Supplementary materials: W-state Analyzer and Multi-party
Measurement-device-independent Quantum Key Distribution
Changhua Zhu1 2*, Feihu Xu3, & Changxing Pei1
1State
Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, Shaanxi 710071, China
2Department
3Research
of Electrical & Computer Engineering, University of Toronto, Toronto, Ontario M5S 3G4, Canada
Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
*Corresponding author: chhzhu@xidian.edu.cn
I. 16 W4 states
The 16 W4 states are given as follows:
W4, 0  1 2  0001  0010  0100  1000

(S1)
W4, 1  1 2  0001  0010  0100  1000

(S2)
W4, 2  1 2  0001  0010  0100  1000

(S3)
W4, 3  1 2  0001  0010  0100  1000

(S4)
W4, 4  1 2  0000  1100  1010  1001 
(S5)
W4, 5  1 2  0000  1100  1010  1001

(S6)
W4, 6  1 2  0000  1100  1010  1001

(S7)
W4, 7  1 2  0000  1100  1010  1001 
(S8)
W4, 8  1 2  0011  0101  0110  1111 
(S9)
W4, 9  1 2  0011  0101  0110  1111 
(S10)
W4, a  1 2  0011  0101  0110  1111 
(S11)
W4, b  1 2  0011  0101  0110  1111 
(S12)
W4, c  1 2  0111  1011  1101  1110
(S13)

1
W4, d  1 2  0111  1011  1101  1110

(S14)
W4, e  1 2  0111  1011  1101  1110

(S15)
W4, f  1 2  0111  1011  1101  1110

(S16)
The state W4, 0
given by equation (S1) is a standard W4 state and belongs to one family states
The states W4, 1 , W4, 2
defined in Ref. 1.
and W4, 3
can be constructed by performing unitary
operations IZZI , ZIZI and ZZII on state W4, 0 , respectively, where I is the unit operator and Z is
the Pauli Z operator. The states W4, c , W4, d , W4, e and W4, f
XXXX operations on the states W4, 0 , W4, 1 , W4, 2
X operator. The state W4, 4
0
1

0

0
0

0
0

0
U 
0
0

0
0

0
0

0

0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
and W4, 3 , respectively, where X is the Pauli
can be obtained by performing unitary operation U
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
on W4, 0 . The states W4, 5 , W4, 6 and W4, 7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0 
0

0
0

0
0

0

0
0

0
0

0
0

0

1
can be obtained by performing ZIIZ , ZIZI and ZZII
operations on state W4, 4 , respectively. The state W4, 8
operation on state W4, 4 . The state W4, 9
The states W4, a
can be constructed by performing
and W4, b
can be obtained by performing XXXX
can be obtained by performing IIZZ operation on state W4, 8 .
can be obtained by performing  XXXX on states W4, 6
and W4, 5 ,
respectively. All these W4 states form a group of orthogonal bases in the 16-dimensional Hilbert space.
Any 4-qubit state can be expressed as a linear combination of these 16 W4 states.
2
II. Details of W-state preparation based on entanglement swapping
Let the 4 Bell states prepared by Alice, Bob, Charlie and David are  

BB
 1
2  00
BB 
 11
BB 

, 
CC 
1
2  00
 11 CC   and
CC 

AA
DD
2  00
1
2  00
1
AA
DD
 11
AA
,
 11 DD  ,
respectively. The state of the system is
 S  1 4  00
AA
 1 4  0000
 0011
 0110
 1001
 1100
 1111
 11
ABCD
ABCD
ABCD
ABCD
ABCD
ABCD
AA
 00
0000
0011
1100
1111
AB C D 
AB C D 
AB C D 
BB 
 0001
  00
ABCD
 1010
 1101
ABCD

0001
1010
1101
AB C D 
AB C D 
0111
ABCD
ABCD
 11 CC    00
CC 
0100
ABCD
 0111
AB C D 
AB C D 
 11
 0100
AB C D 
0110
1001
BB 
AB C D 
AB C D 
AB C D 
DD 
 0010
 0101
 1000
 1011
 1110
 11
ABCD
ABCD
ABCD
ABCD
ABCD
DD 

0010
0101
1000
1011
1110
AB C D 

AB C D 
AB C D 
AB C D 
AB C D 


(S17)


The 16 basis states can be represented by 16 W4 states as follows
0001
0010
0100
1000

ABC D


ABC D


ABC D


ABC D
 1 2 W4,0  W4,1  W4,2  W4,3
ABC D
 1 2 W4,0  W4,1  W4,2  W4,3
ABC D
 1 2 W4,0  W4,1  W4,2  W4,3
ABC D
 1 2 W4,0  W4,1  W4,2  W4,3
ABC D
 1 2 W4,4  W4,5  W4,6  W4,7
ABC D
 1 2 W4,4  W4,5  W4,6  W4,7
ABC D
 1 2 W4,4  W4,5  W4,6  W4,7
ABC D
 1 2 W4,4  W4,5  W4,6  W4,7
0000
1100
1010
1001
0011
0101
0110
1111
0111

ABC D




ABC D


ABC D


ABC D


ABC D


ABC D


ABC D


ABC D


ABC D
ABC D
 1 2 W4,8  W4,9  W4, a  W4,b
ABC D
 1 2 W4,8  W4,9  W4, a  W4,b
ABC D
 1 2 W4,8  W4,9  W4, a  W4,b
ABC D
 1 2 W4,8  W4,9  W4, a  W4,b
ABC D
 1 2 W4,c  W4, d  W4,e  W4, f
ABC D
(S18)
(S19)
(S20)
(S21)
(S22)
(S23)
(S24)
(S25)
(S26)
(S27)
(S28)
(S29)
(S30)
3
1011
1101
1110


ABC D


ABC D
ABC D
 1 2 W4,c  W4, d  W4,e  W4, f
ABC D
 1 2 W4,c  W4, d  W4,e  W4, f
ABC D
 1 2 W4,c  W4, d  W4,e  W4, f


(S31)
(S32)
(S33)
ABC D
Then,

 S  1 4 W4,0
ABCD
W4,0
AB C D 
 W4,1
ABCD
W4,1
AB C D 
 W4,2
ABCD
W4,2
AB C D 
W4,3
ABCD
W4,3
AB C D 
 W4,4
ABCD
W4,4
AB C D 
 W4,5
ABCD
W4,5
AB C D 
W4,6
ABCD
W4,6
AB C D 
 W4,7
ABCD
W4,7
AB C D 
 W4,8
ABCD
W4,8
AB C D 
W4,9
ABCD
W4,9
AB C D 
 W4, a
ABCD
W4, a
AB C D 
 W4,b
ABCD
W4,b
AB C D 
W4, c
ABCD
W4, c
AB C D 
 W4, d
ABCD
W4, d
AB C D 
 W4, e
ABCD
W4, e
AB C D 
W4, f
ABCD
W4, f
AB C D 




(S34)


Therefore, from equation (S34) we obtain that when the state of particles A, B, C and D  are
projected into one of the W4 states, the state of particles A, B, C and D will be in the corresponding
W4 state.
III. Details of estimation of Q1 and e1
We compute the gains at different cases, respectively.
(1) Case 1: outputs deriving from four background counts
In this case, there is no successful photon transmission (with probability 1 a 1 b 1 c 1 d  ,
where  i is the transmittance of participant i , i  a, b, c, d ). Emma will give an output when there exist
dark counts in four specific modes according to Table 1 (with probability Y04 ) and there are no dark counts
in other 12 modes (with probability 1  Y0 
12
W4,c
( W4,1
and
W4,d
) . There are 12 (4) detection modes for states W4,0
). Based on the post-selection scheme shown in Table 2, input
states 0000,0001,0010 and 0011 (corresponding to the states W4,0
1111 (corresponding to
and
W4,c
and
W4,d
and W4,1 ), 1100,1101,1110 and
) can be used to generate key.
Each input state
( 0000,0001,...,1110 or 1111 ) is prepared with equal probability, i.e. each one with probability 1 16 . So
4
the gain at this case is

12
12
1 16 4 1  a 1  b 1  c 1  d  12Y04 1  Y0  +4Y04 1  Y0   +



12
12
4 1  a 1  b 1  c 1   d  12Y04 1  Y0  +4Y04 1  Y0  


(S35)
 8 1  a 1  b 1  c 1  d  Y04 1  Y0 
12
While, qubit error will occur when input states are 0000,0011,1100 and 1111 with probability
4 1  a 1  b 1  c 1   d  Y04 1  Y0  .
12
(S36)
(2) Case 2: outputs deriving from one photon count and three background counts
In this case, one count derives from a successful photon transmission and the other three counts derive
from background. For an input photon, it can arrive at the SPD in different spatial and temporal modes, e.g.
the state aa†,t 0
0
of Alice’s photon can be mapped into the state


aa†,t     as†,t  ei as†,t  iei au†,t  iei  au†,t  iav†,t

iei av†,t  ei aw† ,t  ei  aw† ,t 
and the state aa†,t 0
1

,
(S37)
of Alice’s photon can be mapped into the state

aa†,t     as†,t  ei as†,t  iei au†,t  iei  au†,t  iav†,t

iei av†,t  ei aw† ,t  ei  aw† ,t 
.
(S38)
When Alice’s photon, e.g. in state aa†,t 0 , is detected and the input state is 0000 , the output
0
probability of Emma’s device according to Table 1 is

a 1  b 1  c 1  d  1 4
2

2
 22 10  22  3  22 10  22  3 
22 10  22  3  22 10  22  3 Y03 1  Y0 
12
(S39)
 208 32a 1  b 1  c 1  d  Y03 1  Y0 
12
In equation (S39), 10 denotes the number of operators as†,t , au†,t , av†,t aw† ,t in the detection modes of the
0
W states W4,0
1
0
1
and W4,1 ; 3 denotes the number of operators as†,t , au†,t , av†,t , aw† ,t in the detection modes
of the W states W4,0
1
2
1
2
and W4,1 . Similarly, we can obtain the output probability of other participants’
photons’ counts under different inputs state. So, the gain in this case is
5

1 16 8  208 32 a 1  b 1  c 1  d   b 1  a 1  c 1  d   
 4  208 32  4  96 32  c 1  a 1  b 1  d   d 1  a 1  b 1  c  Y03 1  Y0 
12
(S40)

1 16 52 a 1  b 1  c 1  d   b 1  a 1  c 1  d   

38 c 1  a 1  b 1  d   d 1  a 1  b 1  c   Y03 1  Y0 
12
In this case the error will occur with probability

1 16 26 a 1  b 1  c 1  d   b 1  a 1  c 1  d  

(S41)
19 c 1  a 1  b 1  d   d 1  a 1  b 1  c  Y03 1  Y0 
12
(3) Case 3: outputs deriving from two photon counts and two background counts
In this case, two counts derive from successful photon transmission and two counts derive from
background. Any state of two input photons can evolve into a state superposed by different spatial and time
modes, e.g. state aa†,t ab†,t 0
0
0
of Alice’s and Bob’s photons evolves into the state

aa†,t ab†,t            i ei  au†,t




 ei  au†,t aw† ,t  i ei  aw† ,t



 
a 
ei  as†,t au†,t  iei  as†,t aw† ,t  iei  au†,t av†,t  ei  av†,t aw† ,t  i ei  as†,t

ei  as†,t av†,t  i ei  au†,t


 
 ei  au†,t aw† ,t  i ei  av†,t

 i  e i 

ei as†,t au†,t  iei as†,t aw† ,t  iei au†,t av†,t  ei av†,t aw† ,t  i  as†,t

as†,t av†,t  i  av†,t


†
w, t




(S42)

  

From Table 1, we can obtain that there are 6, 6, 4, 4, 6 and 6 detection modes for states
as†,t0 au†,t1 , as†,t0 av†,t0 , as†,t0 aw† ,t1 , au†,t1 av†,t0 , au†,t1 aw† ,t1 and av†,t0 aw† ,t1 , respectively, corresponding to the states
W4,0
and
W4,1 . Based on equation (S42) the output probability is
2
12
1 32  ab 1  c 1  d   64  6  6  4  4  6  6  Y02 1  Y0 
12
 2ab 1  c 1   d  Y02 1  Y0 
(S43)
Similarly, we can obtain the output probability of other participants’ photons counts under different
inputs state. So, the gain in this case is
1 16 16ab 1  c 1   d   10  ac 1  b 1   d   b d 1   a 1  c   
9 ad 1  b 1  c   bc 1   a 1   d    8c d 1   a 1  b Y02 1  Y0 
12
(S44)
In this case the error will occur with probability
6
1 32 16ab 1  c 1   d   10  ac 1  b 1   d   b d 1   a 1  c   
(S45)
9 ad 1  b 1  c   bc 1   a 1   d    8c d 1   a 1  b Y02 1  Y0 
12
(4) Case 4: outputs deriving from three photon counts and one background count
In this case, three counts derive from successful photon transmission and one count derives from
background. Any state of three input photons can evolve into a state superposed by different spatial and
time modes, e.g. the state aa†,t ab†,t ac†,t 0 of Alice’s, Bob’s and Charlie’s photons evolves into the state
0
0
0
(partial terms are omitted for simplification)



aa†,t ab†,t ac†,t         ei  au†,t


iei  aw† ,t



...  iei av†,t



aw† ,t


 iei  au†,t
  e a a
 i  a     a  a
 iei  as†,t au†,t


†
s , t
i 
†
s ,t

†
s , t
†
u ,t
†
v , t



aw† ,t  ei  au†,t aw† ,t



 
    a   
aw† ,t  i ei  as†,t aw† ,t
 ias†,t av†,t


†
v , t

(S46)

In equation (S46), the states which may lead to successful outputs in Emma’s device include:
as†,t0 au†,t1 av†,t0 , as†,t0 au†,t1 av†,t1 , as†,t0 au†,t1 aw† ,t1 , as†,t0 au†,t1 aw† ,t2 , as†,t0 au†,t2 av†,t0 , as†,t1 au†,t1 aw† ,t1 , as†,t1 av†,t0 aw† ,t1 , au†,t1 av†,t1 aw† ,t1 ,
au†,t2 av†,t0 aw† ,t1 ,
as†,t0 av†,t0 aw† ,t1 ,
as†,t0 av†,t0 aw† ,t2 ,
au†,t1 av†,t0 aw† ,t1 ,
as†,t0 au†,t2 aw† ,t1 ,
as†,t0 av†,t1 aw† ,t1 ,
as†,t1 au†,t1 av†,t0
and


au†,t1 av†,t0 aw† ,t 2 . Their coefficients are all 16 23  23  23 . In the former 12 terms, each one corresponds to 2
detection modes. In the later 4 terms, each one corresponds to 1 detection mode according to Table 1.
When the input state is aa†,t ab†,t ac†,t 0 the output probability of Emma’s device is
0
0
0
16 2  2  2   12  2  4   1  Y 1  Y 
3
3
3
2
12
a b c
d
0
0
(S47)
Similarly, we can obtain the output probabilities of other participants’ photons counts under different
inputs state. So, the gain in this case is

1 256 34 abc 1  d   abd 1  c  

15 acd 1  b   bcd 1  a   Y0 1  Y0 
12
(S48)
In this case the error will occur with probability

1 256 17 abc 1  d   abd 1 c  

7 acd 1  b   bcd 1 a  Y0 1  Y0 
12
(S49)
7
(5) Case 5: outputs deriving from four photon counts
In this case, there exist successful outputs when the input state is one of
1110 . State
0001
can be projected into states
identification probabilities of W4,0
and W4,1
W4,0
or
0010 ，0001 ，1101 or
with probability 1 4 . Let the
W4,1
be D p 0 and D p1 , respectively. When input state is
0001 the output probability is
1 4abcd  Dp 0  Dp1  1  Y0 
12
(S50)
So, the output probability in the case is
1 16  4 1 4abcd  Dp 0  Dp1  1  Y0 
12
(S51)
 1 16abcd  Dp 0  Dp1  1  Y0 
12
There is no error in this case.
By adding all output probabilities in five cases together, the gain is
Q1 =8 1   a 1  b 1  c 1   d  Y04 1  Y0  
12

1 16 52  a 1  b 1  c 1   d   b 1   a 1  c 1   d   

38 c 1   a 1  b 1   d    d 1   a 1  b 1  c   Y03 1  Y0  
12
1 16 16 ab 1  c 1   d   10  ac 1  b 1   d   b d 1   a 1   c   
9  a d 1  b 1  c   bc 1   a 1   d    8 c d 1   a 1  b Y02 1  Y0  
12
(S52)

1 256 34  abc 1   d    ab d 1   c   

15  ac d 1  b   bc d 1   a   Y0 1  Y0  
12
1 16 abc d  D p 0  D p1  1  Y0 
12
, and the QBER is
e1  4 1  a 1  b 1  c 1  d  Y04 1  Y0  
12

1 16 26  a 1  b 1  c 1   d   b 1   a 1  c 1   d   

19 c 1   a 1  b 1   d    d 1   a 1  b 1  c   Y03 1  Y0  
12
1 32 16 ab 1  c 1   d   10  ac 1  b 1   d   b d 1   a 1  c   
.
(S53)
9  a d 1  b 1  c   bc 1   a 1   d    8c d 1   a 1  b Y02 1  Y0  
12

1 256 17  abc 1   d    ab d 1  c   

7  ac d 1  b   bc d 1   a   Y0 1  Y0 
12
We assume that four optical fiber links and four detectors are identical, i.e. a  c  d  b   . So,
8
equations (S52) and (S53) can be simplified as
12
4
3
2
Q1  1  Y0  1024 1    Y04  1440 1    Y03  496 2 1    Y02 

49 3 1    Y0  8  D p 0  D p1  4  128
e1 
1
12
4
3
2
1  Y0  64 1    Y04  90 1    Y03  31 2 1    Y02 
16Q1
3 1    Y0 
(S54)
(S55)
3
1. Verstraete, F., Dehaene, J., De Moor, B. & Verschelde, H. Four qubits can be entangled in nine different
ways. Phys. Rev. A 65, 052112 (2002).
9