1. Analytical Approach to Optimal Discrimination with Unambiguous Measurements University of Tabriz M. Rezaei Karamaty Academic member of Tabriz University Email: Karamaty@tabrizu.ac.ir

2. 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

3. Let’s see whether the discrimination of Non-Orthogonal States is p ossible. 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.

4. POVM A set of operators is POVM iff the following conditions are present: 1) 2)

5. in which only the orthogonality condition is added when compared to POVM, but its success rate is less. Another way is the Projective Measurement in which the following conditions are present.

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

7. 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

8. Inconclusive rate:. Success probability:

9. Optimal USD 1. The POVM is a USD measurement on 2. The inconclusive rate is minimal where the minimum is taken over all USDM.

10. Analytical Method for Obtaining Optimal USD For distinguishing set let Where From the positivity of and the normalization conditions of

11. Let and then Where and The feasible region is achieved through the following condition

12. 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 :

13. Feasible Region

14. Exact Calculations. For the tangent of the line S and the feasible region the following should exist to prove the condition :

15. Substitution in feasible region equation Then

16. Analytical Solutions The positivity of and The minimum value of inconclusive result

17. Solutions We conclude that :

18. Geometric Overview Let and are two pure states on Bloch Sphere

19. USD Regarding to figure 2 we see that is equal to sphere center and the line connecting and

20. 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. Feasible Region

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

23. Exact Calculations for Three States And the result is as follows : Only give the acceptable values for

24. 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

25. Example 2) Let Then we have the following : 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.

27. Lewenstein Sanpera Method 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

28. LSD 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 :

29. LSD Via Opt USD 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.

30. LSD Analytical Solution for Two States For two pure states with the priori probabilities if then and if

31. LSD Analytical Solution for Three States For three pure states with priori probabilities If then

32. LSD Analytical Solution for Three States ******

33. And our final method is the Approximate Optimal USD * Convex optimization: *Where *

34. 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.

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