Analytical Approach to Optimal Discrimination with

Unambiguous Measurements

University of Tabriz

M. Rezaei KaramatyAcademic member of Tabriz University

Email: Karamaty@tabrizu.ac.ir

For the Optimal State Discrimination

We propose three different methods which I am trying to illustrate. -Analytical solution for optimal USD

-Lewenstein-Sanpera decomposition as an optimal USD

-Approximating USD by linear programming

arXiv/0708.2323

M. A. Jafarizadeh, M. Rezaei, N. Karimi, A. R. Amiri

17 Sep 2007

Let’s see whether the discrimination of Non-Orthogonal States is possible.

• We know that single instances of non-orthogonal quantum states cannot be distinguished with certainty.

• We also know that this is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications.

• Then we can learn how to distinguish quantum states for sure.

POVM

• A set of operators is POVM iff

the following conditions are present:

1)

2)

k

Hk 0

Ik

kk 2

kjjjk in which only the orthogonality condition is added when comparedto POVM, but its success rate is less .

Another way is the Projective Measurement in whichthe following conditions are present.

POVM

von Neumannmeasurement

Comparison

At 0, the von Neumann strategy has a discontinuity-- only then can you succeed regardless of measurement choice.

Unambiguous State Discrimination

1) For N signals the set of measurement is

• The element is related to an inconclusive result and the other elements correspond to the identification of one of the states , i = 1, ...,N.

2) No states are wrongly identified, that is

Nkk 0

iii

kkkk

ki

p

NkiTr

~~...10][

0

][ 0 ii

iTrQ

Inconclusive rate:

. Success probability:

QP 1

Optimal USD

1. The POVM is a USD measurement on

2. The inconclusive rate is minimal

where the minimum is taken over all USDM.

optk i

)(min)( koptk QQ

Analytical Method for Obtaining Optimal USD

iiia

}{ i

0

i

• For distinguishing set let

• Where

• From the positivity of and the normalization conditions of

][ 0Tr

}{ i

n

iii

ii

i

a

a

XwhereGXX

pats

p

12

:1

1.

1max

Let and

then

Where and

The feasible region is achieved through the following

condition

XDYGGDD

21

21

iip

0)ˆ( IGDet

n

D

00

00

00

: 2

1

Analytical Solution for Two Pure States

For two pure states and with the arbitrary prior probabilities η1 and η2

Where

Then the feasible region is defined :

1 2

2112 a

221

*12

2112

1

2

1

12*

12

2

1

11

11

10

01

1

11

0

01

ˆ

a

a

a

aG

0)1()(0)1()(2

122121

2

12212

21 appppa

Feasible Region

Exact Calculations

. For the tangent of the line S and the feasible region

the following should exist to prove the condition :

)]1()([)(2

1221212211 apppppp

2112 11 pandp

21

12

a

Substitution in feasible region equation

Then

122

1212

1

21 11 apandap

Analytical Solutions

• The positivity of and

• The minimum value of inconclusive result

2p1p

121

212

1

aa

12212211 2)(1 appQ

Solutions

We conclude that :

01

0

1121

2

2121

2

pa

If

paIf

121

22

1221

121

2121221

121

22

1212

1

12

aifaQ

aaifaQ

aifaQ

Geometric Overview

Let and are two pure states on Bloch Sphere1 2

).(2

1

).(2

1

222

121

nI

nI

• USD

• Regarding to figure 2 we see that is equal to sphere center and the line connecting and

21

).(2

1

).(2

1

1)(,1)(

0)(,0)(

'222

'121

2211

1221

nI

nI

TrTr

TrTr

1'2

2'1

nn

nn

cos11 1221 app

QR cos2

121 For

Analytical Solution for Three Pure States

For three pure states and with the arbitrary prior probabilities η1, η2 and η3

Where

Then the feasible region is defined through the following

21 , 3

jiija

132

*23

31

*13

32

23

121

*12

31

13

21

12

1

3

2

1

*23

*13

23*12

1312

3

2

1

1

1

1

100

01

0

001

1

1

1

100

01

0

001

ˆ

aa

aa

aa

aa

aa

aa

G

01)2()1()1()1(

)(

132312223

213

2121

2232

2133

212

323121321

aaaaaapapapa

ppppppppp

Feasible Region

2

1

6

2231312 aandaa

1p

2p

3p

332211: pppS

For the tangent of the surface S and the feasible region the following should exist to prove the condition

Exact Calculations for Three States

And the result is as follows :

Only give the acceptable

values for

]Re[)( 332211 EquationgionFeasibleppp

32122121

22133131

12233232

1)(

1)(

1)(

apppp

apppp

apppp

321

22311312

21

2313)(

0

aaaaaand

ip

Some Examples

Example 1) Here let

Where they are all the optimal values of and the minimum value of inconclusive result

WBE:

• Where is the minimum eigenvalue of Frame operator

sppp 1321

saaa 231312

sQ

,0

jiforsaij

sQsppp n 121

ip

ip

Example 2) Let

Then we have the following :

22311312 , sasaa

)2(3

1

10

)2(3

2

1

2163

22

21

232

2

212

1

211232

1121

21

ss

sQ

spps

ssp

ssQsspp

spsss

One of the answers above which gives smaller Q has to lie in feasible region. If not, one of the foremost positions on planes or vertices will be optimal.

Lewenstein Sanpera Method

i

n

iip

1

'

Lemma

ρ: a hermitian density matrix

ρ = ρ′+(1−p)δρ: the decomposition of this density matrix

: one part of density operator ρ and

Such that

iii ~~

11

n

iip

LSD

i

Then the set of {pi}, which are maximal with respect to the density matrix ρ and the set of the projection operators form a manifold which generically has the dimension (n−1) and is determined via the following equation :

0...)1(...1...

...

21

21

nji

iiiiii

nkji

ijkijji

jiiji

ii pppDpppDppDpD

n

n

jiij

nnnn

n

n

a

aaa

aaa

aaa

D ~~~,

~~~

~~~

~...~~

:

21

22221

11211

LSD Via Opt USD

0~~1

0

ii

n

iipI

This equation determines the feasible region via reciprocal states which is the same as the one introduced with the previous method

Inconclusive result:

LSD method

LSD is the same as Opt USD and we use LSD in order to obtain the elements of the optimal POVM.

0~~11

ii

n

ii

n

ii pI

LSD Analytical Solution for Two States

2

122211

122

111

22

122211

121

222

1 ~~~

~~

,~~~

~~

aaa

aa

paaa

aa

p

For two pure states with the priori probabilities

if

then

and if

21 ,

21,

2

12

2

11

2

1112

12

2

11

2

12

~~

~

~~

~

aa

a

aa

a

21

12 22

21 ~1

,0a

pp

0,~1

211

1 pa

p

LSD Analytical Solution for Three States

For three pure states with priori probabilities

If

then

321 , and

321 ,,

0,,,

~

~~)~1(

~

~~)~1(

~

~~)~1(

321

33

2

232

2

131333

22

2

233

2

121222

11

2

133

2

122111

ppp

a

apapap

a

apapap

a

apapap

)~~~~(~)~~~~(~)~~~(~

)~~~~()~~~~()~~~(

)~~~~(~)~~~~(~)~~~(~

)~~~~()~~~~()~~~(

)~~~~(~)~~~~(~)~~~(~

)~~~~()~~~~()~~~(

31223221132331332112

2

23332211

132223113

213222312

3

22

122211

3

31223221131332332112

2

13332211

311232112

313323321

1

22

133311

2

31223221312331332112

2

22332211

312232211

323313321

1

22

223322

1

aaaaaaaaaaaaaa

aaaaaaaaaaa

p

aaaaaaaaaaaaaa

aaaaaaaaaaa

p

aaaaaaaaaaaaaa

aaaaaaaaaaa

p

LSD Analytical Solution for Three States

*

*

*

2

12

2

11

2

1112

12

2

11

2

12

~~

~

~~

~

aa

a

aa

a

2

13

2

11

2

1112

13

2

11

2

13

~~

~

~~

~

aa

a

aa

a

2

23

2

22

2

2212

23

2

22

2

23

~~

~

~~

~

aa

a

aa

a

2

233322

233

222

3

2

233322

232

333

2

1

~~~

~~

~~~

~~

0

aaa

aa

p

aaa

aa

p

p

2

133311

133

111

3

2

2

133311

131

333

1

~~~

~~

0

~~~

~~

aaa

aa

p

paaa

aa

p

0

~~~

~~

~~~

~~

3

2

122211

122

111

2

2

122211

121

222

1

p

aaa

aa

p

aaa

aa

p

And our final method is the Approximate Optimal USD

12

1.

max

1222

22

21

2211

a

ppts

pp

* Convex optimization:

*Where

*

21

STATE-DISCRIMINATION SUMMARY

-Unambiguously distinguishing between linearly independent quantum states is a challenging problem in quantum information processing.

-An exact analytic solution to an optimum measurement problem involving an arbitrary number of pure linearly independent quantum states is presented.

-The relevant semi-definite programming task is reduced to a linear programming one with a feasible region of polygon type which can be solved via simplex method.

-The strength of the method is illustrated through some explicit examples.

-Also using the close connection between the Lewenstein-Sanpera decomposition and semi-definite programming approach, the optimal positive operator valued measure for some of the well-known examples is obtain via Lewenstein-Sanpera decomposition method.

State-discrimination References

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[16] M. A. Jafarizadeh, M. Rezaee, and S. Ahadpour, Phys. Rev. A 74, 042335 (2006).

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