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Impossible Operations in the
Quantum World-I and II

Debasis Sarkar
dsappmath@caluniv.ac.in
Department of Applied Mathematics,
University of Calcutta
A List of Impossible Operations
(No Go Principles)
   1. Impossibility of Exact Cloning (No Cloning)
   2. Impossibility of Exact Deletion (No Deleting)
   3. Stronger No-Cloning
   4. Non-Existence of Universal Exact
Flipper (No Flipping)
   5. No Partial Cloning
   6. No Partial Erasure
   7. Non-existence of Universal Hadamard Gate
   8. Impossibility of Probabilistic Cloning
   9. Impossibility of Broadcasting mixed states
   10. No Splitting
   11. No Hiding
   Etc….!
Quantum Systems
Physical System- associated with a separable
complex Hilbert space
acting on the Hilbert space
States are represented by density operators acting
on the Hilbert space
   Measurements are governed by two rules
   1. Projection Postulate:- After the measurement
of an observable A on a physical system
represented by the state ρ, the system jumps into
one of the eigen states of A.
   2. Born Rule:- The probability of obtaining the
system in an eigen state      is given by
   Tr(ρP[      ]).

   The evolution is governed by an unitary operator
or in other words by Schrodinger’s evolution
equation.
States of a Physical System
   Suppose H be the Hilbert space associated
with the physical system.
   Then by a state ρ we mean a linear, Hermitian
operator acting on the Hilbert space H such
that
   It is non-negative definite and
   Tr(ρ)= 1.
   A state is pure iff ρ2 = ρ and otherwise mixed.
   Pure state has the form ρ=|, |H.
Composite Systems

   Consider physical systems consist of two or more
number of parties A, B, C, D, ……
   The associated Hilbert space is HAHB HC
HD …
   States are then classified in two ways
   (I) Separable:- have the form,
ρABCD =wi ρiA ρiB ρiC ρiD with 0 wi 1,
and wi =1.
   (II) All other states are entangled.
Pure Bipartite System
States are represented by unique Schmidt form

here
A
is the Schmidt rank,
   AB
B

and    are the Schmidt coefficients, such that
.    Schmidt


vector of this State is   ( ,  , )
1   2      d   
Pure Bipartite Entanglement
   Entanglement of pure state is uniquely measured by von
Nuemann entropy of its subsystems,

   States are locally unitarily connected if and only if they
have same Schmidt vector, hence their entanglement
must be equal.
   Thermo-dynamical law of Entanglement : Amount of
Entanglement of a state cannot be increased by any
LOCC (local operations performed on subsystems
together with classical communications between the
subsystems).
Some Use of Quantum
Entanglement

   Quantum Teleportation,(Bennett et.al., PRL,
1993)

   Dense coding, (Bennett et.al., PRL, 1992)

   Quantum cryptography, (Ekert, PRL, 1991)
Physical Operation
Suppose a physical system is described by a state 
Krause describe the notion of a physical operation
defined on       as a completely positive map
, acting on the system and described by

where each Ak is a linear operator that
satisfies the relation
Separable Super operator
If                      then the operation Ak is
trace preserving. When the state is shared between a
number of parties, say, A, B, C, D,. ....
and each Ak has the form

A    B    C     D
with all of   Lk , Lk , Lk , Lk ,
are linear operators then the operation      is said to
be a separable super operator.
Local operations with classical
communications (LOCC)
Consider a physical system shared between a
number of parties situated at distant laboratories.
Then the joint operation performed on this system
is said to be a LOCC if it can be achieved by a set
of some local operations over the subsystems at
different labs together with the communications
between them through some classical channel.
A result : Every LOCC is a separable
superoperator.

But whether the converse is
also true or not ?
It is affirmed that there are
separable superoperators which could
not be expressed by finite LOCC.

Bennett et. al. Phys. Rev. A 59, 1070 (1999).
   The number of Schmidt coefficients of a pure bipartite
entangled states cannot be increased by any LOCC.
   Measure of Entanglement:-
   1. For pure bipartite state entanglement is measured by
the Von Neumann entropy of any of it’s subsystems.
This is the unique measure for all pure bipartite states.
   2. For mixed bipartite entangled states, there is no
unique way to define entanglement of a state.
   Two useful measure of entanglement:
   Entanglement of Formation and Distillable
Entanglement.
   For multipartite case the situation is very complex.
Cloning
   Basic tasks: Is to copy quantum information
exactly in (arbitrary) quantum states.
   If there exists a machine which can perform
such tasks, then we call it a quantum cloning
machine.
   The question is: whether there exists quantum
cloning machine which can copy exactly
arbitrary quantum information or not?
The No-Cloning Theorem
   Arbitrary quantum information cannot be
copied.
   To prove it, suppose we have provided with two
states  ,  and assume that there is an exact
cloning machine. Then we could write its action
as:
 b M    M
 b M    M
where |b>,|M> are the blank and machine states.
Sketch of the proof
   Use the Unitarity of quantum operations, i.e.,
consider U be the unitary operation responsible
for cloning two above states. Unitarity implies,
    =       M M
2
   

If the states are different, then the above relation
immediately implies, both must be orthogonal.
One may also prove the theorem using linearity.
Some points to Note
   Quantum mechanics prohibits exact cloning of
arbitrary information encoded in quantum
states. This does not imply inexact cloning is not
possible.
   For qubit system, there exists optimal universal
isotropic quantum cloning machine with fidelity
5/6. If it is restricted further to states in a great
circle of Bloch sphere, then fidelity is increased
further and is ½+¼().
   There is a nice relation between impossibility of
discriminating non-orthogonal states with
certainty and no cloning theorem (exercise!).
   There are lot of results already in literature to
study the possibility of doing inexact quantum
cloning in different quantum systems(see, arXiv)
   One can also prove no-cloning theorem using
some other physical constraints on the system.
We will do some later.
   Ref:-Wootters & Zurek, Nature, 299(1982)802, Diecks, Phys. Lett. A, 92(1982)271,
   Yuen, Phys. Lett. A, 113(1986)405, Scarani et al, Rev. Mod. Phys., 77(2005)1225,
   N. Gisin & S. Massar, PRL, 79(1997), 2153, Bruss et al, PRA, 57(1998)2368.
Quantum Deleting
   Here the task is: Given two copies of a quantum
state (unknown) |, is it possible to delete the
information of one of the part.
   In other words, given two distinct states  , 
whether there is any quantum operation which
could perform the following operation:
|   |   |   | 0
|   |   |   | 0
No Deleting theorem
   Given two copies of an unknown quantum state,
it is impossible to delete the information
encoded in one of the copy.
   Suppose we assume there is a quantum
operation which can perform the following task:

0    0    M       0         b       M0
1 1 M            1 b            M1

Ref.:-Pati & Braunstein, Nature, 404(2000)164.
Sketch of the proof
   Use linearity of quantum operations; i.e., as the
operation is linear, the above task is also
possible for the states, |, where
2 2 .
   Then, we have, from   M   b M
and using other two relations,
{ 2 00    ( 01  10 )   2 11 } M
 ( 2 0bM 0   2 1bM 1 )    2  
 ( 0b   1b ) M 
   Clearly, the above expressions imply it is
impossible to delete exactly arbitrary quantum
information encoded in a state.
   Exercise:- What would be the status of ancilla
states|M0, |M1 and |M?
   One may prove the result with other ways also;
e.g., considering some constraints on the
systems.
   Observe the differences between cloning and
deleting operations. Have there any dual kind of
relation between them?
Stronger No-Cloning Theorem
   Here the question is:
   “How much or what kind of additional quantum
information is needed to supplement one copy
of a quantum state in order to be able to
produce two copies of that state by a physical
operation?”
   i.e., what information is needed initially to clone.
   Ref.:-R. Jozsa, quant-ph/0305114, quant-ph/0204153.
 Consider a finite set of non-orthogonal states
{|i} and a set of states{i}(generally mixed).
 Then stronger no-cloning theorem states that:

There is a physical operation

 i  i  i   i  i   i  i
if and only if there is a physical operation

i   i  i
i.e., full information is needed to clone.
Outline of the proof
   Use the lemma that: Given two sets of pure
states {|i}, {|i}, if they have equal matrices
of inner products, ijij for all i and j,
then there is a unitary operation U on the direct
sum of the state spaces with U|i=|i, for all i
and vice-versa.
   Now consider first i be a pure state. Then
prove the result using lemma. After that using i
as mixture of pure states, prove the result in
general.
Some consequences
   For cloning assisted by classical information (i
are required to be mutually commuting),
supplementary data must contain full identity of
the states as classical information.
   The proof of no-deleting theorem could be
done using the lemma that used in stronger no
cloning theorem. Both no-go theorems together
establish permanence of quantum information.
Spin Flipping
   Here the task is: Whether it is possible to flip
spin directions of any arbitrary qubit or not?
   In other words, is it possible to construct a
quantum device which could take an
arbitrary(unknown) qubit and transform it into
the orthogonal qubit or not? i.e., possibility of
following operation:

   0   1     * 0  * 1
No-Flipping theorem
   It is not possible to flip arbitrary (unknown) spin
directions.
   To prove it, we must careful about the nature of
the operation we are trying to do. Unlike,
cloning here it is not possible to prove the result
by considering only two non-orthogonal states.
   Why?
   Because flipping is possible for states lying on a
great circle of the Bloch sphere. (exercise)
Sketch of the proof
   One way is to show the operation we have
written is in general an anti-unitary operation
and proof is complete as anti-unitary operations
are not Physical!!(Not a CP map!)
   Other way is to consider three states not lying
on a great circle and prove the impossibility.
   Consider states like the following:
| 0
|    a | 0  b | 1
|    c | 0  dei | 1
   Then consider there is a flipping machine which
could act on those three states.
   Now try to prove the operation is impossible.
   One may also prove by considering some
constraints on the quantum systems.
   Ref.:-Buzek et al, PRA, 60(1999) R2626,
   Martini et al, Nature, 419(2002)815.
Probabilistic Quantum Cloning
   Here the task is copying/cloning exactly but not
with certainty. i.e., the condition is relaxed one.
   Interesting fact is that if one try to copy a set of
states exactly but probabilistically, then the set
should be a linearly independent one.
   Ref.:-Duan & Guo, PRL, 80 (1998)4999.
   The intuitive proof follows from a constraint on
the system.(Pati, PLA, 270(2000)103; Hardy &
Song, PLA, 259(1999)331.
Partial Cloning
   Here we have to study the possibility of cloning
partially, i.e., whether there is any quantum
device which could perform the following
operation or not:
 b M   F ( ) M

where    0   1 ,  are arbitrary and F
is a function of the original.
Ref:- Pati, PRA, 66(2002)062319.
Outline of the proof
 Consider some cases of F first. Suppose,
|F(, where K is a unitary or anti-
unitary operator. Then using linearity and anti-
linearity of the operation it is straight forward to
prove.
 Then consider K as a combination (linear)
unitary and anti-unitary operator and prove the
result.
 One may also prove in general way.
Non-Existence of Universal
   There are two ways of defining it:
   Check the possibility of the following
operations:         1
         (   )
2
1
       (   )
2
   Or,             1
     ( i )
2
1
     (i    )
2
   Result: There is no Hadamard gate of the above
kind for arbitrary unknown qubits.
   The proof is straightforward, if we consider any
two distinct qubits and then taking inner
products of them and their orthogonals before
and after the operations. (exercise)
   Ref:- Pati, PRA, 66(2002)062319.
Partial Erasure
   Reversibility of a quantum operation(unitary)
prohibits complete erasure of state (say, qubit).
i.e., to transform any qubit to a standard state by
unitary evolution is not possible.
   However, one may ask the following:
   Whether it is possible to erase partially quantum
information, even by using irreversible ones?
   What do we mean by partial erasure?
   A partial erasure is a trace preserving completely
positive map that maps all pure states of a n-
dimensional Hilbert space to pure states in a m-
dimensional subspace (m<n) via a constraint.

   In other words, partial erasure reduces the
dimension of the parameter domain and does
not leave the state entangled with other system.
No-partial Erasure
   Theorem 1. Given any pair of non-orthogonal
qudits, in general, there is no physical operation
that can partially erase them.
   Linearity of quantum theory gives us even more
than the above.
   Theorem 2. Any arbitrary qudit cannot be
partially erased by an irreversible operation.

   Ref.:-A.K.Pati & B.C.Sanders, PLA, 359(2006)31.
Outline of the proofs
 For theorem-1, Consider two states  , 
in a n-dim Hilbert space and suppose there is a
partial erasure with a constraint K.
 For simplicity, choose the constraint such that it
reduces the parameter space at least 1 dim. The
reduced dim. states then may not have in general
equal inner product with the initial states.
 Now attach ancilla to see the effect of the
quantum evolution and considering unitarity
take inner product of both side.
   For theorem-2, take an orthonormal basis
{|i}, i= 1,…,d. Then partial erasure machine
yields,
 i A   i A   i

Where|A> is the ancilla state.
 Now consider an arbitrary state |. Then,
d
n           d
n
 A   e Cos i n
 n A   e Cos i n
 n A

n            2           n              2             n

   Clearly, the resultant state of the system is mixed
that contradicts the definition of erasure to be
pure.
Some Consequences
   Check the result of theorem-2 for d=2 not using
ancilla.
   For d=2 in theorem-1 an immediate
consequence is “A universal NOT gate is
impossible”, i.e., No-flipping theorem (Check it)
   Also from theorem-2 the no-splitting of
quantum information  ,    follows.
   There are also some other consequences.
No-Hiding
 Perfect Hiding is a process  i   o , where  i
is arbitrary input state and  o be the fixed
output state.
 The input states forms a subspace of a larger
Hilbert space and output state resides in a
subspace where it has no dependence on input.
 No-Hiding theorem states, “Quantum
information can run, but it can’t hide”.
Sketch of the proof
   Use linearity and unitarity of quantum process.
   Consider only pure input states using linearity.
   Unitarity allows suitable choice of ancilla making
the space larger. Then,         d
    i
      pk k   o
 Ak ( )
k 1

   Examining physical nature of hiding, we have,
d
   i
  pk k   o
 ( qk    0)
k 1
   So, perfect hiding is impossible, as ancilla
contains full information of hiding state, which
by definition impossible.
   One may also study the imperfect hiding
processes and found the implication of the result
of no-hiding theorem.
   Further, it has severe implication in Black-hole
   Ref.:-Braunstein & Pati, PRL, 98(2007)080502.
Constraints over Physical
Operations
   1) No-Signalling
   2) Non-increase of Entanglement under
LOCC
   3) Impossibility of inter conversion by
deterministic LOCC of a pair of Incomparable
states.
No-Signalling
Status : It is a very strong constraint over any physical system.
This is not only a quantum mechanical constraint.

Power : This is very powerful restriction and senses easily in
most of the settings and for almost all kind of impossible
operations. Even not using quantum mechanical formalism, e.g.,
linearity and unitary dynamics.

Impossible Operations detected by it :
UNIVERSAL EXACT CLONING,
UNIVERSAL EXACT DELETING,
UNIVERSAL EXACT SPIN-FLIPPING, e.t.c.
Non-Increase of
Entanglement under LOCC
Status : It is quite similar to the thermo-dynamical
constraint over physical systems. This is entirely
a constraint of quantum information
processing.
The physical reason behind this restriction entirely
depends on the existence of entangled states of a
physical system. Entanglement is a measurable
physical resource that can be applied to perform
various kinds of information and computational
tasks. As it may be viewed as an amount
of non-locality of the system, therefore we have
a principle that entanglement can not be increased
by local operations with classical communications.
Power :    This is also a very powerful restriction
over any quantum mechanically allowed
Evolution of any physical system and it detects
many impossible operations of quantum
Information processing.
As it is purely a quantum
mechanical constraint so we can use it to detect
any local operation to be physical or not and we
have the facility of using the operation to act
linearly on superposition level of quantum states.
Impossible Operations detected by using the
principle of non-increase of entanglement by
LOCC :
UNIVERSAL EXACT CLONING - In a pure state
setting, by Horodecki et.al. In a much more relaxed
mixed state scenario, by I. Chattopadhyay et. al. (in
preparation).
UNIVERSAL EXACT DELETING – By Horodecki et.al.,
considering the constraint that,
`In a closed system the entanglement remain unchanged
under LOCC’.
Further, using `Non-increase of entanglement under
LOCC’, in a very simple setting, with linearity assumption
only on density matrix level and applying the deletion over
only two arbitrary qubits, by I. Chattopadhyay et. al.
UNIVERSAL EXACT SPIN-FLIPPING -
Assuming the
existence of exact flipping operation acted on
only three states not lying in a great circle of the
Bloch sphere and applying this operation locally
on a joint system using a less restricted linearity
assumption the entanglement content of the system
can be increased which establishes the impossibility
of universal exact spin-flipping.

***In all the settings we assume linearity of the operations
only on density matrix level
Quantum Deleting and Signalling
   Alice-Bob share 2 copy of Singlet state and Bob has 1-2
copy exact deletion machine acting on the arbitrary
qubit
  M   b M
      cos 0  sin  1 as,
  M   b M
  M  
  M  

   Bob apply the deleting machine on joint state

12

34
M ;     1
2
        
   If Alice measure her particles on either {0,1} or in
qubit basis and Bob traced out Machine state, then the
remaining 2 qubit mixed state depends on choice of
basis() Which can be distinguished exactly from =0.

 24  tr135  out   12345
 out    
1
4

 I2  b       4
b  tr5        245
    
  tr5    245
   

   Thus assumption of perfect deletion machine
   Ref.:- Pati & Braunstein, PLA, 315(2003)208.
No Signalling and Probabilistic
Quantum Cloning
   Alice, Bob share a large no. of entangled state AB
N                               N
   AB
   1
N   
n 1
n   A
Bn   B
   1
N    n
n 1
A
BN  n   B

BnB are arbitrary (not necessarily orthogonal)
   If Bob have a 1-2 copy PQCM then he have a 1 to 
copy PQCM, for a large  and for a less (but > 0)
probability as
(  1)        prob  0                
Bn b                   
  Bn

for some values n=1,2,···,N and N+1
   Alice has 1 cbit information, corresponding to it she
make a measurement on each of her particle, but do not
communicate to Bob. BOB apply PQCM on his
particle.

Cbit Measurement Basis of          Input set for Bob’s PQCM
value Alice
0   A1 : {1, 2,····, N}   B1={B1, B2,····, BN}

1   A2 :{1´, 2´,····, N´} B2={BN+1, BN+2,····, B2N}
   For general input Bk Bob’s output
N 1

 c Bl
                               
k
l          d k  k where  k Bl            0  l  1,2,, N  1
i 1
 Bob succeeds to clone exactly  copies of
particle B, if input is one of Bn ;
n=1,2,….,N+1
 P0(A1) = n=1N Prob(n| A1) = 1 , P1(A1) =
P(N+1| A1) = 0,
P0(A2)= N P(n| A2)  1- P(N+1| A2) , P1(A2)=
P(N+1| A2)  0
After many repetition P0(A2)→1, Bob can
correctly know the bit without any actual
communication imply superluminal signaling.
   Ref.:- L. Hardy & D. D. Song, PLA 259(1999) 331. See also
Pati, PLA, 270(2000)103.
   Impossibility of Exact Cloning
   Consider a mixed state ρABC with A in one location and B,
C are in another location so that the system may be
considered as a bipartite system.
 00  11                           00  11 
 ABC     |   |  P           (1   ) |   |  P 
    2  BC                             2   
   where = a0+b1 and = c0+dei 1 are two non-
orthogonal qubit state on which we assume exact cloning is
possible.
   The logarithmic negativity (an upper bound of distillable
entanglement) of the state is
1
LN (  )   N (  ) 
1  bc  ad   1
2                  2
   If exact cloning is possible then we can extract one ebit of
entanglement, however in AB:C cut LN() 1.
   Partial Cloning
   We define the partial cloning machine on two qubit
states ,  as follows:
|   | b |   | F ( )
|   | b |   | F ( )
   where F(), F() are some functions of ,  resp.
   Suppose two distant parties A, B share a state,
1
|  AB      | 0 A | 0  B  |1 A |1  B   | b B
2
   where the first particle is with A and other two with B and
bB is the blank state.
   Applying partial cloning machine the final state has the form:
|   AB
f

1
2
| 0 A | 0 F ( ) B  ei |1 A |1 F ( ) B 
   Tracing out the last two qubits of initial and final
   states, the reduced state between A and B are

 AB  P 00  P 11   |   2 | 0011|  |   2 |1100 |
i   1
2
   And
 P  00  P 11   |   F   | F   | 0011|
1
   f
AB
2
  |   F   | F   |1100 |

   So, whenever   0, i.e., two states remain non-
orthogonal, the entanglement content of the final
reduced state is always greater than the initial one.

   Comment: violation of principle of non-increase of
entanglement.
Quantum Deletion
   Consider a pure entangled state shared between three
parties A, B, C.
1
|  ABC       |   A |1 B |1 C  |   A | 0 B | 0 C 
2
   Assume A has a exact deleting machine for two non-
orthogonal states , , acting as,
|   |   |   | 0
|   |   |   | 0

   Final state is then,
1
|   ABC
d
    |  0 A |1 B |1 C  |  0 A | 0 B | 0 C 
2
   Tracing out first two qubits of A, the reduced state
between B, and C are,

 BC  TrA |  ABC  |

   and

1
2
P 00  P 11   |  2 | 0011|  |   2 |1100 |
 d BC  TrA |  d  ABC  d |

1
2
P 00  P 11   |   | 0011|  |   |1100 |
   The concurrence (used to measure entanglement of
formation) of initial and final states are,
   C() =2 and C() = 
   therefore,        C (  )  C ( )
   which implies, E (  )  E ( )
f              f
Spin Flipping
   Consider three arbitrary qubit states not lying in
one great circle of the Bloch sphere,
| 0
|    a | 0  b | 1
|    c | 0  dei | 1

   where a0, c0, 0, b,d are real nos., such that
a2 +b2 =1, c2 +d2 =1 and 0, 1 are orthogonal to
each other.
   Assume the most general flipping operation for those
three states as: 0 M 1 M0,
    Mei   M ,  M ei M .
   Consider a bipartite pure entangled state shared between
two spatially separated parties A, B:
   |i AB =(13)[|0A |0B+|1A|B+|2A|B ] |MB
   where A holds 3-dim. system with basis {|0A , |1A,|2A}.
   Applying exact flipper on the side of B the state takes the
form
   |f AB =(13)[|0A |1M0B+ ei |1A|MB+
              ei|2A|MB ]
   Now if we calculate the reduced density matrices of A’s
side, then it is easy to check that the initial and final density
matrices are different unless when b=0, or, d=0, or
sin()=0.
   Conclusion: No-signalling implies no-flipping
   Note:- the above setting does not show any change in
entanglement.
   Next consider a bipartite state shared between A, B with
A has a 2-dim. system and B has four qubit system(say,
1,2,3,4),
   |i AB =(18)[(|000+ |111)A12 |1034 –
            (|010+|100+|101)A12|34 –
            (|011+|110+|001)A12|34]  |MB
   Assume B has a exact flipping machine that acts on last
qubit(4-th). Then the final joint system takes the form:
   |f AB =(18)[(|000+ |111)A12 |11M0 34B
–
ei(|010+|100+|101)A12|M34B –
ei(|011+|110+|001)A12|M34B]
   Now if we calculate the reduced density matrices
of A’s side we find that the largest eigen values
of initial and final density matrices are related by:
    f i, which implies E(|fA:B ) E(|iA:B)
   i.e., entanglement has increased by local
operations in A:B cut.
Incomparability ~ As a New
Constraint over Pure Bipartite
System
Incomparability of two pure bipartite
entangled states imply that either of the
two states can not be converted to the other
by LOCC with certainty. We propose
this as a constraint over the joint system through
LOCC. Thus if a local operation can inter-convert
an incomparable pair of states, then the operation
is certainly an unphysical operation.
Nielsen’s criteria
Let  AB and  AB be any two bipartite pure
states of Schmidt rank d with Schmidt vectors,
and
respectively, where
1  i ,  j  0  i, j    ,  i  1    j    and

(PRL, 83, 436 (1999))
Then the state  can be deterministically
AB
transformed to the state          by LOCC (denoted
by               ) if and only if   majorized by
(denoted by              ) i.e., if
 If this criteria is not satisfied for a pair of states
then we denote it as  AB   AB 
 Though it may happen that                        .
 The pair of states                  are said to be
comparable if either  AB   AB or  AB   AB .
 From thermo-dynamical law of entanglement one may
conclude,

   For any pair of states,             implies rank of    AB
is greater or equals to the rank of  AB
.
Incomparability of two bipartite pure
states :-
If for some pair of bipartite pure states we have both |Ψ  |Φ and
|Φ  |Ψ, (i.e., either α1β1 and i=k1 βi  i=k1 αi for some
k2,3,…,n or β1α1 and i=s1 αi  i=s1 βi for some s2,3,…,n Then
(|Ψ, |Φ) is said to be an incomparable pair of states and this
phenomenon is denoted as |Ψ|Φ.
It is to be noted that from the incomparability criteria |Ψ|Φ will
not indicate either E(|Ψ)  E(|Φ) or E(|Φ)  E(|Ψ).
If (|Ψ, |Φ) are mn states, where min{m,n}=3, then the criteria for
|Ψ  |Φ can be expressed as,
either       α1  β1 and α3  β3
or           β1  α1 and β3  α3
Consider an operation ( A) defined on the set of
input states
{         ; i = 1,2,…k}

in the following manner
A(|  i  ) |  'i 
A

Now consider the pure bipartite state

|  AB  i 1i | i A |  i  B
k

shared between two distant parties Alice
and Bob.
Applying locally on Bob’s system the joint state
transforms to another pure bipartite state
|  AB  i 1 i | i A |  'i  B
k

Now the Schmidt coefficients corresponding to
the two pure bipartite states are
   'i ; i  1, 2,....k
  i ; i  1, 2,....k
If the two states |  AB ,|  AB 
are incomparable, then the operation ( A )
is an impossible one.
Universal exact Flipping
via Incomparability
Special feature: Flipping operation is defined only on
a minimum number of three qubit states as in the
earlier cases. | 0
|    a | 0  b | 1
|    c | 0  dei | 1
The flipping operation is described as:
| 0 |1
|    ei |    ei  b | 0  a |1 
|    ei |    ei  de i | 0  c |1 

We omit the machine part as we assume the quantum
formalism.
Now consider a pure bipartite state shared between A, B:
1
|   AB
i
    | 0 A | 00 B  |1 A |   B  | 2 A |   B 
3

After applying the flipping operation on the second qubit of
B, the state takes the form:

|  f  AB 
1
3

| 0 A | 01 B  ei |1 A |   B  ei | 2 A |   B   
Now if we calculate the eigen values of the initial and final
density matrices of A, then we find that they satisfies the
Incomparability criteria unless they are in a great circle.
So incomparability implies no-flipping. Also we find the
reverse case. We illustrating it by examples.
Consider three qubits representing three axes of the Bloch sphere as
|Z =|0 , |x = ½ (|0 +|1) and |Y = ½ (|0 + i|1).
We define our flipping operation on those three state as follows,
|Z         |Z 
|x         eiμ |x              . ……(1)
|Y        eiν |Y 
The flipped state |i  is orthogonal to the initial state |i,  i= z,x,y
and eiμ , eiν are arbitrary phase factors.
Let |χi = 1/ 3 {|0A |ZZ B+|1A |xyB+|2A |yxB} be a 34
( 322 ) state between Alice and Bob situated in two distant labs.
Now if Bob applies the flipping operation defined in (1) on the
2nd qubit of his local system then the joint state between them
will transform as follows,
|χi     |χf = 1/ 3 {|0A |ZZ B + eiμ |1A |xyB
+ eiν |2A|yxB}
Schmidt vector corresponding to |χi and |χf are
λf = {1/ 3 ( 1/ 3 + 1/2 ), 1/3 , 1/ 3 ( 1/ 3 - 1/2 )}
and λi = {2/3 , 1/6 , 1/6 }.
Obviously, 1/ 3 ( 1/ 3 - 1/2 ) < 1/6 < 1/3 <1/ 3 ( 1/ 3 + 1/2 ) < 2/3 , which
imply that the initial and final joint state between Alice and
Bob are incomparable. Hence it is not possible to
transform |χi to |χf by LOCC, though |χf is being
prepared from |χi by applying the flipping operation
locally. Thus the operation defined in (1) is not a valid
(physical) operation.
So we have |χi             |χf            Universal Flipping is not
possible.
Incomparability from No-Flipping
Principle
Lets consider two pure state of 34 system shared between two
spatially separated parties Alice and Bob.
|1 = ·51 |0A |0B + ·30 |1A |1B + ·19 |2A |2B
|2 = ·49 |0A |0B + ·36 |1A |1B + ·15 |2A |2B
Bob’s local system has the following form
|0B= |ΨB1 |ΨB2 , |1B = |ΨB1 |ΨB2 , |2B = |ΨB1 |ΨB2
where |Ψ be a arbitrary qubit state |ΨΨ|= ½ [ I + n · σ ].
Local systems of Bob’s first qubit (B1) corresponding to the two
joint state |1 and |2 between Alice and Bob are,
1= Tr A B2 ( |1 1| ) = ½ [ I + 0.2 n · σ ]
2 = Tr A B2 ( |2  2| ) = ½ [ I - 0.2 n · σ ]
Hence local transformation of |1 to |2 will flip the spin direction
of 1 associated with |Ψ.
Conclusion : If somehow we extend the LOCC
transformation criteria so that either |1     |2 or
|2     |1 then consequently on a subsystem, the direction
of spin-polarization of an arbitrary state 1 will be reversed.
That is equivalent of preparing an exact universal flipping
machine.

So     ``No-Flipping principle’’            |1      |2.

This two result together shows an equivalence between the
constraint on LOCC state transformation criteria and the
No-Flipping principle. The relation is drawn from the
unphysical nature of flipping operation as it is not an unitary
operation but an anti-unitary operation.
Cloning and Deleting via
Incomparability
This work connects the Cloning and
Deleting with the concept of Incomparability as a
constraint imposed over the system by LOCC.
We define the Cloning operation to act
on just two arbitrary input qubits | 0,|  in the
following manner
| 0 | b | 0 | 0
|  | b |  | 
Now consider a pure bipartite state shared
between Alice and Bob situated at distant labs in
the form
1
|  AB         | 1 A (| 00 | 00) B
N
 | 2 A (| 00 | 00 ) B
 | 3 A (| 00 | 00) B   | b B
Let Bob applies the Cloning operation on his last
two qubits. Then the joint state transforms to
another pure bipartite state
1
|  AB        | 1 A (| 00 | 000 ) B
N
 | 2 A (| 000 | 00 ) B
 | 3 A (| 00 | 000) B 
Tracing out Bob’s subsystem we observe the initial
and final form of Alice’s local system and compute
the Schmidt vector corresponding to the two joint
states |  AB ,|  AB are  and  , and find that


|  AB  |  AB
Thus the cloning operation defined over | 0,| 
is unphysical in nature.
If instead of Cloning we consider the Deleting
operation defined on the two arbitrary input
states        in the following way that

Consider now that Alice and Bob will initially share
the joint state     .Then by assuming the existence
of the exact Deleting operation and applying it on
Bob’s system the final joint state will be       . The
incomparability of the pair of states will imply that
the exact deleting operation defined on the two
arbitrary non-orthogonal states does not exists.
Question:- Is it possible to extend the
proof via incomparability for general
anti-unitary operations and inner-
product preserving operations?