DSpace at VNU: Deterministic joint remote state preparation

Physics Letters A 375 (2011) 3570–3573

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Physics Letters A
www.elsevier.com/locate/pla

Deterministic joint remote state preparation
Nguyen Ba An a,∗ , Cao Thi Bich a,b , Nung Van Don a,c
a
b
c

Center for Theoretical Physics, Institute of Physics, 10 Dao Tan, Ba Dinh, Hanoi, Viet Nam
Physics Department, University of Education No. 1, 136 Xuan Thuy, Cau Giay, Hanoi, Viet Nam
Physics Department, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history:
Received 5 August 2011
Accepted 19 August 2011
Available online 25 August 2011
Communicated by V.M. Agranovich

a b s t r a c t
We put forward a new nontrivial three-step strategy to execute joint remote state preparation via
Einstein–Podolsky–Rosen pairs deterministically. At variance with all existing protocols, in ours the
receiver contributes actively in both preparation and reconstruction steps, although he knows nothing

about the quantum state to be prepared.
© 2011 Elsevier B.V. All rights reserved.

Keywords:
Joint remote state preparation
Einstein–Podolsky–Rosen pairs
Unit success probability

1. Introduction

Secure faithful transmission of quantum states encoding quantum information is a primarily important task in quantum information processing, which can now be achieved by dual usage
of local operation and classical communication without physically
transmitting the states themselves, thanks to previously shared
quantum resource (SQR) called entanglement [1]. The first intriguing protocol for such kind of tasks is quantum teleportation
(QT) [2] in which a sender Alice is able to teleport a unknown
quantum state to a space-like receiver Bob. In case full knowledge
of the state (i.e., the complete set of parameters identifying the
state) is known to Alice, the task can be done by a protocol named
remote state preparation (RSP) [3] using the same SQR as in QT but
with simpler local operation and reduced classical communication
cost (CCC). Obviously, however, in RSP all the secret information
encoded in the quantum state is leaked out to the sender Alice. An
idea to get rid of the leakage of information to the sender is that
one more sender enters the game and the state’s parameter set is
split, by some confidential rule, into two subsets, each of which is
given to a sender. By doing so it is impossible for neither sender
alone to infer the secret from his/her own parameter subset. Yet,
to faithfully transmit the full secret to a remote receiver the two
senders must cooperate correctly. Original protocols realizing such
idea were designed some years ago which were referred to as joint

*

Corresponding author.
E-mail address: (N.B. An).

0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2011.08.045

remote state preparation (JRSP) [4] (see also Refs. [5–8] for various extensions). As a rule, all JRSP protocols employ the SQR in
terms of entangled states of which Greenberger–Horne–Zeilinger
(GHZ) trios [9], W states [10] and Einstein–Podolsky–Rosen (EPR)
pairs [11] are typical.
Most JRSP protocols proposed so far succeed with a finite probability which, however, is less than one. That is, they are probabilistic [4–7]. Very recently, a strategy has been introduced in
Ref. [8] to improve JRSP so that the success probability is boosted
to one. The SQR in Ref. [8] are GHZ trios. Here we shall make
use only of EPR pairs as the SQR and unit success probability is
also achieved by adopting a strategy which is different from that
in Ref. [8]. Apart from easier production of the SQR, using EPR
pairs has an advantage in the process of distributing entanglement
among the participants before the execution of the JRSP protocol.
Namely, the presence of unavoidable malicious eavesdroppers necessitates a prior careful checking procedure that sacrifices a large
number of entangled states. Use of SQR in terms GHZ trios as in
Ref. [8] requires the checking to be carried out simultaneously between the three participants, while use of SQR in terms of EPR
pairs as in our protocols requires only pair-wise checking. This
not only simplifies the checking operation but also economizes the
overhead expenses.
In this Letter we put forward a new nontrivial strategy for the
three remote parties to deterministically prepare the most general
single- and two-qubit states without leakage of full information

encoded in the state to either of them, using only EPR pairs (not
GHZ trios) as the SQR. The case of single-qubit states is considered
in Section 2 and that of two-qubit ones in Section 3. Section 4 is
the conclusion.


N.B. An et al. / Physics Letters A 375 (2011) 3570–3573

2. Deterministic joint remote preparation of an arbitrary
single-qubit state

3571

Table 1
The reconstruction operator rln in Eq. (13) versus l and n. I is the identity operator,
σx the bit-flip operator and σz the phase-flip one. Note that rln can generally be
formulated as rln = σzl⊕n σxl .

Let us name the two senders Alice 1 and Alice 2, while the receiver remains Bob. The essence of a JRSP protocol can be grasped
as follows. The complete set S of parameters identifying the state
to be prepared is split into two subsets S 1 and S 2 . The subset S 1
is given only to Alice 1, while S 2 only to Alice 2, so neither Alice 1
nor Alice 2 is able to learn the complete set S. The prerequisite is
that somehow Alice 1, Alice 2 and Bob must a priori share some
kind of entanglement. Usually, the participants execute a two-step
protocol in the following manner [4–7]. In the first step both Alice j ( j = 1 and 2) independently carry out measurements in the
bases determined by S j and publicly broadcast their outcomes.
In the second step Bob performs proper operations depending on
the broadcasted outcomes to obtain the desired state. The success
probability of such customary two-step protocols is always less

than one [4–7].
To achieve unit success probability a strategy composed of three
steps has been chalked out in Ref. [8] which applies when GHZ
trios are served as the SQR. More concretely, in the two first steps
Alice 1 and Alice 2 carry out their measurements sequentially in
such a way that the measurement basis of Alice 2 is defined not
only by her parameter subset S 2 but also by the outcome of Alice 1. It is somewhat surprised to recognize that such a strategy
is not suitable when EPR pairs are used as the SQR. We shall
therefore propose a different strategy to cope with EPR pairs. Our
strategy also undergoes three steps but the content of each step
is more delicate. In the first step Alice 1 and Bob independently
carry out their measurements. The measurement basis of Alice 1 is
determined by S 1 , whereas that of Bob is the computational one.
As for Alice 2, she starts her job only in the second step making
use of S 2 as well as of the outcomes broadcasted by Alice 1 and
Bob in the first step. The target state is then obtained in the third
step by Bob’s actions which are sensitive to the outcomes of both
senders.
In this section we detail our new strategy for JRSP of an arbitrary single-qubit state whose most general form reads

k=

|φ = a|0 + be i ϕ |1 ,

The unitary transformations v (k) in Eq. (9) depends also on S 2 = ϕ :

(1)

with a, b, ϕ ∈ R and a + b = 1. The three parties share two EPR
pairs,

2

|Q

A1 B 1 A2 B 2

where |q

= |q

XY

=

2

A 1 B 1 |q A 2 B 2 ,

√1

2

(2)

(|00 + |11 ) X Y . Each Alice holds a qubit (Al-

ice 1 holds qubit A 1 , Alice 2 qubit A 2 ), while Bob holds two qubits
B 1 and B 2 . Clearly, the complete parameter set identifying |φ is
S = {a, b, ϕ }. There are many ways to split S into two subsets S 1
and S 2 . We find out that to achieve unit success probability the

proper splitting is S 1 = {a, b} and S 2 = ϕ .
In the first step, Alice 1 and Bob independently perform the
following actions. Alice 1 measures her qubit A 1 in the basis
{|ψ0 A 1 , |ψ1 A 1 } determined just by S 1 as

|ψ0
|ψ1

A1
A1

=

a

b

−b a

|0
|1

A1
A1

,

(3)

whereas Bob first applies CNOT B 1 B 2 1 on his two qubits B 1 and B 2

and then measures qubit B 2 in its computational basis {|0 B 2 , |1 B 2 }.
After those actions the state | Q A 1 B 1 A 2 B 2 of the SQR, Eq. (2), becomes | Q A 1 B 1 A 2 B 2 , which can be rewritten as
1
CNOT X Y denotes a controlled-NOT gate acting on two qubits X (control qubit)
and Y (target qubit) as CNOT X Y |i X | j Y = |i X |i ⊕ j Y , where i , j ∈ {0, 1} and ⊕
stands for an addition mod 2.

l
n
rln

0
0
I

Q

A1 B 1 A2 B 2

=

2

1
0

1
1

σz

σ z σx

σx

1

1

1

0
1

|ψl

A 1 |m B 2 |λlm A 2 B 1 ,

(4)

m =0 l =0

where

|λ00

A2 B 1

= a|00 + b|11

A2 B 1

|λ01

A2 B 1

= a|10 + b|01

A2 B 1

|λ10

A2 B 1

= a|11 − b|00

A2 B 1

|λ11

A2 B 1

= a|01 − b|10

A2 B 1

,

(5)

,

(6)

,

(7)

,

(8)

with ⊕ an addition mod 2. As is easily verified from Eq. (4), when
Alice 1 finds qubit A 1 in state |ψl A 1 and Bob finds qubit B 2 in
state |m B 2 , the unmeasured qubits A 2 and B 1 are automatically
projected onto an entangled state |λlm A 2 B 1 , with a probability
plm = 1/4 for any pair of possible l and m. Afterwards, Alice 1
and Bob announce their outcomes by virtue of the public classical
media.
In the second step, Alice 2 starts measuring her qubit A 2 in
a basis that she needs to correctly determine using the parameter subset S 2 = ϕ available to her as well as the outcomes (l, m)
announced by Alice 1 and Bob. There are two choices for the measurement basis labeled by k = 0, 1 as
(k)

|θ0
|θ1(k)

A2

= v (k)

A2

|0
|1

A2
A2

(9)

,

with k = l ⊕ m, i.e.,

0 if l = m = 0 or l = m = 1,
1 if l = 0, m = 1 or l = 1, m = 0.

(10)

1
1 e −i ϕ
v ( 0) = √
−i ϕ ,
2 1 −e
1
e −i ϕ
1
v (1 ) = √
−i ϕ 1 .

2 −e

(11)
(12)

To specify the value of k in states |λlm
(k)

perindex “(k)” to them as |λlm
(k)

{|θn

A2 ; n

A2 B 1

A2 B 1 ,

we assign a su-

and express them through

= 0, 1} in the form
1

1

(k)

λlm

A2 B 1

(k)

=√

2 n =0

θn

r + |φ

A 2 ln

(13)

B1 ,

with rln an operator which is in fact independent of k as listed in
Table 1. Transparently, with an equal probability pn = 1/2, Bob will
(k)

obtain a state |θn A 2 with n = 0, 1 or 2, which he needs also to
announce publicly.
In the third step, depending on the outcomes l and n, Bob’s job
is simply to apply rln on his qubit B 1 to obtain the desired state
|φ B 1 . Since Bob is always able to reconstruct |φ B 1 from |φ B 1 =
+

rln
|φ B 1 , our JRSP protocol is deterministic. Mathematically, its total
success probability p T is
1

1

1

pT =

plm pn = 8 ×
n =0 m =0 l =0

1
4

×

1
2

= 1.

(14)


3572

N.B. An et al. / Physics Letters A 375 (2011) 3570–3573

⎛

In passing we notice the following relationship
(1 )
σx ⊗ I λlm

( 0)

A2 B 1

= λl,m⊕1

A2 B 1

(15)

.
(0)

This allows Alice 2 to use the same basis {|θn A 2 } to measure
qubit A 2 for both l ⊕ m = 0 and l ⊕ m = 1, provided that when
l ⊕ m = 1 she applies σx on A 2 before measuring it.
3. Deterministic joint remote preparation of an arbitrary
two-qubit state
Several authors [6] dealt with JRSP of particular two-qubit
states of the form α |00 + β|11 , with α , β ∈ C and |α |2 + |β|2 = 1.
However, such tasks can be reduced to JRSP of single-qubit states
α |0 + β|1 with an addition application of a CNOT on the collapsed qubit and an ancillary qubit in state |0 at the end by Bob.
Other authors [7] dealt with JRSP of general two-qubit states of the

form
3

a j eiϕ j | j ,

|Φ =

(16)

j =0
3

2
where {a j , ϕ j } ∈ R,
j =0 a j = 1, while |0 ≡ |00 , |1 ≡ |01 , |2 ≡
|10 and |3 ≡ |11 are shorthands for the four two-qubit orthonormal states in their computational basis, but the success probability
in Ref. [7] remains less than 1. In this section we are concerned
with the same most general two-qubit state as defined in Eq. (16),
but we propose JRSP protocols that succeed with unit probability. This really matters because success probability is of paramount
importance for a quantum protocol.
The SQR we exploit consists of four EPR pairs,

2

|Q

A1 B 1 A1 B 1 A2 B 2 A2 B 2

with |q

XY

=

√1

2

=

|q

A j B j |q

j =1

AjBj

(17)

,

(|0 + |3 ) X Y in the abbreviated notations. Qubits

A 1 and A 1 ( A 2 and A 2 ) belong to Alice 1 (Alice 2), while qubits
B 1 , B 2 , B 1 and B 2 to Bob. To perform JRSP of the state (16) deterministically we split its complete parameter set S = {a j , ϕ j } into
S 1 = {a j } and S 2 = {ϕ j }, where without any loss of generality we
set ϕ0 = 0.
Firstly, Alice 1 and Bob independently act as follows. Alice 1
measures qubits A 1 and A 1 in the basis defined by S 1 as

⎛ |Ψ

⎞

⎛

a0
⎜ |Ψ1 A 1 A 1 ⎟ ⎜ −a1
⎝ |Ψ
⎠=⎝
−a2
2 A1 A
1
|Ψ3 A 1 A 1
−a3
0 A1 A
1

a1
a0
a3
−a2

a2
−a3
a0
a1

while Bob applies CNOT B 1 B 2 CNOT B

⎞⎛

|0
a3
a 2 ⎟ ⎜ |1
⎠ ⎝ |2
−a1
|3
a0
1 B2

A1 A1
A1 A1
A1 A1

(18)

⎛

on his qubits followed

⎜ (K )
⎜ |Θ1
⎜
⎜ |Θ ( K )
⎝ 2
|Θ3( K )

=

1
4

3

|Ψl

A 1 A 1 |m B 2 B 2 |Λlm A 2 A 2 B 1 B 1 ,

⎛

⎞ ⎛
|Λ00
|00
⎜ |Λ01 ⎟ ⎜ |10
⎝
⎠=⎝
|Λ02
|20
|Λ03
|30

V ( 0) =

V (1 ) =

V (2 ) =

To

|22
|32
|02
|12

⎞⎛ ⎞
|33
a0
|23 ⎟ ⎜ a1 ⎟
⎠⎝ ⎠,
|13
a2
|03
a3

⎛ |0
⎟
⎟
A2 A2 ⎟
( K ) ⎜ |1
⎟ = V ⎝ |2
A2 A2 ⎠
|3
A2 A2

⎞

a0
⎟ ⎜ a1 ⎟

⎠⎝ ⎠,
a2
a3
⎞⎛ ⎞
a0
⎟ ⎜ a1 ⎟
⎠⎝ ⎠,
a2
a3
⎞⎛ ⎞
a0
⎟ ⎜ a1 ⎟
⎠⎝ ⎠.
a2
a3

(21)

(22)

(23)

A2 A2
A2 A2
A2 A2

⎞
⎟
⎠,

(24)

A2 A2

(25)

(K )

⎞

A2 A2 B 1 B 1

(K )

Λlm

(26)

(27)

(28)

(29)

the choices of measurement bases we write
(K )
as |Λlm A 2 A B 1 B , which can be expressed through
2

|Θn ; n = 0, 1, 2, 3

(20)

⎞⎛

1 e − i ϕ1
e − i ϕ2
e − i ϕ3
−
i
ϕ
−
i
ϕ
1
2
1 ⎜ 1 −e
e
−e −i ϕ3 ⎟
⎝
⎠,
−
i ϕ1
−
i ϕ2
−e
e − i ϕ3
2 1 −e
1 e − i ϕ1
−e −i ϕ2 −e −i ϕ3
⎛ − i ϕ1

⎞
e
1 e − i ϕ3
e − i ϕ2
−
i
ϕ
−
i
ϕ
−
i
ϕ
1 ⎜ −e 1 1 −e 3
e 2 ⎟
⎝ −i ϕ
⎠,
−i ϕ
−e −i ϕ2
2 −e 1 1 e 3
−
i ϕ1
−
i ϕ3
−
i ϕ2
e
1 −e
−e
⎛ − i ϕ2

⎞
e
e − i ϕ3
1 e − i ϕ1
−
i ϕ3
−
i ϕ1
1 ⎜ e − i ϕ2
−e
1 −e
⎟
⎝ −i ϕ
⎠,
e − i ϕ3
1 −e −i ϕ1
2 −e 2
−
i ϕ2
−
i ϕ3
−
i ϕ1
−e
−e
1 e
⎛ − i ϕ3
⎞
e
e − i ϕ2

e − i ϕ1
1
−
i
ϕ
−
i
ϕ
−
i
ϕ
3
2
1
1 ⎜ −e
e
−e
1⎟
⎝ − i ϕ3
⎠.
e
−e −i ϕ2 −e −i ϕ1 1
2
−e −i ϕ3 −e −i ϕ2 e −i ϕ1 1

specify

|Λlm

|11

|01
|31
|21

⎞

A2 A2

⎛

(19)

where

−|22
−|32
−|02
−|12
|11
|01
|31
|21
−|00
−|10
−|20
−|30

More explicitly, K = 0 if {(l, m)} = {(0, 0), (1, 1), (2, 2) or (3, 3)},
K = 1 if {(l.m)} = {(0, 1), (1, 0), (2, 3) or (3, 2)}, K = 2 if {(l.m)} =
{(0, 2), (1, 3), (2, 0) or (3, 1)}, and K = 3 if {(l, m)} = {(0, 3), (1, 2),

(2, 1) or (3, 0)}. As for the unitary transformations V ( K ) in Eq. (24),
they are of the form

3

m =0 l =0

|33
|23
|13
|03
−|00
−|10
−|20
−|30
−|11
−|01
−|31
−|21

⎧
0 if l − m = 0,
⎪
⎨
1 if l + m = 1 or 5,
K=
⎪
⎩ 2 if |l − m| = 2,
3 if l + m = 3.

V (3 ) =

A1 B 1 A1 B 1 A2 B 2 A2 B 2

−|00
−|10
−|20
−|30
−|33
−|23
−|13
−|03
|22
|32
|02
|12

with

2

|Q

|11
⎟ ⎜ |01
⎠=⎝
|31
|21
⎞ ⎛
|22

⎟ ⎜ |32
⎠=⎝
|02
|12
⎞ ⎛
|33
⎟ ⎜ |23
⎠=⎝
|13
|03

(K )

|Θ0

A1 A1

by measuring qubits B 2 and B 2 in their computational bases
{|0 , |1 , |2 , |3 } B 2 B . As a consequence, the SQR state becomes

⎛

Clearly, the qubits A 2 , A 2 , B 1 and B 1 collapse, with a probability P lm = 1/16 for any possible outcomes l, m ∈ {0, 1, 2, 3}, into an
entangled state |Λlm A 2 A B 1 B , if Alice 1 finds |Ψl A 1 A and Bob
2
1
1
finds |m B 2 B . After the measurements Alice 1 and Bob both pub2
lish their outcomes l and m through a classical communication
channel.

Secondly, upon hearing the published outcomes, Alice 2 measures her qubits A 2 and A 2 in an appropriate basis determined not
only by S 2 but also by the outcomes (l, m). There are four choices
for measurement bases determined by

⎞
⎟
⎠,

⎞

|Λ10
⎜ |Λ11
⎝
|Λ12
|Λ13
⎛
|Λ20
⎜ |Λ21
⎝
|Λ22
|Λ23
⎛
|Λ30
⎜ |Λ31
⎝
|Λ32
|Λ33

A2 A2 B 1 B 1

=

1
2

A2 A2

1

as

3

(K )

Θn
n =0

( K )+

A2 A2

R ln

|Φ

B1 B1

(30)

N.B. An et al. / Physics Letters A 375 (2011) 3570–3573

3573

Table 2
(0)
The reconstruction operator R ln in Eq. (30) versus l and n. I is the identity operator and
the phase-flip operator.
l
n
(0)

R ln

0(1, 2, 3)
0(1, 2, 3)

0(1, 2, 3)
1(0, 3, 2)

0(1, 2, 3)
2(3, 0, 1)

0(1, 2, 3)
3(2, 1, 0)

I⊗I

I ⊗ σz

σz ⊗ σz

σz ⊗ I

Table 3
(1,2,3)
The reconstruction operator R ln
in Eq. (30) versus l and n. I is the identity operator and
l
n

σz

σx (σz ) the bit-flip (phase-flip) one.

0
0

0
1

0
2

0
3

1
0

1
1

1
2

1
3

R ln

I⊗I

I ⊗ σz

σz ⊗ σz

σz ⊗ I

I ⊗ σ z σx

I ⊗ σx

σ z ⊗ σx

σ z ⊗ σ z σx

l
n

2
0

2
1

2
2

2
3

3
0

3
1

3
2

3
3

σ z σx ⊗ σ z

σ z σx ⊗ I

σx ⊗ I

σx ⊗ σ z

σ z σx ⊗ σx

σ z σx ⊗ σ z σx

σx ⊗ σ z σx

σx ⊗ σx

(1,2,3)

(1,2,3)

R ln

(K )

with R ln listed in Tables 2 and 3. After her measurement Alice 2
broadcasts the outcome n = 0, 1, 2 or 3 publicly. Transparently, the
(K )
probability that Alice 2 obtains the state |Θn A 2 A is P n = 1/4
2
independent of n.
Finally, as seen from Eq. (30), the desired state (16) can be re(K )
constructed by Bob’s application of R ln on his qubits B 1 and B 1 .
(0)

(1,2,3)

(see
Since for a fixed pair of (l, n), R ln may differ from R ln
Tables 2 and 3), to determine the right reconstruction operator,
Bob should take into account not only the outcomes of Alice 1
and Alice 2, but also that of himself. Precisely, Bob needs first
to calculate the value of K from the values of l and m in accordance with the rule (25), then he looks for the right operator
(K )
R ln from Tables 2 and 3. To simplify Bob’s action, Alice 2 can
(0)

use the same basis {|Θn A 2 A } to measure qubits A 2 and A 2 for
2
all possible outcomes (l, m), provided that when l + m = 1 or 5
(|l − m| = 2, l + m = 3) she needs applying I ⊗ σx (σx ⊗ I σx ⊗ σx ) on
A 2 and A 2 before measuring them. By doing so, Table 3 appears
superfluous and Bob will need only the phase-flip operator (i.e.,
the bit-flip operator is not required), in accordance with Table 2.
Such a simplification comes out from the observation that
(1 )

I ⊗ σx ⊗ I ⊗ I Λlm

( 0)

A2 A2 B 1 B 1

= Λlm

A2 A2 B 1 B 1

,
(31)

σx ⊗ I ⊗ I ⊗ I Λlm

( 0)

A2 A2 B 1 B 1

= Λlm

A2 A2 B 1 B 1

,

m − m = 2,

(32)
(3 )

σx ⊗ σx ⊗ I ⊗ I Λlm

( 0)

A2 A2 B 1 B 1

= Λlm

A2 A2 B 1 B 1

m + m = 3.

,
(33)

Similar to the case of the single-qubit state |φ , our JRSP protocol for the two-qubit state |Φ is also deterministic since its total
success probability P T is
3

3

3

PT =

P lm P n = 64 ×
n =0 m =0 l =0

1
16

×

1
4

= 1.

Acknowledgements

This work is supported by the Vietnam Foundation for Science
and Technology Development (NAFOSTED).
References

m + m = 1 or 5,
(2 )

plays a passive role in the sense that he participates only in the final step for reconstructing the target state, in our protocols Bob’s
role turns out quite active. Namely, here Bob not only performs
the reconstruction operation at the end, but also participates along
with the first sender, Alice 1, at the very beginning of the protocol and his measurement outcome is equally meaningful as that
of the first sender for the second sender (Alice 2) to choose the
right measurement basis. Because of Bob’s active action and the
usefulness of his outcome in the first step, an extra classical communication is needed from him. The total CCC of our deterministic
protocols for JRSP of the single-qubit (two-qubit) state is 3 (6) bits,
i.e., 1 (2) more bit(s) should be communicated in comparison with
the probabilistic ones. In spite of that, the gain of unit success
probability is really a precious figure of merit. Following our new
strategy it is straightforward, though tedious, to perform JRSP of
arbitrary multi-qubit states deterministically.

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4. Conclusion
We have studied JRSP via SQR in terms of EPR pairs. All existing protocols using such SQR are probabilistic. By proposing a new
nontrivial three-step strategy we have made those protocols deterministic. At variance with all the previous protocols using the QSR
in terms of EPR pairs as well as GHZ trios where the receiver Bob

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