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Penny flip game
Introduction
Magic Square Game
Noise in some quantum games.
Piotr Gawron
The Institute of Theoretical and Applied Informatics of the Polish Academy of Sciences
2008-11-16
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
Table of contents
Introduction
Motivation
Noise model
Kraus operators
One-qubit noise
Multiqubit local channels
Penny flip game
The game
Quantum extention
Noise
Magic Square Game
The game
Noise
Success probability
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
Motivation
Motivation
The motivation for studies of noise in quantum games may be manifold:
I
how quantum game reduces to classical case?
I
how the odds change if the noise is present in quantum system
implementing the game?
I
does the knowledge about the noise changes the players’ odds?
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
Noise model
Noise model
In the most general case quantum evolution is described by super
operator Φ, which can be expressed in Kraus form:
X
Ek ρEk † ,
Φ(ρ) =
(1)
k
where
P
k
Ek † Ek = I.
Piotr Gawron
Noise in some quantum games.
2008
Introduction
Penny flip game
Magic Square Game
Noise model
In the literature, the following Kraus operators are considered to create
typical noisy channels:
nq
pα
pα
pα o
I depolarising channel:
1 − 3α
4 I,
4 σx ,
4 σy ,
4 σz ,
i h √ io
nh
1√0
0 α
I amplitude damping:
,
0 1−α , 0 0
nh
i h
io
1√0
0 √0
I phase damping:
,
0 1−α , 0 α
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
Noise model
I
I
I
√
1 − αI, ασz ,
√
√
bit flip and
1 − αI, ασx ,
√
√
bit-phase flip
1 − αI, ασy .
phase flip,
√
Real parameter α ∈ [0, 1] represents here the amount of noise in the
channel, σx , σy , σz are Pauli matrices.
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
Noise model
Multiqubit local channels
In order to apply noise operators to multiple qubits we form new set of
operators acting on larger Hilbert space.
We assume that we have set of n one-qubit Kraus operators {ek }. We
construct new set of nN operators {Ek } that act on Hilbert space of
N
dimension
P 2 . We† can write the action of the extended channel
Φ(ρ) = k Ek ρEk in the following way:
Φ(ρ) =
n
X
ei1 ⊗ ei2 ⊗ . . . ⊗ eiN ρei†1 ⊗ ei†2 ⊗ . . . ⊗ ei†N .
(2)
i1 ,i2 ,...iN =1
When applying equation 2 to sets of the operators listed above we obtain
one-parameter families of local noisy channels.
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
The game
Penny flip game
I
There are two players Alice and Bob.
I
There is a penny hidden in a box.
I
The players can flip the penny.
I
At the beginning the penny is faced up,
The game goes as follows:
I
I
I
I
Alice choses to flip or not flip,
now Bob choses to flip or not to flip,
at last Alice plays,
I
the penny is revealed,
I
Alice wins if the penny is faced down, otherwise Bob wins.
Piotr Gawron
Noise in some quantum games.
2008
Magic Square Game
Penny flip game
Introduction
The game
N
F
NN
-1
1
NF
1
-1
FN
1
-1
FF
-1
1
Table: The game payoff matrix. N means not-flipping and F means flipping.
One means that Alice won, minus one means that Bob won (it is zero-sum
game).
Piotr Gawron
Noise in some quantum games.
2008
Introduction
Penny flip game
Magic Square Game
Quantum extention
Penny flip game
Coin |ci facing heads up is denoted as U = |0i, facing tails up as
D = |1i.
Flip operator F = [ 01 10 ] is σx , not flipping operator N = [ 10 01 ] is identity I.
Winning condition is described by the expectation value of σz operator
hσz i|ci .
We will restrict ourselves only to deterministic strategies. One can easily
see that there is no winning strategy neither for Alice nor for Bob.
Now we will cheat a little bit. We will exchange a qubit for the penny.
The qubit can be in any normalised linear combination α|0i + β|1i. And
what is even more important, it may be transformed by use of any
unitary gate.
Piotr Gawron
Noise in some quantum games.
2008
Introduction
Magic Square Game
Penny flip game
Quantum extention
The cheater in this scheme is Alice. We assume that only she knows that
the game is played with a qubit. So now Alice can rotate the qubit inany
1
√
direction. She may chose to apply the Hadamard gate H =
1
√
2
2
1
√
2
− √12
as
her first and second movement. Bob has still two possibilities: two flip or
not to flip. Lets see what happens in those two cases:
1. HIH|0i → |0i,
2. HNH|0i → |0i.
It means that Bob’s actions have no influence whatsoever on the
outcome of the game. More detailed analysis of this game including
mixed (probabilistic) strategies may be found in [4].
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
Noise
We will assume that the qubit is influenced by the noise. To model this
situation we will assume that the noise takes effect on the system, before
Bob’s movement – Bob has to think a little bit making his decision, Alice
is quick in her decision because she knows that they played with qubit
rather than a penny. The case of depolarising channel was analysed in [1].
During each game we fix the noise type – it means that the qubit is
coupled to the environment. We connect the noise ration α with time
that Bob takes to make his decision.
So now the scheme is as follows:
ρf = HBεα (H|0ih0|H † )B † H † ,
(3)
where B denotes Bob’s decision (I or N) and εα (·) CPTP map
implementing noise parametrised by real parameter α. The expectation
value of Alice win is given by:
h$A i = h0|ρf |0i.
(4)
For Bob it is 1 − h$A i.
Piotr Gawron
Noise in some quantum games.
2008
Magic Square Game
Penny flip game
Introduction
Noise
channel
depolarising
amplitude damping
phase damping
bit flip
phase flip
bit-phase flip
h$A i
1
−
α/2
√
(√1 − α + 1)/2
( 1 − α + 1)/2
1
1−α
1−α
Table: Expectation value of Alice win in noisy penny flip game.
From this simple case we learn two very important lessons: (i) when
playing quantum games it is important to know and use quantum rules,
(ii) the noise can influence the outcome of the game and even change the
odds drastically.
Piotr Gawron
Noise in some quantum games.
2008
Introduction
Penny flip game
Magic Square Game
The game
Magic Square Game
The magic square is a 3 × 3 matrix filled with numbers 0 or 1 so that the
sum of entries in each row is even and the sum of entries in each column
is odd. Although such a matrix cannot exist one can consider the
following game.
There are two players: Alice and Bob. Alice is given the number of the
row, Bob is given the number of the column. Alice has to give the entries
for a row and Bob has to give entries for a column so that the parity
conditions are met. In addition, the intersection of the row and the
column must agree. Alice and Bob can prepare a strategy but they are
not allowed to communicate during the game.
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
The game
There exists a (classical) strategy that leads to winning probability of 98 .
If parties are allowed to share a quantum state they can achieve
probability 1.
In the quantum version of this game[3] Alice and Bob are allowed to
share an entangled quantum state.
The winning strategy is following. Alice and Bob share entangled state
|Ψi =
Piotr Gawron
Noise in some quantum games.
1
(|0011i − |1100i − |0110i + |1001i).
2
(5)
2008
Introduction
Penny flip game
Magic Square Game
The game
Depending on the input (i.e. the specific row and column to be filled in)
Alice and Bob apply unitary operators Ai ⊗ I and I ⊗ Bj , respectively,
0 0 1
−1 −1 1 1 i 1 −1
1 0i −i
1 −ii 11 −1
i
1 1 −1 1
1
0
A1 = √ 0 i 1 0 , A2 =
,
−i , A3 =
1 1
2 −ii 11 −1
2 11 −1
2 1 0 0i
1 −i
−1 −1 −1
"
#
"
#
1 0 0 1
1 1
i 1 i
1 −1
1 −1
1 −ii −i
0 0 1
−i 1 −1
1 i 1 −i
, B2 =
, B3 = √
B1 =
0 1 1 0 ,
1 1 −i i
1 −i 1 i
2 −i
2 −1
2 0 1 −1 0
i 1 1
−i 1 −i
where i and j denote the corresponding inputs.
The final state is used to determine two bits of each answer. The
remaining bits can be found by applying parity conditions.
Piotr Gawron
Noise in some quantum games.
2008
Introduction
Magic Square Game
Penny flip game
Noise
Adding the noise
We analyse the influence of the noise on success probability when the
noise operator is applied before the game gates [2].
The final state of this scheme is ρf = (Ai ⊗ Bj ) Φα (|ΨihΨ|) (A†i ⊗ Bj† ),
where Φα is the superoperator realizing quantum channel parametrized
by real number α. Probability Pi,j (α) is computed as the probability of
measuring ρf in the state indicating success
!
X
Pi,j (α) = Tr ρf
|ξi ihξi | ,
(6)
i
where |ξi i are the states that imply success.
Piotr Gawron
Noise in some quantum games.
2008
Introduction
Penny flip game
Magic Square Game
Success probability
We compute success probability Pi,j (α) for different inputs
(i, j ∈ {1, 2, 3}) and different quantum channels.
Our calculations show
P
that mean probability of success, P(α) = i,j∈{1,2,3} Pi,j (α), heavily
depends on the noise level α.
Piotr Gawron
Noise in some quantum games.
2008
Magic Square Game
Penny flip game
Introduction
Success probability
0.9
0.8
0.7
0.6
0.5
0
0.2
0.6
0.8
0.8
0.7
0.6
0
0.2
0.4
0.6
Error rate α
Piotr Gawron
Noise in some quantum games.
depolarizing channel
classical treshold
0.9
0.8
0.7
0.6
0.5
0
0.2
0.8
1
0.4
0.6
0.8
1
0.8
1
Error rate α
1
flipping channels
classical treshold
0.9
0.5
1
Error rate α
1
Success probability P (α)
0.4
Success probability P (α)
1
amplitude damping channel
classical treshold
Success probability P (α)
Success probability P (α)
1
phase damping channel
classical treshold
0.9
0.8
0.7
0.6
0.5
0
0.2
0.4
0.6
Error rate α
2008
Penny flip game
Introduction
Magic Square Game
Success probability
Bibliography
A. P. Flitney and D. Abbott.
Quantum games with decoherence.
Journal of Physics A: Mathematical and General, 38(2):449–459,
2005.
P. Gawron, J. A. Miszczak, and J. Sladkowski.
Noise effects in quantum magic squares game.
International Journal of Quantum Information, 6(1 supp), 2008.
N. David Mermin.
Simple unified form for the major no-hidden-variables theorems.
Phys. Rev. Lett., 65(27):3373–3376, Dec 1990.
E. W. Piotrowski and J. Sladkowski.
An invitation to quantum game theory.
Int. J. Theor. Phys., 42:1089 – 1099, 2003.
Piotr Gawron
Noise in some quantum games.
2008
Penny flip game
Introduction
Magic Square Game
Success probability
And now a question
We would like to investigate the whole space of strategies.
General equation for penny flip game is:
ρf = A2 εγ (Bεβ (A1 εα (ρ)A†1 )B † )A†2 .
Let α be one-parameter family of quantum channels. How to find a
decomposition of this channel such that:
εγ (εβ (εα (ρ))) = εx (ρ)
what is the relation between the parameters:
α + β + γ = x?
Piotr Gawron
Noise in some quantum games.
2008
Introduction
Penny flip game
Magic Square Game
Success probability
Thank you for your attention.
Piotr Gawron
Noise in some quantum games.
2008