Multi-Receiver Quantum Dense Coding with Non-Symmetric Quantum Channel

0.1 MB·English

by  Chang-Bao Fu

Multi-Receiver Quantum Dense Coding with Non-Symmetric
Quantum Channel
∗
Chang-Bao FU and Shou ZHANG
Department of Physics, College of Science,
Yanbian University, Yanji, Jilin 133002, PR China
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0 Ke LI
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2 Purchasing Office, Yanbian University, Yanji, Jilin 133002, PR China

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J Kyu-Hwang YEON

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1 Department of Physics, Institute for Basic Science Research,

1 College of Natural Science, Chungbuk National University, Cheonju, Chungbuk 361-763
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2 Chung-In UM
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Department of Physics, College of Science, Korea University, Seoul 136-701
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A two-receiver quantum dense coding scheme and an N-receiver quantum dense
/
h
p coding scheme, in the case of non-symmetric Hilbert spaces of the particles of the
-
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n quantum channel, are investigated in this paper. A sender can send his messages
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u to many receivers simultaneously. The scheme can be applied to quantum secret
q
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v sharing and controlled quantum dense coding.
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r PACS numbers: 03.67.Hk, 03.65.Ud, 03.67.-a
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Keywords: Non-symmetric Hilbert space, Quantum dense coding, Unitary transformation,
Projective measurement
The quantum entanglement among the quantum systems can be used to perform many
tasks, such as quantum cryptography [1, 2], quantum secret sharing [3, 4, 5, 6] and so
on. Quantum dense coding is also one of the applications of entanglement in quantum
communication. Since Bennet and Wiesner [7] first proposed the quantum dense coding
(QDC) scheme, different QDC schemes have been presented. For example, Lee et al. [8]
∗
E-mail: [email protected]
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have studied QDC scheme among multiparties. Bose et al. [9] have studied QDC scheme
with distributed multiparticle entanglement. Zhang et al. [10] have studied QDC scheme for
continuous variable. Hao et al. [11] and Fu et al. [12] have proposed a controlled quantum
dense coding scheme by using a three-particle state and a four-particle state, respectively.
Liu et al. [13] have presented a QDC protocol with a multi-level entangled state. Recently,
Bruß et al. [14] have presented a two-receiver quantum dense coding protocol in a four-
particle Greenberger-Horne-Zeilinger (GHZ) state. However, in these schemes, the quantum
channels are symmetric, that is to say, the dimension of the Hilbert space of the particle
with sender is the same as that of the particle with receiver.
On the other hand, Yan et al. [15] have given a QDC scheme by using bipartite entangled
state. Fan et al. [16, 17] have given two QDC schemes by using the direct product state.
Fu et al. [18] have presented a QDC scheme by using multipartite entangled state. In these
schemes, the quantum channels are non-symmetric, but the messages from the sender can
only be obtained by one receiver.
In this paper, we study a two-receiver quantum dense coding and an N-receiver quantum
dense coding in the case of non-symmetric Hilbert spaces of the particles of the quantum
channel.
SupposeAlice, Charlie, andBobinitiallyshareamaximallyentangledstatein3 3 2 2-
× × ×
dimensional Hilbert space in the following form:
1
µ = ( 0000 + 1111 ) , (1)
1 1234
| i √2 | i | i
where particles (1, 2) belong to Alice in 32-dimensional Hilbert space, particle 3 belongs
to Charlie in 2-dimensional Hilbert space, and particle 4 belongs to Bob in 2-dimensional
Hilbert space.
Firstly, Alice encodes one of her messages on qutrit 1 by performing one of the six unitary
3
transformations as follows:
1 0 0 1 0 0 0 0 1
     
U00 = 0 1 0 ,U01 = 0 1 0 ,U10 = 1 0 0 ,
−
     
 0 0 1  0 0 1   0 1 0 
     
     
     
(2)
0 0 1 0 1 0 0 1 0
−
     
U11 = 1 0 0 ,U20 = 0 0 1 ,U21 = 0 0 1 .
     
0 1 0  1 0 0   1 0 0 
     
 −     
     
Secondly, Alice encodes the rest of her messages on qutrit 2 by performing one of the
three unitary transformations U ,U ,U . Then, the collective unitary transformations
00 10 20
{ }
on qutrit 1 and qutrit 2 can be written as
U+ = U U , U− = U U , U+ = U U ,
1 00 00 1 01 00 2 10 00
⊗ ⊗ ⊗
U− = U U , U+ = U U , U− = U U ,
2 11 00 3 20 00 3 21 00
⊗ ⊗ ⊗
U+ = U U , U− = U U , U+ = U U ,
4 00 10 4 01 10 5 10 10
⊗ ⊗ ⊗
(3)
U− = U U , U+ = U U , U− = U U ,
5 11 10 6 20 10 6 21 10
⊗ ⊗ ⊗
U+ = U U , U− = U U , U+ = U U ,
7 00 20 7 01 20 8 10 20
⊗ ⊗ ⊗
U− = U U , U+ = U U , U− = U U .
8 11 20 9 20 20 9 21 20
⊗ ⊗ ⊗
The above collective unitary transformations on qutrit 1 and qutrit 2 will transform the
state in Eq. (1) into the corresponding state, respectively
1
± ±
U µ = ( 0000 1111 ) = µ , (4)
1 1 1234 1
| i √2 | i±| i | i
1
± ±
U µ = ( 1000 2111 ) = µ , (5)
2 1 1234 2
| i √2 | i±| i | i
1
± ±
U µ = ( 2000 0111 ) = µ , (6)
3 1 1234 3
| i √2 | i±| i | i
1
± ±
U µ = ( 0100 1211 ) = µ , (7)
4 1 1234 4
| i √2 | i±| i | i
4
1
± ±
U µ = ( 1100 2211 ) = µ , (8)
5 1 1234 5
| i √2 | i±| i | i
1
± ±
U µ = ( 2100 0211 ) = µ , (9)
6 1 1234 6
| i √2 | i±| i | i
1
± ±
U µ = ( 0200 1011 ) = µ , (10)
7 1 1234 7
| i √2 | i±| i | i
1
± ±
U µ = ( 1200 2011 ) = µ , (11)
8 1 1234 8
| i √2 | i±| i | i
1
± ±
U µ = ( 2200 0011 ) = µ . (12)
9 1 1234 9
| i √2 | i±| i | i
In the above equations, the notes correspond to the superscripts for the collective unitary
±
transformation and the state composed of particles (1, 2, 3, 4). These 18 states above
are orthonormal: hµji|µji′′i = δii′δjj′, where |µjii (or |µji′′i) is one of the above 18 states in
Eqs. (4) (12); i,i′ [0, 9]; j,j′ +, .
− ∈ ∈ { −}
Thirdly, after performing one of these 18 collective unitary transformations in Eq. (3),
Alice sends her particle 1 to Charlie and sends her particle 2 to Bob. Then, Charlie and Bob
share these 18 states in Eqs. (4) (12) with equal probabilities. In order to decode, Charlie
−
makes a measurement with the projectors P = 00 00 + 11 11 , P = 10 10 + 21 21 ,
1 2
| ih | | ih | | ih | | ih |
P = 20 20 + 01 01 , and communicates the measurement result to Bob. If P (P , P )
3 1 2 3
| ih | | ih |
± ± ±
clicks, they know that the state they share is among the three groups µ , µ , µ
1 4 7
{| i | i | i}
± ± ± ± ± ±
( µ , µ , µ , µ , µ , µ ). Now Bob performs a measurement with the same
2 5 8 3 6 9
{| i | i | i} {| i | i | i}
projectors P , P , P , and communicates the measurement result to Charlie. Depending on
1 2 3
the outcomes of the projective measurements, they know that the state they share is among
± ± ± ± ± ± ± ± ±
which of the three groups µ , µ , µ ( µ , µ , µ , µ , µ , µ ).
1 4 7 2 5 8 3 6 9
{| i | i | i} {| i | i | i} {| i | i | i}
Note that none of the above projective measurements disturbs the shared state. Lastly,
depending on the outcomes of the previous projective measurements, Charlie performs a
measurement under the basis ( 00 11 ) , ( ( 10 21 ) , ( 20 01 ) ); Bob
13 13 13
{ | i ± | i } { | i ± | i } { | i ± | i }
performs a measurement under the basis ( 00 11 ) , or ( 10 21 ) , or ( 20
24 24
{ | i±| i } { | i±| i } { | i±
01 ) . Then, they communicate their measurement results with each other, which will
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| i }
help them obtain the messages from Alice. Because Charlie and bob communicate their
measurement results with each other, the amount of classical information transmitted from
Alice to Charlie and Bob is less than log 18 bits. If the maximally entangled state in
2
Eq. (1) is shared between Alice and Charlie (Bob), where particles (1, 2) belong to Alice
in 32-dimensional Hilbert space, particles (3, 4) belong to Charlie (Bob) in 22-dimensional
5
qu(cid:13)b(cid:13)ti(cid:13)(cid:13)(cid:13) N(cid:13)+1(cid:13) (cid:13) PM(cid:13) (cid:13) CM(cid:13) (cid:13)
qu(cid:13)tr(cid:13)ti(cid:13) 1(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K(cid:13)
qu(cid:13)b(cid:13)ti(cid:13)(cid:13)(cid:13) N+(cid:13) (cid:13)2(cid:13) PM(cid:13) (cid:13) CM(cid:13) (cid:13)
qu(cid:13)tr(cid:13)ti(cid:13) 2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L(cid:13)
qu(cid:13)b(cid:13)ti(cid:13)(cid:13)(cid:13) N+(cid:13) (cid:13)3(cid:13) PM(cid:13) (cid:13) CM(cid:13) (cid:13)
qu(cid:13)tr(cid:13)ti(cid:13) 3(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L(cid:13)
.(cid:13) .(cid:13) .(cid:13) .(cid:13)
.(cid:13) .(cid:13) .(cid:13) .(cid:13)
.(cid:13) .(cid:13) .(cid:13) .(cid:13)
qu(cid:13)b(cid:13)ti (cid:13)2(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N(cid:13)
PM(cid:13) (cid:13) CM(cid:13) (cid:13)
qu(cid:13)tr(cid:13)ti(cid:13)(cid:13)(cid:13)(cid:13) N(cid:13) L(cid:13)
FIG.1: QuantumcircuitimplementingN-receiver quantumdensecoding. K U00,U01,U10,U11,
∈{
U20,U21 , L U00,U10,U20 ; PM P1,P2,P3 ; CM (00 11 ),(10 21 ),( 20 01 ) .
} ∈ { } ∈ { } ∈ { | i±| i | i±| i | i±| i }
Hilbert space, the amount of classical information transmitted from Alice to Charlie (Bob)
is equal to log 18 bits.
2
The above scheme can be directly generalized to the case that a number of receivers are
involved. The specific steps of the process are depicted in FIG. 1. Let us follow the state
through this circuit. We assume that (N +1)-parties initially share a maximally entangled
state in 3N 2N-dimensional Hilbert space in the following form:
×
1 2N 2N
′
µ = 0 + 1 , (13)
1 m m
| i √2 m=1| i m=1| i !
Y Y
where particles (1, 2, ..., N) belong to a sender in 3N-dimensional Hilbert space, and each
of the particles (N +1,N +2,...,2N) correspondingly belongs to each of the receivers (1, 2,
..., N) in 2-dimensional Hilbert space. Now we describe the scheme that a sender sends his
messages to the N-receivers by using the non-symmetric quantum channel in Eq. (13). The
scheme is composed of five steps.
(i) The sender encodes one of his messages on qutrit 1 by performing one of the six
unitary operations U ,U ,U ,U ,U ,U .
00 01 10 11 20 21
{ }
(ii) The sender encodes each of the rest of his messages on each of the rest of his qutrits
by performing one of the three unitary operations U ,U ,U .
00 10 20
{ }
(iii) After performing his unitary operations, the sender sends each of the particles (1, 2,
..., N) correspondingly to each of the receivers (1, 2, ..., N).
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(iv) In order to decode, the receiver 1, who receives the qutrit 1 from the sender, makes
a measurement with the projectors P = 00 00 + 11 11 , P = 10 10 + 21 21 ,
1 2
| ih | | ih | | ih | | ih |
P = 20 20 + 01 01 ; each of the receivers (2, 3, ..., N) performs a similar measurement
3
| ih | | ih |
with the same projectors P , P , P . Then, all the receivers communicate their projective
1 2 3
measurement results with one another.
(v) Finally, according to the outcomes of the previous projective measurements, the
receiver 1 performs a measurement under the basis ( 00 11 ) , ( ( 10 21 ) ,
1,1+N 1,1+N
{ | i±| i } { | i±| i }
( 20 01 ) ); eachofthereceivers (2,3,..., N)performsasimilar measurement under
1,1+N
{ | i±| i }
the basis ( 00 11 ) , or ( 10 21 ) , or ( 20 01 ) , where i [2,
i,i+N i,i+N i,i+N
{ | i± | i } { | i± | i } { | i± | i } ∈
N]. Then, all the receivers communicate their measurement results with one another, which
will help them obtain the messages from the sender.
In this way, the upper bound that the amount of classical information transmitted from
the sender to all the receivers by using the non-symmetric quantum channel in Eq. (13) can
be expressed as
C = log (3N 2) = 1+N log 3. (14)
2 2
×
It must be stressed that our scheme is constructive. (i) We have investigated a two-
receiver quantum dense coding and generalized it to an N-receiver quantum dense coding
in the case of non-symmetric Hilbert spaces of the particles of the quantum channel. (ii)
Compared with previous schemes, the sender can send his messages to more receivers at
the expense of some amount of classical information in our scheme. (iii) Comparing the
two-receiver quantum dense coding with the one-receiver quantum dense coding by the
quantum channel in Eq. (1), the amount of classical information transmitted in the one-
receiver quantumdensecodingistheupperboundinthetwo-receiver quantumdensecoding.
(iv) If and only if each of the receivers agrees to collaborate, all the receivers can obtain
the messages from the sender. So each of the receivers may be as a controller during the
quantum dense coding. Obviously, our scheme can be applied to quantum secret sharing
and controlled quantum dense coding.
In conclusion, we have investigated a two-receiver quantum dense coding and an N-
receiver quantum dense coding in the case of non-symmetric Hilbert spaces of the particles
of the quantum channel. The sender can send his messages to many receivers. The scheme
7
can be applied to quantum secret sharing and controlled quantum dense coding.
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