N-qubit system in a pure state: a necessary and sufficient condition for unentanglement
Alain Deville Yannick Deville






N-qubit system in a pure state: a
condition for unentanglement




























Alain Deville IM2NP UMR 7334, Aix-Marseille Université, CNRS, F-13397 Marseille,
France

alain.deville@univ-amu.fr

ORCID: 0000-0001-5246-8391







Yannick Deville IRAP (Institut de Recherche en Astrophysique et Planétologie),
Université de
Toulouse, UPS, CNRS, CNES, 14 avenue Edouard Belin, F-31400 Toulouse, France

yannick.deville@irap.omp.eu

ORCID: 0000-0002-8769-2446




















N-qubit system in a pure state: a necessary
and sufficient condition for unentanglement
    Alain Deville
Yannick Deville
    Received: date / Accepted: date
=======================================================================================





If a pure state of a qubit pair is developed over the four 

basis states,










an equality
between the four coefficients of that development, verified if and only if
that state is unentangled, is already known.











T






his paper considers an arbitrary pure state of an N-qubit system,
developed over the 2^N 

basis states.





It is
shown that the state is unentangled if and only if a well-chosen collection
of [2^N-(N+1)] equalities between the 2^N coefficients of that
development is verified. The number of these equalities is large a soon as N≳ 10, but it is shown that this set of equalities may be classified
into (N-1) subsets, which should facilitate their manipulation. This
result should be useful e.g. in the contexts of Blind Quantum Source
Separation (BQSS) and Blind Quantum Process Tomography (BQPT), with an aim
which should not be confused with that found when using the concept of
equivalence of pure states through local unitary transformations.
























§ INTRODUCTION 


If both parts S_1 and S_2 of a bipartite quantum system S are
initially prepared in a pure state, then, at a time scale allowing one to
neglect any coupling between S and the rest of the universe, S may be
described as being in a time-dependent pure state |Ψ (t)> which
obeys the Schrödinger equation. But, at that time scale, if an internal
coupling transiently exists between S_1 and S_2, then, after it
disappeared, |Ψ (t)> often cannot be equated with a tensorial
product |Ψ _1(t)>⊗|Ψ _2(t)> describing S_1
and S_2 respectively: |Ψ (t)> is entangled (or not separable) 
<cit.>, <cit.>. Entanglement plays a
significant role in Quantum Information (QI) <cit.>, e.g. in
the context of Quantum Computing. The idea of a Quantum Computer (QC) dates
back to the 1980s <cit.>, <cit.>, 
and the word qubit to name the basic cell of the Quantum Computer
appeared in 1995 (cf. <cit.> and its acknowledgements). The
basic components of the future QC should be the qubit, the quantum register
- a quantum device consisting of several qubits- and the quantum gate - a
device aimed at controlling qubits and registers. It is presently possible
to claim that Quantum Computing research is coming of age <cit.>
, and qubits, registers and gates on one side, quantum algorithms using
abstract qubits on the other are under development. Qubits are generally
supposed to have been initially prepared in a pure state, which is not
always easily achieved with physical qubits, as shown e.g. for nuclear spins
in <cit.> (cf. its page 324). Coupling between qubits
induces entanglement between initially unentangled qubit pure states. In the
context of QI, entanglement may be either desired, e.g. when considering a
QC, because it may allow some form of parallel computing (Deutsch speaks of
quantum parallelism <cit.>), or avoided, because e.g. coupling
with the environment may cause decoherence, then stopping a calculation. It
should then be useful to be able to know whether a given pure state of a
system of abstract qubits is entangled or not.

The aim of this paper is to establish a necessary and sufficient
unentanglement condition (for breviety, an iff condition) for a
system consisting of N distinguishable qubits and which is in a pure
state. This iff condition should generalize a result which we recently
established for N=2 qubits <cit.>, in the context of a
subfield of QI, based on Blind Source Separation (BSS), which started around
1985 in a classical context and which is now a mature field. In BSS <cit.>, <cit.>, typically, a set of users (the Writer)
presents a set of simultaneous signals (input signals, called sources) at
the input of a multi-user communication system (the Mixer). The sources,
compelled to possess some general properties (e.g. mutual statistical
independence), are then combined (mixed, in the BSS sense) in the Mixer.
Another set of users (the Reader) receives the signals arriving at the Mixer
output. The Writer possibly knows the sources, but the Reader does not know
them, and cannot access the inputs of the Mixer. That Mixer uses one or
several parameter values, unknown to the Reader, who only knows some of its
general properties. The Reader's final task is the restoration of the
sources (possibly up to some so-called acceptable indeterminacies) from the
signals at the Mixer output, during the “inversion
phase”. An intermediate task is the determination of the
unknown parameters of the Mixer, or of its inverse, made during an
“adaptation phase”. Since 2007, we have
been developing a quantum version of BSS, namely Blind Quantum Source
Separation (BQSS). In our previous papers (<cit.> and
references therein, we considered two distinguishable qubits
numbered 1 and 2, first prepared in a pure unentangled state |Ψ
(t_w)>, by the Writer, at time t_w. The qubit pair is isolated from
the rest of the world between t_w and the time t_r when the Reader
can operate. At time t_r, the qubit pair is therefore still in a pure
state |Ψ (t_r)> but, because of an undesired coupling between the
qubits (viewed as the action of a Mixer), |Ψ (t_r)> is generally
 entangled. The Reader's task is the restoration of |Ψ (t_w)> =
|Ψ _1(t_w)>⊗|Ψ _2(t_w)> from |Ψ
(t_r)> (again possibly up to some acceptable indeterminacies). In 2012,
in this journal, we presented a paper devoted to methods for separating
quantum sources <cit.>. We recently introduced Blind Quantum
Process Tomography (BQPT <cit.>, <cit.>), the
blind version of Quantum Process Tomography (QPT <cit.>), which
is defined in Section <ref>. The iff condition hereafter
established is first aimed at extending the solution of the BQSS and BQPT
problems beyond the qubit pair, but it is hoped that it could be used in
other contexts, e.g that of the QC.

In Section <ref>, we first recall the iff condition for N=2 qubits, and then present already existing criteria related to pure
states or statistical mixtures of quantum systems composed of several parts,
in situations which are more or less different from the present one -
qubits, in arbitrary number. In Section <ref>
the method to be followed for finding this iff condition is presented. This
 iff condition is established in Section <ref>, through an iterative process. The results and
some of their applications are discussed in Section <ref>.
The possibility of using the von Neumann concept is briefly tackled in the
Appendix.



§ ABOUT EXISTING UNENTANGLEMENT CRITERIA


In our previous papers, qubits were supposed to be physically implemented as
spins 1/2. We hereafter first recall the notations used for writing an
arbitrary pure state |Ψ > of a qubit pair.|Ψ >
is developed as:

    |   Ψ
    >
    =
    
    
    
    
    
    
    
    
    ∑_i=1^4c_i| i>,where
       |   1>=| ++>,| 2>=| +->, | 3>=| -+>,
    | 4>=|–>

where | ++> is an abbreviation for | 1,+>⊗| 2,+>, |
1,+> being the eigenstate for the eigenvalue 1/2 in the standard basis of
qubit 1. We now consider an arbitrary N-qubit system, and generalize this
writing: for qubit number k in an N-qubit system:


    s_zk| k,± >=±1/2| k,± >

and for a pure state |Ψ > of this N-qubit system:

    |Ψ >=∑_i=1^2^Nc_i| i>

where | 1> = | ++....++> (all qubits in state | +>),|
2^N>=|—....–> (all qubits in state | ->). Qubit 1 is in
state | +> in the first 2^N-1 states, and in state | -> in the 
2^N-1 remaining states. | 2^N-1+1> =| -+..++> (all qubits in
state | +>, except qubit 1, in state | ->), | 2^N-2> =|—....+-> (all qubits in state | ->, except qubit N-1, in state | +>).

We immediately prove (for c_1≠ 0) the following property, which we
established in more detail (including the c_1=0 case) in <cit.>: a pure state |Ψ > of a qubit pair is unentangled if
and only if the c_i coefficients obey the equality:

    c_1c_4=c_2c_3.

If c_1≠ 0, |Ψ > can always be written as:

    |Ψ >=c_1(| ++>+c_2/c_1| +->+c_3/c_1
    | -+>+c_4/c_1|–>).

When c_1c_4=c_2c_3, |Ψ > can then be written as:

    |   Ψ >=c_1(| ++>+c_2/c_1| +->+c_3/c_1
    | -+>+c_2c_3/c_1^2|–>) 
       =   c_1(| +>+ c_3/c_1| ->)⊗ (| +>+
    c_2/c_1| ->)

which proves that when N=2 and c_1c_4=c_2c_3, |Ψ > is unentangled.

Conversely, if |Ψ > is unentangled, it may be written as:

    |Ψ >=(a_1| +>+b_1| ->)⊗ (a_2| +>b_2| ->).

Eq. (<ref>) must be consistent with Eq. (<ref>), which imposes the following relations:

    c_1=a_1a_2,   c_2=a_1b_2,  c_3=b_1a_2,
      c_4=b_1b_2

and therefore c_1c_4=c_2c_3.

In this paper, the proposed iff condition will take the form of a
set of equalities between the c_i coefficients. In Section <ref> it will e.g. be shown that |Ψ > is unentangled, when N=3, if and only if the following two subsets
(S1, S2) of equalities, corresponding to a total of four independent
equalities, are simultaneously verified:

    S_1   :     c_1/c_2=c_3/c_4=c_5/
    c_6=c_7/c_8
    
    S_2   :      c_2c_8=c_4c_6
    .

Coming now to existing criteria, one should first mention the Schmidt and
the Peres-Horodecki ones. The so-called Schmidt decomposition allows one to
express an iff condition for pure states of a bipartite system, in
which the dimension of each part of the system is not restricted to 2.And an extension of the concept of entanglement to statistical mixtures is
somewhat possible through the notions of separability and of entanglement
witnesses <cit.>. The Peres-Horodecki criterion <cit.> is an iff condition for the separability of the
density matrix describing a state of a bipartite system, valid when the
dimensions of the state spaces of S_1and S_2 are low. In <cit.>, it was explained why these criteria are not appropriate for
the context of BQSS. They are both restricted to bipartite systems, and
therefore should presently be discarded, since this paper is devoted to the
general case N>2. With these presently strong restrictions, their aim is
not very different from the one in this paper.

On the contrary, in the context of quantum communications, it is usual to
speak of Local Unitary (LU) transformations of pure states, and when it is
spoken of equivalent states the aim is different from the one in e.g. BQSS.
If |Ψ (1,2,...N)> is some pure state of a system composed of N
distinguishable particles, and U a unitary operator acting on |Ψ
(1,2,...N)>, then producing a transformed pure state U|Ψ
(1,2,...N)>, the transformation is said to be local if U=U(1)⊗
U(2)...⊗ U(N), where U(1), U(2), ... U(N) are themselves
unitary operators, each acting upon a single particle. The reason for the
choice of the word Local may be guessed if 1) one considers the
transformation of an unentangled state |Ψ (1,2,...N)>, 2a) one
considers two qubits, 1 in space-time zone 1, and 2 in space-time zone 2,
separated by a spacelike interval, 2b) one imagines two experimenters,
A(lice) and B(ob), A being able to access qubit 1 (only) and B qubit 2
(only), 2c) this situation is true for all distinct pairs of an N-qubit
system. On the contrary, an instance of the effect of a non-LU
transformation on an unentangled state is given in Section <ref>. Once LU transformations have been defined, two pure
states are said to be equivalent if they differ by an LU transformation only
(cf. e.g. the review article <cit.>). When trying to define a
degree of entanglement in order to classify all the possible pure states of
a multipartite system, one is led to make no distinction between equivalent
states. On the contrary, in the context of BQSS, this concept of equivalence
through an LU transformation plays no role. If a given pure state |Ψ > is written at the input of the Mixer, and a state |Φ > is
read at its output, the aim is not to get back some state |Ξ >
equivalent to |Ψ >, with the meaning that |Ξ > and |Ψ > differ by an LU transformation, but to get back state |Ψ >
itself, possibly up to some acceptable (in fact, far weaker than an
arbitrary LU transformation) indeterminacies.



§ TOWARDS A GENERALIZATION OF THE RELATION BETWEEN THE C_I
EXISTING FOR A QUBIT PAIR


In order to extend entanglement-based BQSS methods beyond the simplest case,
the qubit pair (N=2), it is highly desirable to find a set of relations,
if it does exist, which, when N>2, could be substituted for the equality c_1c_4=c_2c_3. We have to find a collection of equalities between
the c_i which are obeyed if and only if |Ψ > is unentangled. We
will consider only normalized states: <Ψ |Ψ >=1. Moreover
only the projector |Ψ > <Ψ | has a physical meaning:
one should not distinguish between |Ψ > and e^iη |Ψ > (η is any real number). Therefore, if the complex numbers c_i are written c_i=ρ _ie^iφ _i (ρ _i and φ _i are real numbers and i=1, 2,...2^N), then rather than
the 2^N phases, only e.g. the (2^N-1) phase differences (φ
_i-φ _1) are meaningful when defining an arbitrary pure state |Ψ >. Consequently, an arbitrary pure state |Ψ > of the N
-qubit system is defined by the value of (2^N+1-2) independent real
numbers: (2^N-1) moduli and (2^N-1) phases. However, an unentangled
pure state may be written as:

    |Ψ _ue>=|ψ _1>⊗|ψ _2>⊗ ...|ψ
    _i>⊗ ...⊗|ψ _N>

where each ordered factor describes the state of a given qubit, and
therefore depends upon two real numbers (a modulus, a phase). An
unentangled state |Ψ _ue> therefore depends upon 2N real
numbers only. Then, if |Ψ _ue> is developed following (<ref>), there should exist 2[2^N-(N+1)] relations
between the 2^N+1 real quantities {ρ _i, φ _i}
(besides those expressing that only |Ψ ><Ψ| has a physical
meaning and that |Ψ > is normalized). This result strengthens the
hope that there may exist, between the c_i coefficients, a set of [2^N-(N+1)] equalities which are verified if and only if |Ψ > is
unentangled. This is presently only a hope, since normalization of |Ψ > and the physical meaning of |Ψ > <Ψ | lead to
two constraints between real numbers, not to a constraint between the c_i
themselves. The present paper will not try to directly prove the
existence of such relations between the c_i for unentangled and only
unentangled pure states, but will rather use an iterative approach in order
to try and establish such general relations.



§ FINDING A NECESSARY AND SUFFICIENT CONDITION: AN ITERATIVE APPROACH 



We will first examine the N=3 and N=4 cases in some detail, starting
with N=3. It is supposed that c_i≠ 0 for any i value. The case
when c_i=0 for at least one i value will be discussed in Section <ref>.

When N=3, any pure state may be written as:

    |Ψ >
       
    =
       (| ++>+c_3/c_1| +->+c_5/c_1|
    -+>+c_7/c_1|–>)⊗ c_1| +>  
          
    +( | ++>+c_4/c_2| +->+c_6/c_2| -+>+
    c_8/c_2|–>)⊗ c_2| ->

where | +> in the writing c_1| +> and | -> in the writing c_2| -> refer to a state of qubit no. 3. When

    c_3/c_1=c_4/c_2,c_5/c_1=c_6/
    c_2,c_7/c_1=c_8/c_2

it is possible to write |Ψ > as:

    |Ψ >=|Ψ (1,2)>(| ++>+c_4/
    c_2| +->+c_6/c_2| -+>+c_8/c_2|–>)
    ⊗|Ψ (3)>(c_1| +>+c_2| ->).

When moreover the coefficients of |Ψ (1,2)> obey Eq. (<ref>), which is presently written as:

    c_8/c_2=c_4/c_2.c_6/c_2,i.e.
     c_2c_8=c_4c_6

then |Ψ (1,2)> is an unentangled state, and |Ψ >is unentangled. Therefore, when N=3, if subsets (<ref>)
and (<ref>) are obeyed, |Ψ > is unentangled (subset (
<ref>) is equivalent to the three equations (<ref>)).

Conversely, if |Ψ > is unentangled, it can be written as:

    |Ψ (1,2,3)>=unentangled
    (∑_i=1^4C_i| i>)⊗ (a_3| +>+b_3| ->)

where the C_i coefficients obey the relation C_1C_4 = C_2C_3. Identifying Eq. (<ref>) with Eq. (<ref>) (with N=3), one gets:

    c_1=C_1a_3,c_2=C_1b_3, c_3=C_2a_3,...
    c_8=C_4b_3

and therefore Eqs. (<ref>) and (<ref>) (i.e.
Conditions S_1 and S_2) are obeyed.

When N=3, |Ψ > is therefore unentangled if and only if the two
subsets of equalities (<ref>) and (<ref>),
expressing a total of 4 (independent) equalities, are obeyed. Therefore,
the above-expressed hope of finding an iff condition for
unentanglement using [2^N-(N+1)] relations between the c_i is
satisfied when N=3.

Similarly, when N=4, any pure state may be written as:

    |Ψ >
       
    =
       
    (| +++>+c_3/c_1
    | ++->+...+c_15/c_1|—>)⊗ c_1| +>  
          
    +( | +++>+c_4/c_2
    | ++->+...+c_16/c_2|—>)⊗ c_2| ->

where now | +> in the writing c_1| +> and | -> in the
writing c_2| -> refer to a state of qubit no. 4. When the
following set of equalities is satisfied:

    c_3/c_1=c_4/c_2,c_5/c_1=c_6/
    c_2,...,c_15/c_1=c_16/c_2,

which can also be written as:

    c_1/c_2=c_3/c_4=c_5/c_6=...=c_15
    /c_16,

|Ψ > may be written as:

    |Ψ >=|Ψ (1,2,3)>(| +++>+c_4
    /c_2| ++->+...c_16/c_2|—>)⊗ (c_1|
    +>+c_2| ->).

When moreover the coefficients of |Ψ (1,2,3)> obey subsets (<ref>) and (<ref>), presently written as:

    c_2/c_4   =   c_6/c_8=c_10/c_12=c_14
    /c_16
    
    c_4c_16   =   c_8c_12,

|Ψ > is unentangled. Therefore, when N=4,if the three
subsets of Eqs. (<ref>), (<ref>) and (<ref>
) are verified, |Ψ > is unentangled.

Conversely, if |Ψ > is unentangled, it can be written as:

    |Ψ (1,2,3,4)>=unentangled
    (∑_i=1^8C_i| i>)⊗ (a_4| +>+b_4| ->)

where the C_i coefficients now obey the following sets of relations:

    C_1/C_2   =   C_3/C_4=C_5/C_6=C_7/
    C_8
    
    C_2C_8   =   C_4C_6  
    .

Now identifying Eq. (<ref>) with Eq. (<ref>) (with N=4), one gets:

    c_1=C_1a_4,c_2=C_1b_4, 
    c_3=C_2a_4.....c_16=C_8b_4.

 These equalities were obtained through the same process as in Eq. (<ref>), and therefore lead to the same type of equalities as in Eq.
(<ref>), but with now sixteen and not eight c_icoefficients, namely:

    c_1/c_2=c_3/c_4=...=c_15/c_16.

Moreover, from Eqs. (<ref>), (<ref>) and (<ref>), one gets:

    c_2/c_4   =   c_6/c_8=c_10/c_12=c_14
    /c_16
    
    c_4c_16   =   c_8c_12.

Therefore, when N=4, |Ψ > is unentangled if and only if the
following three subsets of equalities are verified:

    S_1   :       c_1/c_2=c_3/c_4=...=c_15/
    c_16
    
    S_2   :       c_2/c_4=c_6/c_8=c_10/c_12
    =c_14/c_16
    
    S_3   :       c_4c_16=c_8c_12.

When N=4, this iff condition is therefore expressed through 3
subsets of equations, expressing a total of 11 equalities, which is also
again the value of [2^N-(N+1)], now for N=4.

Now taking an arbitrary N value, one may write any pure state |Ψ >
as:

    |Ψ >
       
    =
       
    (| ++...+>+c_3/c_1|
    ++...->+...+c_2^N-1/c_1|–...->)⊗ c_1| +> 
    
          
    +(| ++...+>+c_4/c_2|
    ++..->+...+c_2^N/c_2|–...->)⊗ c_2| ->

which generalizes Eqs. (<ref>) and (<ref>
). The reasoning which led to Eq. (<ref>) and (<ref>) now leads to:

    c_1/c_2=c_3/c_4=...=c_2^N-3/c_2^N-2=
    c_2^N-1/c_2^N.

This subset contains (2^N-1-1) independent equalities. When they are
obeyed, |Ψ > may be written as:

    |Ψ >=|Ψ (1,2,3,...N-1)>⊗|Ψ (N)>.

This state |Ψ > is unentangled if and only if, moreover, |Ψ
(1,2,3,...N-1)> itself is unentangled. If the results obtained for N=3
and N=4 may be generalized, this is obtained when (N-2) other subsets of
inequalities, corresponding to a total of [2^N-(N+1)] - (2^N-1-1)
equalities between the c_i, i.e. (2^N-1-N) independent equalities,
are satisfied. This quantity is equal to 1 if N=3 (subset S_2, Eq. (
<ref>)), and to 4 if N=4 (subsets S_2, Eq. (<ref>), and S_3, Eq. (<ref>)).

We now momentarily suppose that the property established for N=3 and N=4
is true for N-1 (with N-1≥ 4), i.e. that |Ψ > is unentangled
if and only if the following (N-2) subsets of equalities are
simultaneously verified:

    S_1   :     c_1/c_2=c_3/c_4=...=c_2^N-1-3
    /c_2^N-1-2=c_2^N-1-1/c_2^N-1
    
    S_2   :     c_2/c_4=c_6/c_8=...=c_2^N-1-2
    /c_2^N-1
    
    S_3   :     c_4/c_8=c_12/c_16=...=
    c_2^N-1-4/c_2^N-1
          .............  
    
    S_N-3   :     c_2^N-4/c_2∗ 2^N-4=c_3∗ 2^N-4
    /c_4∗ 2^N-4=c_5∗ 2^N-4/c_6∗ 2^N-4=
    c_7∗ 2^N-4/c_8∗ 2^N-4
    
    S_N-2   :     c_2^N-3c_4∗ 2^N-3=c_2∗ 2^N-3 c_3∗
    2^N-3.

In order to help the reader anxious to see more explicitly the meaning of
the dots in Eq. (<ref>), and in Eq. (<ref>) hereafter, the 6 subsets for N=7 are all written at
the end of this section (four of them again with dots, but then with a
rather obvious meaning). We now establish that any ket |Ψ > =∑_i=1^2^Nc_i| i> describing a pure state of an N-qubit
system is unentangled if and only if the N-1 following subsets are all
obeyed: 

    S_1   :     c_1/c_2=c_3/c_4=...=c_2^N-3/
    c_2^N-2=c_2^N-1/c_2^N
    
    S_2   :     c_2/c_4=c_6/c_8=...=
    c_2^N-2/c_2^N
    
    S_3   :     c_4/c_8=c_12/c_16=...=c_2^N-4
    /c_2^N
          .............  
    
    S_N-2   :     c_2^N-3/c_2∗ 2^N-3=c_3∗ 2^N-3
    /c_4∗ 2^N-3=c_5∗ 2^N-3/c_6∗ 2^N-3=
    c_7∗ 2^N-3/c_8∗ 2^N-3
    
    S_N-1   :     c_2^N-2 c_4∗ 2^N-2=c_2∗ 2^N-2c_3∗
    2^N-2.


The approach already used for N=3 and 4 is applied to an N-qubit
system. Any pure state may be written as:

    |Ψ >
       
    =
       
    (| ++...+>+c_3/c_1|
    ++...->+...+c_2^N-1/c_1|–...->)⊗ c_1| +> 
    
          
    +(| ++...+>+c_4/c_2|
    ++...->+...+c_2^N/c_2|–...->)⊗ c_2| ->

where now | +> in the writing c_1| +> and | -> in the
writing c_2| -> refer to a state of qubit no. N. When each c_2^k-1/c_1  quantity (for k=2, 3...N) is equal to c_2^k/c_2, this collection of relations can collectively be written
as S_1 subset (<ref>), and |Ψ > may be
expressed as:

    |Ψ >=|Ψ (1,2,...N-1)>(| ++
    ...
    +>+
    c_4/c_2| ++->+
    ...+
    c_2^N/c_2|–...->)⊗ (c_1|
    +>+c_2| ->).

If moreover the c_2^k/c_2 coefficients in |Ψ
(1,2,...N-1)> obey all the equalities expressed in Eq. (<ref>) to (<ref>), |Ψ
(1,2,...N-1)> is unentangled, and |Ψ > itself is therefore
unentangled. For instance, Eq. (<ref>) presently
takes the form:

    c_2^N-2c_4∗ 2^N-2=c_2∗ 2^N-2 c_3∗ 2^N-2

which is subset (<ref>). More generally, Eq. (<ref>) to (<ref>) presently take
the form of Eq. (<ref>) to (<ref>)
respectively. Any reader aiming at establishing these equalities should
appreciate that the c_i coefficients in Eq. (<ref>)
are generic quantities. For instance, c_1 is the coefficient for |
+++> if N=3, whereas it is the coefficient for | +++++> if N=5.

Conversely, if |Ψ (1,2,...,N> is unentangled, it can be written as:

    unentangled(∑_i=1^2^N-1C_i| i>)
    ⊗ (a_N| +>+b_N| ->)

where the C_i coefficients obey equalities expressed through subsets (
<ref>) to (<ref>),  with
the C_i instead of the c_i coefficients. Then, the method used for N=3 and N=4 is again used for expressing each c_i coefficient in
expression |Ψ >=∑_i=1^2^Nc_i| i> as a function of
both a C_i coefficient and a_N or b_N. This allows us to show
that subsets (<ref>) to (<ref>) are
obeyed. Therefore, if it is true that a ket |Ψ (1,2,...N-1)> is
unentangled if and only if subsets (<ref>) to (<ref>) are all verified, then it is true that a ket |Ψ (1,2,...N> is unentangled if and only if subsets (<ref>) to (<ref>) are all verified.

Considering successively N=2, 3 and 4, it was shown above that |Ψ > is unentangled if and only if a collection of equalities, structured
into (N-1) subsets, is obeyed. These results and this iterative
discussion from (N-1) to N finally allow us to claim that subsets (
<ref>) to (<ref>) do collectively
express an iff condition for a ket |Ψ > of an N-qubit system to
be unentangled.

If e.g. N=7 (the dimension of the state space is 128 then), it is
tedious but quite possible, through successive iterations, to get the
explicit expressions of the six subsets of equalities expressing
unentanglement. They are respectively

    S_1   :    c_1/c_2=c_3/c_4=...=c_125/c_126
    =c_127/c_128
    
    S_2   :     c_2/c_4=c_6/c_8=...=c_126
    /c_128
    
    S_3   :     c_4/c_8=c_12/c_16=...=c_124/
    c_128
    
    S_4   :     c_8/c_16=c_24/c_32=...=c_120/
    c_128
    
    S_5:       c_16/c_32=c_48/c_64=c_80/c_96=
    c_112/c_128
    
    S_6   :     c_32c_128=c_64c_96.

They have been written here in order to help the reader interpret the dots
in Eq. (<ref>) and (<ref>).

Our results for N=2, 3 and 4 suggest that there are [2^N-(N+1)]
independent equalities in Eqs. (<ref>) to (<ref>). We now establish this result, again using
mathe­matical induction. We first suppose that it is true that Eqs. (<ref>) to (<ref>), for an (N-1)
-qubit system, contain (2^N-1-N) such equalities. Now considering an 
N-qubit system, we may claim that the corresponding number of such equalities
is the sum of two quantities: (2^N-1-1), the number of equalities
associated with S_1 (cf. Eq. (<ref>)) and (2^N-1-N)
 new equalities expressing that |Ψ (1,2,3,...N-1)> in Eq. (<ref>) is itself unentangled, which does lead to a
total of [2^N-(N+1)] such equalities. This expression, which was already
known to be valid for N=2, 3 and 4, is therefore valid for any N≥
2.



§ DISCUSSION


It is possible to build other sets of equalities which are obeyed if and
only if an arbitrary pure state |Ψ > is unentangled. When N=3,
for instance, keeping the same approach, it is easy to replace (<ref>) with condition c_1c_7=c_3c_5. Then, with the
same approach for N>3,all even c_i coefficients in the subsets S_k
with k>1 are suppressed. For N=6, e.g. the S_5 subset becomes c_1c_49=c_17c_33. Use of even indices leads to simpler
expressions, which explains the choice made in this paper.

In <cit.>, with N=2, if at least one of the c_i
coefficients is equal to 0, it was shown that condition c_1c_4=c_2c_3 is still valid. In Section <ref> of the present paper, it was assumed that c_i≠ 0 for any i. When N=3, if e.g. c_5=0, then in Eq. (<ref>) the (c_5/c_1)| -+> term is absent and|Ψ
> is then unentangled only if c_6=0 (cf. the presence of the (c_6/c_2)| -+> term in Eq. (<ref>)). When N>3, if 
c_5=0, then in all the subsets expressing unentanglement, the c_6
terms will be absent. The reason is that in Eq. (<ref>) the (unexplicitly written) c_5/c_1 term of
qubits 1 to (N-1) is associated with the c_1| +> state of qubit N, and the corresponding state of qubits 1 to (N-1) associated with the 
c_2| -> state of qubit N has a c_6/c_2 coefficient. This
reasoning may also be used if more than one c_i coefficient is equal to 0.

When N=20, the dimension of the state space ℰ_20 is 2^20,
which is roughly 10^6. An unentangled normalized state then depends upon 
40 real numbers only. In the context of Quantum Information Processing,
it is generally considered that the wealth of the quantum behaviour
originates in the existence of entanglement, but it may be important to be
able to decide whether a given pure state is entangled or not, e.g. in order
to achieve BQSS or BQPT, and finding an iff condition is therefore
significant. The present paper has shown that, when |Ψ > is
unentangled, there exist [2^N-(N+1)] independent equalities between the c_i coefficients, the value of which is itself roughly 10^6 when N=20. But it has also been found that these equalities may be classified
into only (N-1) subsets, e.g. 19 subsets when N=20. It is hoped that
this classification should allow tractable operations in numerical
simulations or calculations.

We now come to recent papers making use of a Local Unitary (LU)
transformation. Ninety years after the building of modern Quantum Mechanics
(QM), there is a vast literature devoted to its foundations (see e.g. <cit.>, <cit.>, <cit.>), while most physicists
use QM without discussing its deep content. The following lines just aim at
drawing a link between these recent papers and this literature. Paty <cit.> has stressed that, historically, well before the 1935 EPR paper,
Einstein, at the 1927 Solvay Congress <cit.> (p.
256), exposed his concern about what he would later on call the
incompleteness of QM. In his 1995 paper, Paty clearly and convincingly
asserts that "it is only recently, indeed, that the concept of
non-locality as a fundamental feature of quantum mechanics has been fully
appreciated, and commentators have seldom realized that this was one of
Einstein's main points", and that "it is in the
Einstein-Podolsky-Rosen's paper itself that non-locality is described and
that emphasis is put on it". At the same 1927 meeting, Einstein stated
that the interpretation of |Ψ| ^2 as a probability density
for a single particle (rather than for an ensemble of particles) implied,
for him, "a contradiction with the principle of relativity".
 After 1945, a deepening of the foundations of QM partly overlapped with
the development of Classical and Quantum Information. Bell's contributions
to these foundations, from his 1966/1964 papers down to his death in 1990 
<cit.>, especially stimulated the development of both experimental
tests and so-called quantum communications. In the context of quantum
communications, speaking of LU transformations (cf. Section <ref>) is usual. An LU transformation U=U(1)⊗ U(2)
transforms a pure unentangled state |Ψ (1>⊗|Φ (2)>into the unentangled state (U(1) |Ψ (1>) ⊗ (U(2)|Φ (2)>). On the contrary, a simple calculation shows that e.g. the
non-LU transformation U=e^ias_1xs_2x acting on the unentangled state 
| +->,(a  is  some dimensional real constant) transforms it into
the entangled state  (cf. Eq. (<ref>))

    e^ia/4/4(| +x,+x>-| -x,-x>)-e^-ia/4/4(|
    +x,-x>-| -x,+x>)

where e.g. | +x,-x> means | 1,+x>⊗| 2,-x>, and |
i,+x>  (resp. | i,-x>) is the eigenket for s_ix for the
eigenvalue 1/2 (resp. -1/2), with i=1, 2. LU transformations should
therefore be distinguished from entanglement-inducing transformations. The
latter transformations are faced e.g. in BPQT and BQSS, which are important
quantum information processing problems due to their applications, including
those presented hereafter.

Blind or non-blind QPT may be defined as the identification (i.e.
estimation) of a given quantum process or gate, called the direct process or
gate hereafter, which receives a “source
state”. As discussed e.g. in <cit.>
, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, (B)QPT is a major quantum information
processing tool, since it especially allows one to characterize the actual
behavior of quantum gates, which are the building blocks of the quantum
computers considered in Section <ref>. The
usual, i.e. non-blind, version of QPT requires one to know, hence to
precisely control (i.e. prepare), the specific quantum source states used as
inputs of the quantum gate to be characterized. The blind version of this
tool, i.e. BQPT, then provides an attractive extension of QPT, since it
allows one to use quantum source states whose values are unknown and
arbitrary, except that they are requested to meet some general properties.
These properties e.g. consist of unentanglement <cit.>,
which is one of the motivations for analyzing unentanglement in the present
paper (more details about the operation of BQPT are available e.g. in <cit.>, <cit.>).

BQSS may be seen as a quantum information processing problem where one aims at
handling the altered quantum state available at the output of a direct
quantum process / gate which typically involves undesired coupling between
its qubits, this process and its input being initially unknown. This BQSS
problem is e.g.[
Other approaches perform BQSS directly, i.e. without first resorting to BQPT.
] handled by (i) first identifying that direct gate with BQPT, thus using
only the output of that gate and unentanglement or other properties, (ii)
then deriving a quantum gate that performs the inverse transform of that of
the direct gate, and (iii) then feeding that “inverse
gate” with the altered states available at the output of
the direct gate during final operation, so as to restore the corresponding
source, i.e. non-altered, states. One may anticipate that this approach will
be useful e.g. in situations where data are stored in a register of qubits
of a quantum computer, for subsequent use. Due to non-idealities of the
physical implementation of that register, the qubits which form it may be
coupled (e.g. when qubits are implemented as the spins of electrons which
are close to one another). As time goes on, the register state will
therefore evolve in a complicated way due to qubit coupling, thus making the
final value of that register state not directly usable in the target
application of the quantum computer. BQSS may then be used to restore the
initially stored register state, before providing it to the part of the
quantum computer which uses these nonaltered data to perform the target task
of that computer.



§ CONCLUSION


In the 2009 review article devoted to entanglement <cit.>, the
Horodecki team noticed that "it appears that this new resource is
complex and difficult to detect". Experimental and theoretical aspects were
both involved. If one focuses this comment on the idea that establishing
whether a pure state is entangled or not is a cumbersome task, the following
remarks may be made. In the present paper, devoted to an arbitrary number, N, of distinguishable qubits, it has been shown that if a pure state of
that N-qubit system is developed over the 2^N basis states of the
generalized standard basis (or of some arbitrary well-defined basis) as |Ψ >=∑_ic_i | i>, one is then led to introduce (N-1)
subsets of equalities, which are verified if and only if |Ψ > is
unentangled. It should however be realized that if N=20, these 19
subsets together collect [2^N-(N+1)] equalities, which is here roughly
equal to 2^20,i.e. approximalely 10^6. While the complexity of the
problem is reflected in the fact that the number of equalities roughly grows
as 2^N, it is hoped that the necessary and sufficient condition
established in this paper, which introduced a systematic ordering within
these equalities, through a classification into (N-1) subsets, may in
practice help in the manipulation of the entanglement concept.





§ 





THE VON NEUMANN ENTROPY AND THE ESTABLISHMENT OF THE IFF
CONDITION


The entropy concept, which did not appear in this paper yet, is briefly
considered here. The von Neumann entropy of a quantum system in a pure or
mixed state described by a density operator ρ is the trace S=-Tr(ρ
Lnρ ). This concept cannot directly be used in an attempt to find an 
iff condition for the unentanglement of a pure state |Ψ >
of an N-qubit system, since its von Neumann entropy is zero for both
unentangled and entangled pure states. But this N-qubit system can be
viewed as a bipartite system Σ , composed of parts Σ _A and Σ _B, and if Σ is described by ρ , one may first
introduce reduced density operators ρ _A=Tr_Bρ and ρ
_B=Tr_Aρ (see e.g. <cit.>). From now on, we focus on
the situation when ρ =|Ψ ><Ψ| . Both Σ _A and Σ _B possess orthonormal basis states |φ _i^A> and |χ _i^B> allowing to write any pure state |Ψ > of Σ as |Ψ >=∑_iλ _i|φ _i^A>⊗|χ _i^B> (Schmidt decomposition), where the sum of the squares
of the real non-negative so-called Schmidt coefficients λ _i is
equal to 1 (see e.g. <cit.>). Moreover, ρ _A and ρ
_B have the same eigenvalues, equal to λ _i^2 <cit.>. One introduces the entropies for Σ _A and Σ
_B, respectively S_A=-Tr_A(ρ _ALnρ _A) and
 S_B=-Tr_B(ρ _BLnρ _B), and, as a result
of both the Schmidt decomposition and the just mentioned property of the
eigenvalues of ρ _A and ρ _B, S_A=S_B=-∑_iλ
_i^2Lnλ _i^2. Then S=0, while S_A=S_B≥ 0, and |Ψ > is unentangled if and only if S_A = S_Bis equal to
zero. A means of establishing an iff condition through the reduced entropy
concept therefore does in principle exist. But the fact that the reduced
entropy of a bipartite system is related to the Schmidt decomposition
immediately suggests that, if this concept is used as a tool for
establishing an iff condition for the c_i introduced in this paper,
the difficulty will be at least as great as the one found with the Schmidt
criterion, already discussed in Section <ref>.

Let us first examine the two-qubit case: A is qubit 1 and B qubit 2.
Then, keeping our previous notations, |Ψ >=∑_i=1^4 c_i| i>, one has first to express the condition S_A=0 as a
function of the c_i coefficients, but this means: 1) calculating the
expression of ρ _A , 2) calculating its eigenvalues, 3)
calculating S_A and solving the equation S_A=0. The reader may
verify that a tedious calculation leads to our well-known result: 

    (c_1c_4-c_2c_3)=0.

 The next simplest situation is N=3, and one may first introduce ρ
_3, the reduced entropy for qubit no. 3, and focus on the corresponding
reduced entropy S_3= -Tr_3(ρ _3Lnρ _3), which is zero 
iff |Ψ > is unentangled. This necessitates first to
calculate all the elements of the reduced density matrix ρ _3, each
one a complicated sum involving our c_i coefficients, and secondly to
find an analytical expression for the eigenvalues of ρ _3. But, once
this is done, one knows that if and only if one and only one eigenvalue is
non-zero, and therefore equal to one, then the state is unentangled.
Considering the reduced entropy S_3, i.e manipulating sums of quantities
involving logarithms, is therefore unnecessary.





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