A simple test of quantumness for a single system
Robert Ali
ki
Institute of Theoreti
al Physi
s and Astrophysi
s, University of Gda«sk, Wita Stwosza 57, PL 80-952 Gda«sk, Poland
Ni
holas Van Ryn
S
hool Of Physi
s, Quantum Resear
h Group, University of KwaZulu-Natal,
Westville Campus, Private Bag x54001, Durban, South Afri
a.
November 17, 2018
Abstra
t
We propose a simple test of quantumness whi
h 
an de
ide whether for the given set of a

essible experimental
data the 
lassi
al model is insu�
ient. Take two observables A,B su
h that for any state ψ their mean values
satisfy 0 ≤ 〈ψ|A|ψ〉 ≤ 〈ψ|B|ψ〉 ≤ 1. If there exists a state φ su
h that the se
ond moments ful�ll the inequality
〈φ|A2|φ〉 > 〈φ|B2|φ〉 then the system 
annot be des
ribed by the 
lassi
al probabilisti
 s
heme. An example of
an optimal triple (A,B, φ) in the 
ase of a qubit is given.
Although we are 
on�dent that the proper theory des
ribing all physi
al phenomena is the quantum theory, there
are many situations where the 
lassi
al des
ription in terms of fun
tions and probability distributions over a suitable
"phase-spa
e" is su�
ient. In parti
ular, the systems 
onsisting of a large number of parti
les and/or emerging
in quantum states 
hara
terized by large quantum numbers are supposed to behave 
lassi
ally. The standard
explanation of this fa
t refers to the stability properties of quantum states with respe
t to the intera
tion with an
environment. For large quantum systems the intera
tion with the environment is so strong that most quantum
states rapidly de
ay (de
ohere) and the remaining manifold of experimentally a

essible states 
an be des
ribed by

lassi
al models. However, the a
tual border between quantum and 
lassi
al worlds is still a topi
 of theoreti
al
debate and experimental e�orts [1℄. This question is parti
ularly important in the �eld of quantum information.
Useful large s
ale quantum 
omputations would demand preservation of some quantum properties for rather large
physi
al systems, say at least for 103 qubits. Moreover, some promising implementations of a qubit are based on
mesos
opi
 systems whi
h lie on the aforementioned border. In parti
ular, the so-
alled super
ondu
ting qubits are
systems 
omposed of 108− 109 parti
les (Cooper pairs). Therefore, it is important to �nd a simple operational test
of quantumness whi
h 
ould be applied to a single system implementing a qubit. It is generally believed that the
simplest operational and model independent test of quantumness (or stri
tly speaking non-
lassi
ality) is based on
Bell inequalities whi
h involve systems 
omposed of at least two parts [2℄. However, this test is problemati
 for
systems whi
h 
annot be spatially well-separated ("lo
ality loophole").
We propose here a mu
h simpler operational test whi
h 
an be applied to a single system and formulated in
terms of inequalities between mean values of properly 
hosen observables. In 
ontrast to Bell inequalities this test
does not refer to the notion of lo
ality, asso
iated with the spa
e-time, but involves only the most fundamental
di�eren
es between 
lassi
al and quantum observables. It is based on the observation that for any two real fun
tions
f, g satisfying
0 ≤ f(x) ≤ g(x) (1)
and any probability distribution ρ(x) the following inequality holds
〈f2〉ρ ≡
2(x)ρ(x)dx ≤
2(x)ρ(x)dx ≡ 〈g2〉ρ . (2)
On the other hand, as illustrated below for the 
ase of a single qubit, for quantum systems we 
an always �nd a
pair of observables A,B su
h that for all states ψ their mean values satisfy 0 ≤ 〈ψ|A|ψ〉 ≤ 〈ψ|B|ψ〉 but there exists
a state φ su
h that for the se
ond moments 〈φ|A2|φ〉 > 〈φ|B2|φ〉.
The above statement follows from the interesting and nontrivial mathemati
al result in the theory of C∗-algebras
[3℄, whi
h is presented below for 
ompleteness.
Consider an abstra
t formalism where (bounded) observables are elements of a 
ertain C∗-algebra A, and states
are positive normalized fun
tionals A ∋ A 7→ 〈A〉ρ with 〈A〉ρ denoting the mean value of an observable A in a state
http://arxiv.org/abs/0704.1962v3
ρ. For all pra
ti
al purposes we 
an restri
t ourselves to two extreme 
ases: the �rst being a 
lassi
al model where
A is an algebra of fun
tions on a 
ertain "phase-spa
e" with 〈A〉ρ =
A(x)ρ(x)dx, where ρ(x) is some probability
distribution, and the se
ond being a �nite quantum model in whi
h A is an algebra of matri
es and 〈A〉ρ = Tr(ρA),
where ρ is some density matrix. For any pair of observables A,B ∈ A, the order relation A ≤ B means that
〈A〉ρ ≤ 〈B〉ρ for all states ρ (in fa
t it is enough to take all pure states). Now we 
an formulate the following:
Theorem. If the following impli
ation
0 ≤ A ≤ B =⇒ A2 ≤ B2 (3)
always holds then the algebra A is 
ommutative, i.e. isomorphi
 to the algebra of 
ontinuous fun
tions on a 
ertain

ompa
t spa
e.
As a 
onsequen
e of the above theorem, for any quantum system there exists a pair of observables (identi�ed
with matri
es) (A,B) su
h that the eigenvalues of A,B and B−A are nonnegative but the matrix B2−A2 possesses
at least one negative eigenvalue.
In order to apply our test of quantumness in an experiment, one should �rst guess a pair of observables A,B
with nonnegative values of possible out
omes and perform a statisti
al test of the inequality 〈A〉ρ ≤ 〈B〉ρ with as
large as possible number of di�erent, generally mixed, initial states ρ. These states should be as pure as possible,
otherwise the quantumness 
ould be not dete
ted. Then one should sear
h amongst these for any states σ satisfying
the relation for the se
ond moments 〈A2〉σ > 〈B2〉σ. If the violation of the 
lassi
al relation (2) holds, it means
that the system exhibits some quantum 
hara
teristi
s. Although there are always in�nitely many triples (A,B, σ),
the e�e
ts of external noise a
ting on the system and measuring apparatus 
an easily wash out the deviations from
"
lassi
ality". Therefore, instead of a random guess it is useful to �nd the examples of su
h triples whi
h maximally
violate 
lassi
ality and 
an be used to optimally design the experimental setting. This 
an easily be a
hieved in the

ase of a qubit whi
h is the most important example for quantum information.
We sear
h for a pair of 2× 2 matri
es A,B and a pure state φ whi
h satisfy
0 ≤ A ≤ B ≤ I , 〈A2〉φ ≡ 〈φ|A2|φ〉 > 〈B2〉φ ≡ 〈φ|B2|φ〉 , (4)
where the observables A and B are normalized in su
h a way that their upper bound is the identity. Su
h a triple
is given as an example as
ξ∗ a2
, B =
, φ =
. (5)
The matrix B is 
hosen to be diagonal, with the identity as its upper bound, and this 
hoi
e of basis 
an be made
sin
e the solution to this problem is unique up to unitary equivalen
e. From the upper bound it 
an be seen that
both eigenvalues of B should be at most 1, and to maximize the violation of (3) one of the eigenvalues is 
hosen to
be �xed at 1. The usual 
ondition |α|2 + |β|2 = 1 applies for the parameters of the state φ. The positivity of the
observables A, B and (B − A) is ensured by the requirement that both their diagonal elements and determinants
are positive. These 
onditions are expressed below as;
0 ≤ b ≤ 1, (6)
0 ≤ a1 ≤ 1, (7)
0 ≤ a1a2 − |ξ|2, (8)
0 ≤ (1 − a1)(b− a2)− |ξ|2. (9)
In order for this triple to satisfy equation (4), it is su�
ient that one of the eigenvalues of (B2 −A2) is found to be
negative while remaining within the 
onstraints listed above.
The eigenvalues of the matrix (B2 −A2) are found to be
b2 + 1− a2
− 2|ξ|2 ±
(b2 − 1 + a2
)2 + 4(a1 + a2)2|ξ|2
. (10)
In order to �nd the most negative value whi
h one of the eigenvalues 
an attain, one need only 
onsider one of the
two. Sin
e the square root is always positive, one eigenvalue always remains greater than the other. A numeri
al
te
hnique is used to 
al
ulate the maximal violation of the inequality (3) sin
e we are unable to �nd an exa
t
solution to this optimisation problem. In �nding the optimal set of parameters a1, a2, b and real ξ, it 
an be seen
that the maximal violation of the inequality (3) arises when the 
onditions (8) and (9) are equalities rather than
inequalities. Using
a1a2 = |ξ|2, (11)
(1− a1)(b− a2) = |ξ|2 (12)
redu
es the problem from four unknowns to two, and the triplet 
an then be written as
a1a2√
a1a2 a2
, B =
, φ =
. (13)
The parameters whi
h result in one of the eigenvalues of (B2 −A2) attaining it's most negative value, and thus
maximally violating inequality (3), while still remaining within the 
onstraints give us the triplet
0.724 0.249
0.249 0.0854
, B =
0 0.309
, φ =
0.391
0.920
. (14)
In this example, the value of 〈φ|B|φ〉 − 〈φ|A|φ〉 is 0.0528, a positive value, whereas it 
an be seen that 〈φ|B2|φ〉 −
〈φ|A2|φ〉 = −0.0590 whi
h 
learly demonstrates the quantum nature of this example. The eigenve
tors and 
orre-
sponding eigenvalues of A using these parameters is 
al
ulated to be
0.946
0.325
= 0.809
0.946
0.325
−0.325
0.946
. (16)
To give a 
on
rete example, one 
an apply these results to the polarization of a single photon. Choosing
a polarization basis as |H〉, |V 〉 and attributing the values 1 to |H〉 and 0.309 to |V 〉 we obtain the observable
B. The observable A 
orresponds to a rotated polarization basis |H ′〉 = cos(19◦)|H〉 + sin(19◦)|V 〉 , |V ′〉 =
− sin(19◦)|H〉 + cos(19◦)|V 〉 with the eigenvalues 0.809 and 0, respe
tively. The maximal violation of 
lassi
ality
should be observed in the neighborhood of the state |φ〉 = cos(67◦)|H〉+ sin(67◦)|V 〉.
In prin
iple, the proposed test with the parameters obtained above 
ould be used to support the quantum pi
ture
for di�erent implementations of a qubit in
luding, for example, "super
ondu
ting qubits" with a still questionable
quantum 
hara
ter [4℄.
A
knowledgements. The authors thank P. Badzi�ag, M. Horode
ki, R. Horode
ki, W.A. Majewski and M.
�ukowski for dis
ussions. Finan
ial support by the POLAND/SA COLLABORATION PROGRAMME of the
National Resear
h Foundation of South Afri
a and the Polish Ministry of S
ien
e and Higher Edu
ation and by the
European Union through the Integrated Proje
t SCALA is a
knowledged.
Referen
es
[1℄ E. Joos, H.D. Zeh, C. Kiefer, D. Giulini, J. Kups
h, and I,-O. Stamates
u, De
oheren
e and the Appearan
e of
a Classi
al World in Quantum Theory, 2nd ed. Springer,Berlin 2003.
[2℄ J.S. Bell, Rev. Mod. Phys. 38, 447 (1966)
[3℄ R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory,
A
ademi
 Press, New York ,1983
[4℄ R. Ali
ki, quant-ph/0610008
http://arxiv.org/abs/quant-ph/0610008

  We propose a simple test of quantumness which can decide whether for the
given set of accessible experimental data the classical model is insufficient.
Take two observables $ A,B$ such that for any state $\psi$ their mean values
satisfy $0\leq <\psi|A|\psi>\leq <\psi|B|\psi>\leq 1$. If there exists a state
$\phi$ such that the second moments fulfill the inequality $<\phi|A^2|\phi>
><\phi|B^2|\phi>$ then the system cannot be described by the classical
probabilistic scheme. An example of an optimal triple $(A,B,\phi)$ in the case
of a qubit is given.

A simple test of quantumness for a single system
Robert Ali
ki
Institute of Theoreti
al Physi
s and Astrophysi
s, University of Gda«sk, Wita Stwosza 57, PL 80-952 Gda«sk, Poland
Ni
holas Van Ryn
S
hool Of Physi
s, Quantum Resear
h Group, University of KwaZulu-Natal,
Westville Campus, Private Bag x54001, Durban, South Afri
a.
November 17, 2018
Abstra
t
We propose a simple test of quantumness whi
h 
an de
ide whether for the given set of a

essible experimental
data the 
lassi
al model is insu�
ient. Take two observables A,B su
h that for any state ψ their mean values
satisfy 0 ≤ 〈ψ|A|ψ〉 ≤ 〈ψ|B|ψ〉 ≤ 1. If there exists a state φ su
h that the se
ond moments ful�ll the inequality
〈φ|A2|φ〉 > 〈φ|B2|φ〉 then the system 
annot be des
ribed by the 
lassi
al probabilisti
 s
heme. An example of
an optimal triple (A,B, φ) in the 
ase of a qubit is given.
Although we are 
on�dent that the proper theory des
ribing all physi
al phenomena is the quantum theory, there
are many situations where the 
lassi
al des
ription in terms of fun
tions and probability distributions over a suitable
"phase-spa
e" is su�
ient. In parti
ular, the systems 
onsisting of a large number of parti
les and/or emerging
in quantum states 
hara
terized by large quantum numbers are supposed to behave 
lassi
ally. The standard
explanation of this fa
t refers to the stability properties of quantum states with respe
t to the intera
tion with an
environment. For large quantum systems the intera
tion with the environment is so strong that most quantum
states rapidly de
ay (de
ohere) and the remaining manifold of experimentally a

essible states 
an be des
ribed by

lassi
al models. However, the a
tual border between quantum and 
lassi
al worlds is still a topi
 of theoreti
al
debate and experimental e�orts [1℄. This question is parti
ularly important in the �eld of quantum information.
Useful large s
ale quantum 
omputations would demand preservation of some quantum properties for rather large
physi
al systems, say at least for 103 qubits. Moreover, some promising implementations of a qubit are based on
mesos
opi
 systems whi
h lie on the aforementioned border. In parti
ular, the so-
alled super
ondu
ting qubits are
systems 
omposed of 108− 109 parti
les (Cooper pairs). Therefore, it is important to �nd a simple operational test
of quantumness whi
h 
ould be applied to a single system implementing a qubit. It is generally believed that the
simplest operational and model independent test of quantumness (or stri
tly speaking non-
lassi
ality) is based on
Bell inequalities whi
h involve systems 
omposed of at least two parts [2℄. However, this test is problemati
 for
systems whi
h 
annot be spatially well-separated ("lo
ality loophole").
We propose here a mu
h simpler operational test whi
h 
an be applied to a single system and formulated in
terms of inequalities between mean values of properly 
hosen observables. In 
ontrast to Bell inequalities this test
does not refer to the notion of lo
ality, asso
iated with the spa
e-time, but involves only the most fundamental
di�eren
es between 
lassi
al and quantum observables. It is based on the observation that for any two real fun
tions
f, g satisfying
0 ≤ f(x) ≤ g(x) (1)
and any probability distribution ρ(x) the following inequality holds
〈f2〉ρ ≡
2(x)ρ(x)dx ≤
2(x)ρ(x)dx ≡ 〈g2〉ρ . (2)
On the other hand, as illustrated below for the 
ase of a single qubit, for quantum systems we 
an always �nd a
pair of observables A,B su
h that for all states ψ their mean values satisfy 0 ≤ 〈ψ|A|ψ〉 ≤ 〈ψ|B|ψ〉 but there exists
a state φ su
h that for the se
ond moments 〈φ|A2|φ〉 > 〈φ|B2|φ〉.
The above statement follows from the interesting and nontrivial mathemati
al result in the theory of C∗-algebras
[3℄, whi
h is presented below for 
ompleteness.
Consider an abstra
t formalism where (bounded) observables are elements of a 
ertain C∗-algebra A, and states
are positive normalized fun
tionals A ∋ A 7→ 〈A〉ρ with 〈A〉ρ denoting the mean value of an observable A in a state
http://arxiv.org/abs/0704.1962v3
ρ. For all pra
ti
al purposes we 
an restri
t ourselves to two extreme 
ases: the �rst being a 
lassi
al model where
A is an algebra of fun
tions on a 
ertain "phase-spa
e" with 〈A〉ρ =
A(x)ρ(x)dx, where ρ(x) is some probability
distribution, and the se
ond being a �nite quantum model in whi
h A is an algebra of matri
es and 〈A〉ρ = Tr(ρA),
where ρ is some density matrix. For any pair of observables A,B ∈ A, the order relation A ≤ B means that
〈A〉ρ ≤ 〈B〉ρ for all states ρ (in fa
t it is enough to take all pure states). Now we 
an formulate the following:
Theorem. If the following impli
ation
0 ≤ A ≤ B =⇒ A2 ≤ B2 (3)
always holds then the algebra A is 
ommutative, i.e. isomorphi
 to the algebra of 
ontinuous fun
tions on a 
ertain

ompa
t spa
e.
As a 
onsequen
e of the above theorem, for any quantum system there exists a pair of observables (identi�ed
with matri
es) (A,B) su
h that the eigenvalues of A,B and B−A are nonnegative but the matrix B2−A2 possesses
at least one negative eigenvalue.
In order to apply our test of quantumness in an experiment, one should �rst guess a pair of observables A,B
with nonnegative values of possible out
omes and perform a statisti
al test of the inequality 〈A〉ρ ≤ 〈B〉ρ with as
large as possible number of di�erent, generally mixed, initial states ρ. These states should be as pure as possible,
otherwise the quantumness 
ould be not dete
ted. Then one should sear
h amongst these for any states σ satisfying
the relation for the se
ond moments 〈A2〉σ > 〈B2〉σ. If the violation of the 
lassi
al relation (2) holds, it means
that the system exhibits some quantum 
hara
teristi
s. Although there are always in�nitely many triples (A,B, σ),
the e�e
ts of external noise a
ting on the system and measuring apparatus 
an easily wash out the deviations from
"
lassi
ality". Therefore, instead of a random guess it is useful to �nd the examples of su
h triples whi
h maximally
violate 
lassi
ality and 
an be used to optimally design the experimental setting. This 
an easily be a
hieved in the

ase of a qubit whi
h is the most important example for quantum information.
We sear
h for a pair of 2× 2 matri
es A,B and a pure state φ whi
h satisfy
0 ≤ A ≤ B ≤ I , 〈A2〉φ ≡ 〈φ|A2|φ〉 > 〈B2〉φ ≡ 〈φ|B2|φ〉 , (4)
where the observables A and B are normalized in su
h a way that their upper bound is the identity. Su
h a triple
is given as an example as
ξ∗ a2
, B =
, φ =
. (5)
The matrix B is 
hosen to be diagonal, with the identity as its upper bound, and this 
hoi
e of basis 
an be made
sin
e the solution to this problem is unique up to unitary equivalen
e. From the upper bound it 
an be seen that
both eigenvalues of B should be at most 1, and to maximize the violation of (3) one of the eigenvalues is 
hosen to
be �xed at 1. The usual 
ondition |α|2 + |β|2 = 1 applies for the parameters of the state φ. The positivity of the
observables A, B and (B − A) is ensured by the requirement that both their diagonal elements and determinants
are positive. These 
onditions are expressed below as;
0 ≤ b ≤ 1, (6)
0 ≤ a1 ≤ 1, (7)
0 ≤ a1a2 − |ξ|2, (8)
0 ≤ (1 − a1)(b− a2)− |ξ|2. (9)
In order for this triple to satisfy equation (4), it is su�
ient that one of the eigenvalues of (B2 −A2) is found to be
negative while remaining within the 
onstraints listed above.
The eigenvalues of the matrix (B2 −A2) are found to be
b2 + 1− a2
− 2|ξ|2 ±
(b2 − 1 + a2
)2 + 4(a1 + a2)2|ξ|2
. (10)
In order to �nd the most negative value whi
h one of the eigenvalues 
an attain, one need only 
onsider one of the
two. Sin
e the square root is always positive, one eigenvalue always remains greater than the other. A numeri
al
te
hnique is used to 
al
ulate the maximal violation of the inequality (3) sin
e we are unable to �nd an exa
t
solution to this optimisation problem. In �nding the optimal set of parameters a1, a2, b and real ξ, it 
an be seen
that the maximal violation of the inequality (3) arises when the 
onditions (8) and (9) are equalities rather than
inequalities. Using
a1a2 = |ξ|2, (11)
(1− a1)(b− a2) = |ξ|2 (12)
redu
es the problem from four unknowns to two, and the triplet 
an then be written as
a1a2√
a1a2 a2
, B =
, φ =
. (13)
The parameters whi
h result in one of the eigenvalues of (B2 −A2) attaining it's most negative value, and thus
maximally violating inequality (3), while still remaining within the 
onstraints give us the triplet
0.724 0.249
0.249 0.0854
, B =
0 0.309
, φ =
0.391
0.920
. (14)
In this example, the value of 〈φ|B|φ〉 − 〈φ|A|φ〉 is 0.0528, a positive value, whereas it 
an be seen that 〈φ|B2|φ〉 −
〈φ|A2|φ〉 = −0.0590 whi
h 
learly demonstrates the quantum nature of this example. The eigenve
tors and 
orre-
sponding eigenvalues of A using these parameters is 
al
ulated to be
0.946
0.325
= 0.809
0.946
0.325
−0.325
0.946
. (16)
To give a 
on
rete example, one 
an apply these results to the polarization of a single photon. Choosing
a polarization basis as |H〉, |V 〉 and attributing the values 1 to |H〉 and 0.309 to |V 〉 we obtain the observable
B. The observable A 
orresponds to a rotated polarization basis |H ′〉 = cos(19◦)|H〉 + sin(19◦)|V 〉 , |V ′〉 =
− sin(19◦)|H〉 + cos(19◦)|V 〉 with the eigenvalues 0.809 and 0, respe
tively. The maximal violation of 
lassi
ality
should be observed in the neighborhood of the state |φ〉 = cos(67◦)|H〉+ sin(67◦)|V 〉.
In prin
iple, the proposed test with the parameters obtained above 
ould be used to support the quantum pi
ture
for di�erent implementations of a qubit in
luding, for example, "super
ondu
ting qubits" with a still questionable
quantum 
hara
ter [4℄.
A
knowledgements. The authors thank P. Badzi�ag, M. Horode
ki, R. Horode
ki, W.A. Majewski and M.
�ukowski for dis
ussions. Finan
ial support by the POLAND/SA COLLABORATION PROGRAMME of the
National Resear
h Foundation of South Afri
a and the Polish Ministry of S
ien
e and Higher Edu
ation and by the
European Union through the Integrated Proje
t SCALA is a
knowledged.
Referen
es
[1℄ E. Joos, H.D. Zeh, C. Kiefer, D. Giulini, J. Kups
h, and I,-O. Stamates
u, De
oheren
e and the Appearan
e of
a Classi
al World in Quantum Theory, 2nd ed. Springer,Berlin 2003.
[2℄ J.S. Bell, Rev. Mod. Phys. 38, 447 (1966)
[3℄ R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory,
A
ademi
 Press, New York ,1983
[4℄ R. Ali
ki, quant-ph/0610008
http://arxiv.org/abs/quant-ph/0610008
