polarizations.
According to (70) only the events with double clicks in odd number of detector pairs contribute
to each of the three probabilities in the formula (69).
For n = 2 this type of experiment has already been performed with two two-photon
sources producing entangled states [25]. However, the sources were considered to provide

118 Sub– and super–fidelity as bounds for quantum fidelity
the same state on average rather than two different ones. A similar reasoning was used in
another recent experiment, in which photon polarization and momentum degrees of freedom
were used to estimate the concurrence where additional strong assumption about purity of
each copies were also used [26]. In general, measurements schemes of quantities like purity,
concurrence, sub– and super–fidelity in the collective framework like the one presented here
requires the assumption that the sources producing states are stationary and memoryless.
Quite remarkably this is the same assumption one makes in quantum tomography. As dis-
cussed in [29, 30] there may be difficulties with satisfying it in real experimental scenarios for
instance due to classical correlations between the consecutive copies of the system. In other
words the condition of having the global state in ̺N form (which mathematically corresponds
to quantum de Finnetti condition [31]) may not be obeyed. This point requires more analysis,
and leads, in general to nontrivial issues. It seems however that stationarity and memoryless
character of the source can usually be satisfied approximately. Then the present measure-
ment would serve (similarily like quantum tomography does, though may be in a way more
sensitive to source correlations) as an approximative, coarse-grained-like characteristic of the
states sources under reasonable physical assumptions.
6.3 Quantum networks
There is yet another method of detection of quantities that may be considered here. This is a
method based on quantum networks. It is known that a unitary operation U acting on state
σ but controlled by a qubit in the superposition state |+〉 ≡ 12 (|0〉+ |1〉) reproduces the value
Re(trUσ) directly as a mean value of the Pauli matrix 〈σx〉 measured on the controlled qubit
[24, 20]. This fact allows us to measure certain nonlinear functions of the state. To get trρk
one takes k copies of the state, σ = ρ⊗k and takes for U the operator of cyclic permutation,
in full analogy to V π used in previous subsection. To measure the overlap of two matrices,
ρ1, ρ2, one takes σ = ρ1 ⊗ ρ2 and uses the SWAP operator, U = V .
The corresponding network is already provided explicitly in Ref. [20], so we shall not
write it down here. Such a network allows one to measure all three quantities needed to
reproduce the super–fidelity G. Indeed, the network produces directly (as mean values of σz
on controlled qubit) all three mean values: trρiρj , i = 1, 2 provided that states forming the
input of the controlled part of the network are ρi ⊗ ρj . Some alternative constructions of
programmable networks designed to measure super–fidelity are also possible.
Similarly, sub–fidelity E can also be estimated with a network-based experimental scheme.
Following reasoning from Ref. [21] one constructs the following programmable quantum
network – see Fig. 4. Depending on the program state |Ψ12〉 as a mean value of σz of the
measured controlling qubit one gets
(i) trρ1ρ2 if |Ψ12〉 = |0〉|0〉,
(ii) trρ1ρ2ρ1ρ2 if |Ψ12〉 = |1〉|0〉,
(iii) 12
(
trρ1ρ2ρ1ρ2 − (trρ1ρ2)2
)
if |Ψ12〉 = (|0〉|1〉+ |1〉|0〉)/
√
2 i.e. if it is in Bell state.
The last quantity up to the factor (− 14 ) is just the quantity that occurs under the square
root in the formula for E. In general to estimate the sub–fidelity E we may ”run” the
first ”program” and either the second or the third one. Alternatively, we may run all three

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 119
|Ψ12〉
• • •
• • •
|0〉 X  •  X  •   • • •
|0〉 H π • • • • • α H "%#$M
ρ1 × × ×
ρ2 × × ×
ρ1 × ×
ρ2 × ×
Fig. 4. Example of programmable network allowing to measure in particular the quantities
tr(ρ1ρ2ρ1ρ2) and 12 tr(ρ1ρ2ρ1ρ2) − tr(ρ1ρ2)2. The state |Ψ12〉 represents the program (see the
main text). Symbol M corresponds to the measurement of Pauli matrix σz . The last phase gate
usually is chosen to be α = 0 unless the interferometric picture with visibility is needed (cf. Ref.
[20]). See [8] for the description of quantum gates used in this circuit.
programs and use the data to verify the accuracy of the experiment by comparing the two
partially independent estimates of E obtained in that way. It is easy tu see, that the same
network can be used to estimate super–fidelity G if one puts as an input ρi ⊗ ρi ⊗ ρj ⊗ ρj ,
j = 1, 2.
7 Distances and geometry of the space of states
7.1 Hilbert-Schmidt distance and flat geometry
The geometry of the space of quantum states depends on the metric used [32, 33, 34, 9]. The
set ΩN of mixed states of size N reveals the Euclidean (flat) geometry if it is analyzed with
respect to the Hilbert-Schmidt distance,
DHS(ρ1, ρ2) =
√
tr[(ρ1 − ρ2)2]. (71)
To demonstrate this property let us first concentrate on the simplest case, N = 2. Making
use of the notion of a coherence vector ~τ any state of a qubit can be written in the Bloch
representation
ρ = IN + ~τ ·
~λ. (72)
Here ~λ denotes the vector of three rescaled traceless Pauli matrices {σx, σy , σz}/
√
2, which are
orthogonal in the sense of the Hilbert-Schmidt scalar product, 〈λk|λm〉 = tr(λk)†λm = δkm.
Together with λ0 = I/
√
2 they form an orthonormal basis in the space of complex density
matrices of size two. Due to Hermiticity of ρ the three-dimensional Bloch vector ~τ is real.
Positivity condition implies |~τ | ≤ 1/
√
2 = R2 with equality for pure states, which form
the Bloch sphere of radius R2. Representation (72) implies that for any state of a qubit
trρ2 = 1/2 + |τ |2.
Consider two arbitrary density matrices and express their difference ρ1 − ρ2 in the Bloch
form. The entries of this difference consist of the differences between components of both

120 Sub– and super–fidelity as bounds for quantum fidelity
Bloch vectors ~τ1 and ~τ2. Therefore Hilbert-Schmidt distance induces the flat (Euclidean)
geometry of Ω2,
DHS
(
ρ~τ1 , ρ~τ2
)
= DE(~τ1, ~τ2), (73)
where DE is the Euclidean distance between both Bloch vectors in R3.
It is worth to add that expression (73) holds for an arbitrary N . In this case ~τ is a real
vector with N2 − 1 components, while the vector ~λ = {λk}N2−1k=1 in (72) denotes the set of
N2 − 1 traceless generators of the group SU(N). Positivity of ρ implies that the length of
the Bloch vector is limited by
|~τ | ≤ DHS(I/N, |ψ〉〈ψ|) =
√
N − 1
N =: RN . (74)
For N = 2 the condition |~τ | ≤ R2 is sufficient to imply that the corresponding matrix is
positive and represents a state, while for N ≥ 3 it is only a necessary condition [9]. This is
related to the fact that with respect to the flat, H–S geometry the set Ω2 forms a full 3-ball,
while for larger N the set ΩN forms a convex subset of the (N2−1)-dimensional ball of radius
RN centered at ρ∗ = I/N .
7.2 Bures distance and the geometry it induces
The notion of fidelity, introduced in (1), can be used to define the Bures distance [35, 2]
DF (ρ1, ρ2) =
√
2− 2
√
F (ρ1, ρ2). (75)
or the Bures length [36] (later called angle in [8]),
D′F (ρ1, ρ2) := arccos
√
F (ρ1, ρ2) =
1
2
arccos
(
2F (ρ1, ρ2)− 1
)
. (76)
For any pair of pure states the Bures length coincides with their Fubini–Study distance,
D′F
(
ρψ, ρφ
)
= dFS
(
|ψ〉, |φ〉
)
= arccos|〈ψ|φ〉|.
The Bures metric is distinguished by its rather special properties: it is a Riemannian,
monotone metric [37], Fisher adjusted metric [33], closely related to the statistical distance
[32].
It is not difficult to describe the geometry of the set of mixed states of a single qubit
induced by the Bures metric. Consider a mixed state ρ ∈ Ω2 and its transformation proposed
in [4]
ρ(x, y, z) →
(
x, y, z, t =
√
1
2
− x2 − y2 − z2
)
. (77)
It blows up the Bloch ball B3 of radius R2 = 1/
√
2 into a hyper-hemisphere 12S3 of the same
radius. The original variables (x, y, z) denote the parameters of the state in the Bloch vector
representation. The auxiliary variable reads t =
√
1/2− |τ |2 in terms of the Bloch vector, so
that t2 + trρ2 = 1. The maximally mixed state ρ∗ = (0, 0, 0), is mapped into a hyper-pole.
It is equally distant from all pure states located at the hyper-equator S2, which form the
boundary of Ω2.
Any state ρ is uniquely represented by an ’extended Bloch vector’, ~v = (x, y, z, t) of length
R2. The auxiliary variable reads t =
√
1/2− |τ |2 in terms of the Bloch vector, so that

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 121
t2 + trρ2 = 1. Consider two states ρ1 and ρ2, described by two vectors ~v1 and ~v2 ∈ R4,
which form the angle ϑ. Since for any one-qubit states the bound (13) becomes an equality,
we see that fidelity between them reads F (ρ1, ρ2) = trρ1ρ2 +
√
t21t22 = 1/2 + ~τ1 · ~τ2 + t1t2.
This can be rewritten with the use of extended vectors ~vi and the angle between them,
F = 1/2 + ~v1 · ~v2 = 1/2 +R22 cosϑ. Since R22 = 1/2 we find
ϑ = arccos
(
2F − 1
)
= 2D′F (ρ1, ρ2), (78)
which shows that the Bures length (76) between any two mixed states is proportional to the
Riemannian distance between the corresponding points at the Uhlmann hemisphere.
Making use of the fidelity F one can also define other distances in the space of quantum
states. For instance, Gilchrist et al. [38] have shown that the root infidelity
C(ρ1, ρ2) =
√
1− F (ρ1, ρ2) (79)
satisfies the triangle inequality and thus introduces a metric. This very quantity can be used
to bound the trace distance Dtr(ρ1, ρ2) = 12Tr|ρ1 − ρ2| ≤ C(ρ1, ρ2) from above [39, 9]. Note
that the Bures distance, Bures length and root infidelity are functions of the same quantity,
so they generate the same topology.
7.3 Modified Bures length
In analogy to (75) and (76) one may ask whether
DG(ρ1, ρ2) =
√
2− 2
√
G(ρ1, ρ2). (80)
and
D′G(ρ1, ρ2) := arccos
√
G(ρ1, ρ2) (81)
define distances. This is obvious for N = 2 because F = G. The situation changes for N ≥ 3,
for which the DF and DG do differ and only F ≤ G is valid.
We do not know, whether DG and D′G are distances. However, it can be proved that a
direct analogue of the root infidelity (79)
C′(ρ1, ρ2) =
√
1−G(ρ1, ρ2) (82)
is a genuine distance. The same is true for the modified Bures length,
D′M (ρ1, ρ2) = arccosG(ρ1, ρ2). (83)
Proof. Let us call L the direct sum of the real linear space of all Hermitian operators
and the 1-dimensional space of real numbers. Its elements are {H,x}, H Hermitian, x a real
number. L becomes Euclidean (i.e. a real Hilbert space) by defining the scalar product
({H1, x1}, {H2, x2}) = trH1H2 + x1x2. (84)
Let us denote by B(L) the unit ball of L and by S(L) the unit sphere. Our proof rests on
the embedding of the Hermitian operators
BN =
{
H | trH = 1, trH2 ≤ 1
}
(85)

122 Sub– and super–fidelity as bounds for quantum fidelity
into S(L) by
H → ξH :=
{
H,
√
1− trH2
}
. (86)
Clearly, (ξH , ξH) = 1, and from (84) we get
(ξH , ξH′) = G(H,H ′). (87)
Now it is obvious that
√
2− 2G is the Euclidean distance between ξH and ξ′H , provided H
and H ′ belong to BN . Because the density operators form a subset of BN , (82) is a distance.
From (87) we get G(H,H ′) = cosα, where α is the angle from which ξH and ξ′H are seen
from the center of the ball B(L). Thus, arccosG = α and, in particular, (83) is a distance.
2.
Let us now return to the two conditions of (85). They are equivalent with
BN =
{
H | trH = 1, tr(H − 1N I)
2 ≤ N − 1N
}
(88)
and they describe the smallest ball containing the state space. BN is an affine translate
by 1/N of the generalized Bloch-ball, [8]. BN is centered at A = N−1I and is of radius
√
(N − 1)/N .
Above we have embedded BN by the map (86) into the sphere S(L). Just this gives the
opportunity to apply Mielnik’s definition [40] for a transition probability (he also called it
affine ratio) of two extremal states of a compact convex set. In our case the compact convex
set is B(L) and its extremal part is S(L). At the case at hand, Mielnik’s procedure starts
with first choosing an extremal point ξ ∈ S(L) and selecting all affine functions l satisfying
l(ξ) = 1 and 0 ≤ l ≤ 1 on B(L). Any such function can be written
l(η) = a+ (ξ, η) + 1a+ 2 , (89)
with a ≥ 0 and η ∈ L arbitrarily. Now we have to vary over all these affine functions,
pM (η, ξ) := minl l(η) = mina
a+ (ξ, η)
2 + a , (90)
to get Mielnik’s transition probability
pM (ξ, η) =
1 + (ξ, η)
2
. (91)
Returning to H,H ′ ∈ BN , we can write
pM (H,H ′) := pM (ξH , ξH′ ) =
1 +G(H,H ′)
2
(92)
and, in becoming even more special by choosing two density operators for H and H ′ in the
equation above, we arrive at
DM (ρ1, ρ2) = 2
√
1− pM (ρ1, ρ2) = 2 sin
α
2
, (93)
(using 2 cos2 = 1 + cos) and also at
D′M (ρ1, ρ2) = 2 arccos
√
pM (ρ1, ρ2). (94)

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 123
8 Concluding remarks
In this paper we analyzed various bounds for quantum fidelity. Two quantities, we propose to
call sub– and super–fidelity, posses particularly nice properties. On one hand these quantities
form universal lower and upper bounds for the fidelity. Moreover, with respect to the tensor
product they display sub– and super–multiplicativity.
On the other hand, quantities E and G are much easier to calculate than the original
fidelity F . To compute any of these bounds it is enough to evaluate three traces only. Thus
one can expect, the quantities introduced in this paper might become useful for various tasks
of the theory of quantum information processing. Furthermore, under a realistic assumption
that several copies of both states are available, it is possible to design a scheme to measure
experimentally sub– and super–fidelity between arbitrary mixed states. For instance, the
measurement of super–fidelity is possible if one has three copies of each state. In this paper we
have worked out concrete schemes of such experiments concerning the super–fidelity between
any two mixed states representing the polarization of photons.
Acknowledgements
We would like to thank A. Buchleitner for inviting three of us to Dresden in September 2005
for a workshop on Quantum Information, during which our collaboration on this project was
initiated. It is also a pleasure to thank I. Bengtsson and M. Horodecki for inspiring discussions.
J.A.M. would like to thank Iza Miszczak for her help.
We acknowledge financial support by the Polish Ministry of Science and Higher Education
under the grants number N519 012 31/1957 and DFG-SFB/38/2007, by the LFPPI network
and by the European Research Project SCALA.
Note added
After this paper was submitted we learned about a related work by Mendonca et al. [43]
in which the super–fidelity was independently introduced and was called an ’alternative fi-
delity’ measure. In this valuable work the authors provide an alternative proof of super–
multiplicativity of G, discuss its relation to the trace distance and analyze the distance G
induces into the space of mixed quantum states, and prove that G is jointly concave in its
two arguments.
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Appendix A Algebraic facts
In this appendix we collect useful algebraic facts, which are used in the main body of the
paper.
Fact 1 (From corollary IX.5.3 in [7]) For any positive matrices A and B and every uni-
tarily invariant norm ||| · ||| we have
|||AνB1−ν ||| ≤ |||A|||ν |||B|||1−ν , (A.1)
where ν ∈ [0, 1].
Fact 2 (From corollary IX.5.4 in [7]) For any positive matrices A and B and every uni-
tarily invariant norm ||| · ||| we have
|||AνBν ||| ≤ |||I|||1−ν |||AB|||ν , (A.2)
where ν ∈ [0, 1].
Next two facts can be found in [16].
Fact 3 Matrix AB is similar to matrices
√
AB
√
A and
√
BA
√
B.
Fact 4 For positive matrices A and B matrix AB has positive eigenvalues.
Fact 5 If p1 + p2 + · · ·+ pn = 1 and pi ≥ 0 then
1− p21 − p22 − · · · − p2n =
∑
i6=j
pipj . (A.3)
Proposition 4 Let g be defined as
g(x) =
∑
i6=j
√xi
√xj . (A.4)
For x, y ∈ Rn+ such that
k
∏
i=1
xi ≤
k
∏
i=1
yi, for k = 1, . . . , n, (A.5)
with equality for k = n, we have
g(x) ≤ g(y). (A.6)
Proof. We introduce notation
gi(·) =
∂g
∂xi
(·). (A.7)
Direct computation shows that function g satisfies
u1g1(u) ≥ u2g2(u) ≥ · · · ≥ ungn(u), (A.8)

126 Sub– and super–fidelity as bounds for quantum fidelity
for u ∈ Rn such that u1 ≥ u2 ≥ · · · ≥ un ≥ 0. We denote αi = log(xi) and βi = log(yi). Note
that (A.5) can be rewritten as
k
∑
i=1
αi ≤
k
∑
i=1
βi for k = 1, . . . , n, (A.9)
with equality for k = n. We define new function
h(v) = g(ev1 , ev2 , . . . , evn). (A.10)
For a given vector u such that u1 ≥ u2 ≥ · · · ≥ un ≥ 0, and vi = log(ui) we have
v1 ≥ v2 ≥ · · · ≥ vn. (A.11)
Using (A.8) we can write
ev1g1(ev1 , ev2 , . . . , evn) ≥ · · · ≥ ev1gn(ev1 , ev2 , . . . , evn). (A.12)
Now from above and (A.10) we have
h1(v) ≥ h2(v) ≥ · · · ≥ hn(v). (A.13)
Note now that function h satisfies condition from [15, Theorem 3.6] and thus it is Schur-
convex, so
h(α) ≤ h(β). (A.14)
Using (A.10) we can write
g(x) ≤ g(y). (A.15)
Thus the proof is complete. 2.
Fact 6 (Ho¨lder’s inequality [19]) For a > 1, b = a/(a − 1) and positive semidefinite A
and B we have
tr(AB) ≤ (trAa)1/a(trBb)1/b. (A.16)
Fact 7 For density matrices A and B we have
1−
√
trA2
√
trB2 ≥
√
1− trA2
√
1− trB2. (A.17)
Proof. This inequality can be rewritten in equivalent form
1− 2
√
trA2trB2 + trA2trB2 ≥ 1− trA2 − trB2 + trA2trB2, (A.18)
which is equivalent to
√
trA2trB2 ≤ trA
2 + trB2
2
. (A.19)
This completes the proof since for any positive numbers the arithmetic mean is always greater
than or equal to the geometric mean. 2.

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 127
Fact 8 (Maclaurin inequality [42, p. 5]) For a given matrix A of rank r and with r pos-
itive eigenvalues we have
k
√
sk(A)
(r
k
) ≥ k+1
√
sk+1(A)
( r
k+1
) (A.20)
for 1 ≤ k < r.
Appendix B Proof of the lower bound (10)
To prove that sub–fidelity E is not larger than fidelity F , let us take a look at equations (15)
and (27) in which both quantities are expressed in terms of the second symmetric function.
We can rewrite the function s2, which forms fidelity,
s2(
√
A1/2BA1/2) =
∑
i<j
λi(
√
A1/2BA1/2)λj(
√
A1/2BA1/2) (B.1)
=
∑
i<j
√
λi(A1/2BA1/2)
√
λj(A1/2BA1/2) (B.2)
=
∑
i<j
√
λi(AB)
√
λj(AB). (B.3)
The last equality is the consequence of similarity of matrices A1/2BA1/2 and AB. Making
use of subadditivity of square root we obtain
s2(
√
A1/2BA1/2) ≥
√
∑
i<j
λi(AB)λj(AB) (B.4)
=
√
s2(AB). (B.5)
As a consequence we get
F (A,B) = trAB + 2s2(
√
A1/2BA1/2) ≥ trAB + 2
√
s2(AB) = E(A,B). (B.6)
Appendix C Proof of the lower bound (11)
To prove inequality (11) we use Fact 8 (Maclaurin inequality) and obtain


s2
(√
A1/2BA1/2
)
(r
2
)


1/2
≥


sr
(√
A1/2BA1/2
)
(r
r
)


1/r
. (C.1)
Using Fact 3 we get
s2
(√
A1/2BA1/2
)
≥
(r
2
)
(
sr
(√
A1/2BA1/2
))2/r
=
(r
2
)
( r
∏
i=1
λi
(√
A1/2BA1/2
)
)2/r
=
(r
2
)
( r
∏
i=1
√
λi
(
A1/2BA1/2
)
)2/r
=
(r
2
)
( r
∏
i=1
λi (AB)
)1/r
=
(r
2
)
r
√
sr(AB).

128 Sub– and super–fidelity as bounds for quantum fidelity
Now using (15) we write
F (A,B) = trAB + 2s2
(√
A1/2BA1/2
)
≥ trAB + r(r − 1) r
√
sr(AB). (C.2)
Appendix D Proofs of Lemmas
Proof of Lemma 1. Observe that the matrix ρ1/21 ρ2ρ
1/2
1 is similar to ρ1ρ2 and thus
2s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
=
∑
i6=j
λi
(
√
ρ1/21 ρ2ρ
1/2
1
)
λj
(
√
ρ1/21 ρ2ρ
1/2
1
)
=
∑
i6=j
√
λi
(
ρ1/21 ρ2ρ
1/2
1
)
λj
(
ρ1/21 ρ2ρ
1/2
1
)
=
∑
i6=j
√
λi (ρ1ρ2)λj (ρ1ρ2),
where λi(A) denotes ith eigenvalue of a matrix A.
Let us define a function g : Rn → R which acts on a vector ~x of non-negative numbers
g(~x) :=
∑
i6=j
√xixj . (D.1)
It allows one to rewrite
2s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
= g
(~λ(ρ1ρ2)
)
, (D.2)
where ~λ(A) denotes the vector of eigenvalues of A.
From [18, Theorem 3.3.2 and 3.3.4] we obtain
k
∏
i=1
λi(ρ1ρ2) ≤
k
∏
i=1
λi(ρ1)λi(ρ2) for k = 1, . . . , n, (D.3)
with equality for k = n. Making use of Proposition 4 from Appendix A with xi = λi(ρ1ρ2)
and yi = λi(ρ1)λi(ρ2) we obtain
g
(~λ(ρ1ρ2)
)
≤ g
(~λ(ρ1) ◦ ~λ(ρ2)
)
, (D.4)
where ◦ denotes Hadamard product, [18, Definition 7.5.1]. Now making use of (D.2) we get
s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
≤ s2
(
√
diag(~λ(ρ1))diag(~λ(ρ2))
)
. (D.5)
And thus the proof is complete. 2.
Proof of Lemma 2. For given density matrices ρ1, ρ2 with eigenvalues p1, . . . , pn and
q1, . . . , qn respectively. We denote diagonal matrices with entries on diagonal p1, . . . , pn and
q1, . . . , qn as diag(p), diag(q) respectively.
Rewriting the second elementary function s2 we obtain
2s2
(
√
diag(p)diag(q)
)
=
∑
i6=j
√piqi
√pjqj . (D.6)

J. A. Miszczak, Z. Pucha la, P. Horodecki, A. Uhlmann, and K. Zyczkowski 129
On the other hand
2
√
s2(ρ1)s2(ρ2) =
√
(
1−
∑
p2i
)(
1−
∑
q2i
)
. (D.7)
Let us define vectors x, y ∈ Rn2
xi,j =
√pipj(1 − δi,j), yi,j =
√qiqj(1− δi,j), (D.8)
where xi,j = xn(i−1)+j . Using Cauchy–Schwarz inequality
|〈x|y〉| ≤
√
〈x|x〉
√
〈y|y〉, (D.9)
we get
∑
i6=j
√piqi
√pjqj ≤
√
(
1−
∑
p2i
)(
1−
∑
q2i
)
. (D.10)
This completes the proof. 2.
Appendix E The case N = 3
In this section we are going to study the fidelity of two states ρ1 and ρ2 in the case where
the rank r of their product ρ1ρ2 is not greater than 3. As in Section 2 we will denote
eigenvalues of
√
ρ1/21 ρ2ρ
1/2
1 by λi, so eigenvalues of ρ
1/2
1 ρ2ρ
1/2
1 are given by {λ2i } and by
similarity we have that the eigenvalues of ρ1ρ2 are also given by {λ2i }. Since r ≤ 3, not
more than three eigenvalues of ρ1ρ2 are positive, so the third symmetric function (6) reads
s3(ρ1ρ2) = (λ1λ2λ3)2. This is so for any two states of a qutrit, so for N = 3 one has
s3(ρ1ρ2) = det(ρ1ρ2).
Consider now the expression for fidelity (4) which can be rewritten with the use of the
second symmetric function,
F (ρ1, ρ2) = trρ1ρ2 + 2s2
(
√
ρ1/21 ρ2ρ
1/2
1
)
. (E.1)
The square of the symmetric function presented in the above equation, can be written as
(
∑
i<j λiλj
)2
=
∑
i<j λ2i λ2j +R. The reminder R, defined implicitly by this equation, is equal
to zero if r ≤ 2 and the sum consists of a single term only. It is difficult to handle R generally.
But if r = 3 one has
R = λ1λ2(λ2λ3 + λ3λ1) + ... = 2(λ1λ2λ3)(λ1 + λ2 + λ3). (E.2)
However, in the particular case r ≤ 3 discussed here used to (3) one has λ1 + λ2 + λ3 =
√
F
while λ1λ2λ3 =
√
s3(ρ1ρ2). Combining this with (4) we get the equation for fidelity satisfied
for r ≤ 3
F = trρ1ρ2 + 2
√
s2(ρ1ρ2) + 2
√
F
√
s3(ρ1ρ2). (E.3)
In the case r ≤ 2 the third function s3 vanishes, so this equation leads to an expression,
F = trρ1ρ2 + 2
√
s2(ρ1ρ2) = trρ1ρ2 + 2λ1λ2, already discussed in Section 2.

130 Sub– and super–fidelity as bounds for quantum fidelity
Another relation for fidelity is due to the fact that an assumption r ≤ 3 implies that
∑
j<k
λjλk = (λ1λ2λ3)(λ−11 + λ−12 + λ−13 ). (E.4)
Therefore in this case one has
s2
(
√
A1/21 BA1/2
)
=
√
det(AB)F (1/A, 1/B). (E.5)
Lifting for a moment the assumption that the arguments of fidelity have to be normalized,
we arrive therefore at another equation for fidelity satisfied for r ≤ 3,
F (A,B) = trAB + 2
√
det(AB)F (1/A, 1/B). (E.6)