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arXiv:quant-ph/0602016v3 12 Jan 2007
Inseparability of Quantum Parameters
I.Chakrabarty
1∗
,S.Adhikari
2
,Prashant
3
,B.S.Choudhury
2
1
Herita ge Institute of Technology, Kolkata, India
2
Bengal Eng ineering and Science University, Howrah, India
3
Indian Institute of IT and Management, Indi a
Abstract
In this work, we show that ’splitting of quantum inf ormation’ [6] is an impos-
sible task from three different but consistent principles of unitarity of Quantum
Mechanics, no-signalling condition and non increase of entanglement u nder Local
Operation and Classical Communication.
PACS Numbers: 03.67.-a, 0 3.65.Bz, 89.70.+c
1 Introduction
In quantum information theory it is most important of knowing the various differences
between the classical and quantum information. Many operations which are feasible in
digitized information becomes an impossibility in quantum world [1-6]. This may be
probably due to the linear structure or may be due to the unitary evolution in quantum
mechanics. Regardless of their origin, these impossible operations are making quantum
information processing much more restricted than it’s classical counterpart. On the other
hand this restriction on many quantum infor matio n processing tasks is making quantum
information more secure. In the famous land mark paper of Wootters and Zurek it was
∗
indranilc@indiainfo.com
1
shown that a single quantum cannot be cloned [1]. La t er it was also shown by Pati and
Braunstein that we cannot delete either of the two quantum states when we are provided
with two identical quantum states at our input port [2]. In spite of these two famous
’no-cloning’ [1] a nd ’no-deletion’ [2] theorem there are many other ’no-go’ theorems like
’no-self replication’ [3] , ’no-partia l erasure’ [5], ’no-splitting’ [6] and many more which
have come up. R ecent research has revealed that these theorems are consistent with dif-
ferent principles like principle of no-signalling and conservation of entanglement under
LOCC [7-9]. If we put it in a different way it means that if we violate these ’no-go’
theorems we will violate the principle of no-signalling and non increase of entanglement
under LOCC.
No-splitting theorem: It is a well known fact that there are many operatio ns which
are feasible in the classical world but doesn’t hold good in the quantum domain. These
we generally refer as ’General impossible operations’. ”No splitting Theorem” [6] (almost
equivalent to the ’No-Partial Erasure of Quantum Information’ [5])is yet another addition
to this set. It states that ’For an unknown qubit,quantum information cannot be split
into two complementing qubits,i.e. the information in o ne qubit is an inseparable entity.
An important application of splitting of quantum information is a construction of a gate
which can be used to reversibly split a parameter encoded in non orthogonal quant um
states , enabling the necessary quantum information compression and decompression re-
quired for optimal quantum cloning with multiple copies [10].
In this work our obj ective is different from [6] in the sense that here we will investigate
whether we can split two non orthogonal quantum states from three different but consis-
tent principles like, preservation of inner products under unitary evolutio n, the principle
of non increase of entanglement under LOCC and principle of no signalling . In other
words, unlike in ref [6], instead proving the splitting of quantum state from linearity, we
correlate this impossibility with other aspects like, unitarity ,restrictions on entanglement
processing and causality.
2
2 Proo f of No-splitting theor em from three different
principle s:
I. From Unitarity of Quantum Mechanics: First of all we show that the ’No-
splitting theorem’is consistent with the unitary evolution of quantum theory. For this
purpose we will consider a pair of non orthogonal states [|ψ
1
(θ
1
, φ
1
)i, |ψ
1
(θ
2
, φ
2
)i], where
0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. These non o rt hogonal states are represented by points
on the Bloch sphere. Let us assume that the splitting of quantum information into
complementary parts is possible. If we consider a hypothetical machine which can split
quantum information in each of the non-orthogonal states into complementary part s, then
the action of the machine on the non-ortho gonal states [|ψ
1
(θ
1
, φ
1
)i, |ψ
1
(θ
2
, φ
2
)i] is defined
by the set of transformations:
|ψ
1
(θ
1
, φ
1
)i|ψ
2
i → |ψ
1
(θ
1
)i|ψ
2
(φ
1
)i (1)
|ψ
1
(θ
2
, φ
2
)i|ψ
2
i → |ψ
1
(θ
2
)i|ψ
2
(φ
2
)i (2)
where
|ψ
1
(θ
j
, φ
j
)i = cos(
θ
j
2
)|0i + sin(
θ
j
2
)e
(iφ
j
)
|1i (3)
|ψ
2
(φ
j
)i = |ψ
21
i + e
(iφ
j
)
|ψ
22
i (4)
|ψ
1
(θ
j
)i = cos(
θ
j
2
)|ψ
11
i + sin(
θ
j
2
)|ψ
12
i (5)
where j = (1, 2). Here |ψ
11
i, |ψ
12
i, |ψ
21
i, |ψ
22
i are non normalized states independent of
θ, φ. The unitarity of transformations will preserve the inner product .
hψ
1
(θ
1
, φ
1
)|ψ
1
(θ
2
, φ
2
)i =
hψ
1
(θ
1
)|ψ
1
(θ
2
)ihψ
2
(φ
1
)|ψ
2
(φ
2
)i (6)
The above equality will not hold for all values of (θ, φ). The equality will hold if
φ
2
= φ
1
+ nπ and θ
1
± θ
2
= (2m + 1)π where m, n are integers(see appendix).This
equality corresponds to a situation where the quantum states are orthogonal. Thus we
see that the equality does not hold for all values of θ and φ and hence we conclude t hat
this kind of transformation doesn’t exist. We cannot split the quantum info r matio n for
3
two non orthogonal quantum states.
II. From Non Increase of Entanglement under LOCC and No signalling Condi-
tion: Next we show that splitting of quantum information is an imp ossible operation from
the principle of non increase of entanglement under LOCC. Let us consider an enta ng led
state shared by two distant part ies Alice and Bob, of the form
|ψi
AB
=
1
√
2
[|0i
A
|ψ
1
(θ
1
, φ
1
)i
B
+ |1i
A
|ψ
1
(θ
2
, φ
2
)i
B
]|ψ
2
i
B
(7)
where {|ψ
2
i} is the blank state attached to the Bob’s particle.
The reduced density matrix on Alice’s side is given by,
ρ
A
= T r
B
(|ψi
ABAB
hψ|) =
1
2
[I + |1ih0|(hψ
1
(θ
1
, φ
1
)|ψ
1
(θ
2
, φ
2
)i) +
|0ih1|(hψ
1
(θ
2
, φ
2
)|ψ
1
(θ
1
, φ
1
)i)] (8)
Let us assume that Bob is in possession of a machine which will split the quantum infor-
mation of his particle. The transformation describing the action of the machine is given
by equations (1) and (2) . Now after the application of the quantum info r matio n splitting
machine the entangled state (7) takes the form
|ψi
C
AB
=
1
√
2
[|0i
A
|ψ
1
(θ
1
)i
B
|ψ
2
(φ
1
)i
B
+ |1i
A
|ψ
1
(θ
2
)i
B
|ψ
2
(φ
2
)i
B
] (9)
The reduced density matrix on the Alice’s side after the application of the machine is
given by,
ρ
C
A
=
1
2
[I + |1ih0|(hψ
1
(θ
1
)|ψ
1
(θ
2
)i)(hψ
2
(φ
1
)|ψ
2
(φ
2
)i)
+|0ih1|(hψ
1
(θ
2
)|ψ
1
(θ
1
)i)(hψ
2
(φ
2
)|ψ
2
(φ
1
)i)] (10)
The respective la r gest eigen values of these two r educed density matrices are given by
λ
A
=
1
2
+
|p|
2
2
(11)
λ
C
A
=
1
2
+
|q|
2
|r|
2
2
(12)
4
where p = hψ
1
(θ
2
, φ
2
)|ψ
1
(θ
1
, φ
1
)i, q = h ψ
1
(θ
2
)|ψ
1
(θ
1
)i, r = hψ
2
(φ
2
)|ψ
2
(φ
1
)i. To show that
the amount of entanglement E(|ψi
AB
) and E(|ψi
C
AB
) o f the respective entangled states
before and after the splitting doesn’t increase, we must show that λ
A
< λ
C
A
. To show
λ
A
< λ
C
A
this we must show that,
|hψ
1
(θ
2
, φ
2
)|ψ
1
(θ
1
, φ
1
)i| <
|hψ
1
(θ
2
)|ψ
1
(θ
1
)i||hψ
2
(φ
2
)|ψ
2
(φ
1
)i| (13)
LHS : |cos(
θ
1
2
) cos(
θ
2
2
) + e
i(φ
1
−φ
2
)
sin(
θ
1
2
) sin(
θ
2
2
)|
RHS : |cos(
θ
1
2
) cos(
θ
2
2
) + sin(
θ
1
2
) sin(
θ
2
2
)|
|1 + e
i(φ
1
−φ
2
)
| (14)
Let
(φ
1
− φ
2
) = k, cos(
θ
1
2
) cos(
θ
2
2
) = x (15)
sin(
θ
1
2
) sin(
θ
2
2
) = y (16)
where x and y are real quantities.
Now |hψ
1
(θ
2
, φ
2
)|ψ
1
(θ
1
, φ
1
)i| = | x+e
ik
y| = |[x+y cos(k)]+iy[sin(k)]| =
q
[x
2
+ y
2
+ 2xy cos(k)]
and |hψ
1
(θ
2
)|ψ
1
(θ
1
)i||hψ
2
(φ
2
)|ψ
2
(φ
1
)i| = (x + y)
q
2(1 + cos(k)).
Therefore A(say)=[|hψ
1
(θ
2
, φ
2
)|ψ
1
(θ
1
, φ
1
)i|]
2
−[|hψ
1
(θ
2
)|ψ
1
(θ
1
)i||hψ
2
(φ
2
)|ψ
2
(φ
1
)i|]
2
= x
2
+
y
2
+2xy cos(k)−2(x+y)
2
(1+cos(k)) = −[x
2
+y
2
+4xy+2(x
2
+y
2
+xy) cos(k)] > 0 for some
values o f x,y,k. This implies that, λ
A
> λ
C
A
⇒ E(|ψi
C
AB
) > E(|ψi
AB
), as a consequence
of which we can say that the amount of entang lement will increase under local operation.
However we know that enta nglement is non increasing under such operations [ in general
one can only claim that it is conserved under a bilocal unitary operation ]. This gives rise
to contradiction. Therefore, it is clear that the principle of non increase of entanglement
under LOCC doesn’t a llow perfect splitting of nonorthogonal quantum states. This rules
out the existence of a hypothetical quantum information splitting machine, designed to
split the quantum information of a nonorthogonal quantum state .
5
Next we show tha t the splitting of quantum information is not possible from the principle
of no signalling. In other words we can say that if we assume perfect splitting of quantum
information it will violate the principle of no-signalling.
Suppose we have a singlet state shared by two distant parties Alice and Bob . The singlet
state can be written in two different basis as
|χi =
1
√
2
(|ψ
1
i|
ψ
1
i − |ψ
1
i|ψ
1
i)
=
1
√
2
(|ψ
2
i|
ψ
2
i − |ψ
2
i|ψ
2
i) (17)
where {|ψ
1
i, |
ψ
1
i} and {|ψ
2
i, |ψ
2
i} are two sets of mutually orthogonal spin states (qubit
basis). Alice possesses the first particle while Bob possesses t he second particle. Alice can
choose to measure the spin in any one of the qubit basis namely {|ψ
1
i, |ψ
1
i}, {|ψ
2
i, |ψ
2
i}.
The theorem o f no signalling tells us that the measurement outcome of Bob are invari-
ant under local unitary transformation done by Alice on her qubit.The density matrix
ρ
B
= trρ
AB
= tr[(U
A
⊗ I
B
)ρ
AB
(U
A
⊗ I
B
)
†
] is invar ia nt under local unitary operation by
Alice . Hence Bob cannot distinguish two mixtures due to the unitary operation done
at remote place. One may ask if Bob split the quantum information of his particle and
if Alice measure her particle in either of the two basis then is there any possibility that
Bob know the basis in which Alice measure her qubit or in other words, is there any way
by which Bob using a perfect splitting machine can distinguish the statistical mixture in
his subsystem resulting from the measurement done by Alice. If Bob can do this then
signalling will take place, which is impossible. Hence now our ta sk is to show that the
splitting of information is an impossible task from no-signalling principle.
Let us consider a situation where Bob is in possession of a hypothetical quantum informa-
tion splitting machine. The unitary transformation describing the splitting of quantum
information fo r an input state |ψ
i
(θ, φ)i (where i=1,2) is defined as ,
|ψ
i
(θ, φ)i|Σi → |ψ
i
(θ)i|Σ(φ)i
|
ψ
i
(θ, φ)i|Σi → |ψ
i
(θ)i|Σ(φ)i (18)
6
where {|Σi} is the ancilla state a tt ached by Bo b .
After the application of the transformation defined in (18) by Bob on his particle the
singlet state defined by (17) including the ancilla state attached by Bob reduces to t he
form,
|χi|Σi → |χi
S
=
1
√
2
[|ψ
1
(θ, φ)i|
ψ
1
(θ)i|Σ(φ)i
−|ψ
1
(θ, φ)i|ψ
1
(θ)i|Σ(φ)i] =
1
√
2
[|ψ
2
(θ, φ)i|
ψ
2
(θ)i|Σ(φ)i
−|ψ
2
(θ, φ)i|ψ
2
(θ)i|Σ(φ)i] (19)
After Bob applying t he splitting machine on his qubit, Alice can measure her particle
in two different basis. If Alice measures her particle in the basis {|ψ
1
i, |
ψ
1
i}, then the
reduced density matrix in the Bob’s subsystem (including ancilla) is given by,
ρ
BC
= tr
A
(ρ
ABC
)
=
1
2
{|
ψ
1
(θ)Σ(φ)ihψ
1
(θ)Σ(φ)|
+|ψ
1
(θ)Σ(φ)ihψ
1
(θ)Σ(φ)|} (20)
On the other hand if Alice measures her particle in the basis {|ψ
2
i, |
ψ
2
i} then the state
described by the reduced density matrix in the Bob’s side is given by,
ρ
BC
= tr
A
(ρ
ABC
)
=
1
2
{|
ψ
2
(θ)Σ(φ)ihψ
2
(θ)Σ(φ)|
+|ψ
2
(θ)Σ(φ)ihψ
2
(θ)Σ(φ)|} (21)
Since the statistical mixture in (20) and (21) are different, so this would have allowed Bob
to distinguish in which basis Alice has performed measurement, thus allowing for super
luminal signalling. However the criterion of ’No-signalling’ tells us that communication
faster than light is not possible. So we arrive at a contradiction, that is, the transfor-
mation defined in (18) is not possible in quantum world. This rules out the existence of
hypothetical machine like quantum information splitting machine.
7
3 Conclus i on
In this work we show that the total information contained in the quantum state on
the Bloch-sphere cannot be written as the tensor product of the state containing the
information of the azimutha l angle and the state conta ining the information of the phase
angle. Therefore the quantum information can be rega rded as an inseparable entity or
in other words we can say that it is impossible to express the function of θ and φ as the
product of function of θ alone and function of φ alone. To justify the above statement, we
proved the no-splitting theorem from three different principles: (i) Unitarity of quantum
mechanics (ii) principles of non increase of entanglement under LO CC and (iii) Principle
of no signalling.
4 Acknowled gement
I.C acknowledges Prof C.G.Chakraborti, S.N.Bose Professor of Theoretical Physics, De-
partment of Applied Mathematics, University of Calcutta for being the source of inspi-
ration in carrying out research. S.A acknowledges CSIR project no.F.No.8/3(38)/2003-
EMR-1 for providing financial support . I.C and S.A would like to thank Dr.A.K.Pati for
useful discussion.
5 Appendix
hψ
1
(θ
1
, φ
1
)|ψ
1
(θ
2
, φ
2
)i = cos
θ
1
2
cos
θ
2
2
+e
i(φ
2
−φ
1
)
sin
θ
1
2
sin
θ
2
2
, hψ
1
(θ
1
)|ψ
1
(θ
2
)i = cos
θ
1
2
cos
θ
2
2
+
sin
θ
1
2
sin
θ
2
2
, and hψ
2
(φ
1
)|ψ
2
(φ
2
)i = 1+e
i(φ
2
−φ
1
)
. Now we have to show that, hψ
1
(θ
1
, φ
1
)|ψ
1
(θ
2
, φ
2
)i =
hψ
1
(θ
1
)|ψ
1
(θ
2
)ihψ
2
(φ
1
)|ψ
2
(φ
2
)i. This equality is possible only when
tan
θ
1
2
tan
θ
2
2
= −e
i(φ
2
−φ
1
)
(22)
Now by equating the real and imaginary parts of the above expression we get,
tan
θ
1
2
tan
θ
2
2
= cos(φ
2
− φ
1
) (23)
sin(φ
2
− φ
1
) = 0 (24)
8
On simplifying equation (19) we get, (φ
2
− φ
1
) = nπ ,where n = 0, ±1, ±2, ...... Using
(φ
2
− φ
1
) = nπ equation (17) reduces to the form tan
θ
1
2
tan
θ
2
2
= (−1)
n+1
.
Now we consider two cases:
Case1. When n is even, tan
θ
1
2
tan
θ
2
2
= −1 ⇒ (θ
1
− θ
2
) = (2m + 1)π.
Case2. When n is odd, tan
θ
1
2
tan
θ
2
2
= 1 ⇒ (θ
1
+ θ
2
) = (2m + 1)π. (where, m =
0, ±1, ±2, .....) Thus we see that the unitarity of the transformation is preserved only
when φ
2
= φ
1
+ nπ and θ
1
± θ
2
= (2m + 1)π.
6 Reference
[1] W.K.Wootters and W.H.Zurek,Nature 299 (1982) 802-803.
[2] A.K.Pati and S.L.Braunstein Nature 404 (2000) 164.
[3] A.K.Pati and S.L.Braunstein, Quantum mechanical universal constructor,quantph/0303124
(2003).
[4] A.K.Pati, Phys. Rev. A 66, 062319(2002).
[5] A.K.Pati and Barry C. Sanders, No partial erasure of quant um information,quant-
ph/0503138
[6] D.Zhou.et.al, Quantum information cannot be split into complementary parts,quant-
ph/0503168
[7] A.K.Pati and S.L.Braunstein,Phys.Lett.A 315,208-212 (2003)
[8] I.Chattopadhyay.et.al.Phys. Lett. A, 351, 384-387 (2006).
[9] I.Chakrabarty,A.K.Pati,S.Adhikari,B.S.Choudhury, General Impossible Operations as
a consequence of principles of quantum info r matio n (under preparation).
[10] A.Chefles and S.M. Barnett , Phys. Rev. A 60,136 (1999).
9
