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Discrete Phase-Space Structure and MUB Tomography

Discrete Phase-Space Structure and MUB Tomography. Andrei B. Klimov J.L. Romero, G.Bjork, L.L. Sanchez-Soto. Palermo 07. Motivation. Standard N - qubit state tomography: Number of measurement sets: Indirect measurement of DM elements accumulation of errors

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Discrete Phase-Space Structure and MUB Tomography

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  1. Discrete Phase-Space Structure and MUB Tomography Andrei B. Klimov J.L. Romero, G.Bjork, L.L. Sanchez-Soto Palermo 07

  2. Motivation • Standard N - qubit state tomography: • Number of measurement sets: • Indirect measurement of DM elements accumulation of errors • - Only local operations are required K. Korbicz, O. Gühne, M. Lewenstein, H. Häffner, C. F. Roos, R. Blatt Phys. Rev. A 74, 052319 (2006) M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. Körber, W. Hänsel, H. Häffner, C. F. Roos, R. BlattPhys. Rev. Lett. 97, 220407 (2006)

  3. Example: 2 qubits: Measurements in the standard basis diagonal elements of DM Measurements in locally rotated basis non-diagonal elements + diagonal elements non-diagonal elements + diagonal elements Accumulations of errors !

  4. Mutually Unbiased Bases Consider a d -dimensional Hilbert space Two orthonormal basis are mutually unbiased if MUBs provide the most efficient determination of quantum state. MUB sets are maximally incompatible, in the sense that a state producing precise measurement results in one set produces maximally random results in the others. I. D. Ivanovic, J. Phys. A 14, 3241 (1981) W. K. Wootters and B. D. Fields, Ann. Phys. (N.Y.) 191, 363 (1989)

  5. Known facts about MUB • There are at most d+1 MUB in a d-dimensional Hilbert space • If d is a prime, then d+1 MUB can be constructed using the generalized Fourier transform • If d is a power of a prime number, , then d+1 MUBs exist • and can be constructed by use of finite fields I. D. Ivanovic, J. Phys. A 14, 3241 (1981) W. K. Wootters and B. D. Fields, Ann. Phys. (N.Y.) 191, 363 (1989) F. A. Buot, Phys. Rev. B 10, 3700 (1974) J. H. Hannay and M. V. Berry, Physics D 1, 267 (1980)

  6. Quantum state tomography in MUB are MUB projectors, define a complete measurement scheme Direct measurements of the density matrix components where are the measured probabilities: Number of measurement sets: Shortcut: Non local operations are required for MUB construction

  7. Is there a single MUB set for N qubits? There are several types of MUBs, no equivalent under local operations! Is there the “most convenient” set of MUB for quantum tomography purposes?? Is there a regular form for constructing different MUB set for N qubits??

  8. MUB construction Find d+1 sets of disjoint operators, so that each set contains d -1 commuting operators, then the eigenstates of such sets form MUBs Bandyopadhyay S, Boykin P O, Roychowdhury V and Vatan V Algorithmica 34 512 (2002) Simple operations in prime dimensions. There is a unique set of MUB. Finite fields methods are required for power prime dimensions. There are several types of MUB, no equivalent under local operations.

  9. Two qubit MUB There exist several systematic construction algorithms, we will use one based on the simultaneous eigenstates to the identity and the Pauli operators Separable Separable Separable Non-separable Non-separable

  10. For two qubits, this is the only MUB construction that consists of separable or maximally entangled basis states. Any unitary transformation, local or nonlocal, that contains only these types of states, lead to the same structure. Separable Nonseparable (3, 2) -structure

  11. MUB operators ( ) Table of MUB operators Commutativity: Disdjoitness: Main idea: label states and operators of d - dim QS with elements ofGF(d) where point in the discrete phase-space

  12. Elements of the finite field theory Finite field is an Abelian group with respect to sum and multiplication Number of elements of is Trivial example: General case: and - primitive element: root of a minimal (n - th degree) polynomial Example: - primitive polynomial - primitive polynomial

  13. Discrete phase space for qubits Wootters W K Found. Phys. 36 112 (2006) Phase space is a grid endowed with a finite geometry structure Finite geometry: there exists a concept of straight lines which intersect only at a single point or parallel - fixed elements of Equation of a straight line: line-ray: parallel line: some other line: Rays: fixed element of There are rays. For each ray there are parallel lines There are lines. Each line contains points.

  14. Standard set of MUB operators Commuting operators rays in the discrete phase space different for different sets Example:2 qubits 5 rays: Each ray contains 3 points 5 sets of commuting operators Each set contains 3 operators

  15. Non-standard sets of MUB operators Commutative Abelian non-singular curves Commutative curve: operators labeled with points of a curve commute Abelian curve: are points of the same curve are Abelian commutative curves Trivial example: rays We look for a set of commuting operators self intersection Each curve should contain operators Nonsingular curves: curves with no self intersections G. Björk, J.Romero, A. B. Klimov, LL. Sánchez-Soto JOSA B 24. 371 (2007) A. B. Klimov, J.Romero, G. Björk, LL. Sánchez-Soto J.Phys.A (2007)

  16. Single non-singular commutative curve Set of commuting operators Non intersecting curves disjoint set of commuting operators Curve bundle: set of non-intersecting commutative curves Problem: find all Abelian commutative nonsingular curves and separate in bundles of mutually non intersecting (except at the origin) curves Operators labeled with points of curves in each bundle form sets of operators which eigenstates are MUBs Non-equivalent curve bundles MUB with different factorization structure

  17. Example Factorization structures existing in 8 dim space (3 qubits) 9 rays 2 rays + 7 curves 1 ray + 8 curves 9 curves (3,0,6) (2,3,4) (1,6,2) (0,9,0) (m,n,k) separable bi-separable non-separable J. Lawrence, C. Brukner, and A. Zeilinger, Phys. Rev. A 65, 032320 (2002) A.B. Klimov, L. L. Sánchez-Soto, and H. de Guise, J. Phys. A 38, 2747 (2005) J. Romero, G.Bjork, A.B.Klimov, L.L. Sanchez-Soto, Phys.Rev.A 72 062310 (2005)

  18. Rays for 3-qubit system corresponding to (3,0,6) structure θ 7 θ 6 θ 5 θ 4 θ 3 θ 2 θ 0 0 θ θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 factorization structure

  19. Curves for 3-qubit system corresponding to (0,9,0) stricture θ 7 θ 6 θ 5 θ 4 θ 3 θ 2 θ 0 0 θ θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 factorization structure

  20. Basis complexity and optimal MUBs N qubits: possible MUB factorizations: Basis complexity : P – fidelity of the CNOT gate, n – number of CNOT gates to generate corresponding basis total number of CNOT gates

  21. Examples 1. N=3 (0,9,0) is the optimal basis for quantum tomography

  22. Conclusions • MUBs are related to geometric structures in the • finite phase-space • 2. Rays give just one MUB set related to • the factorization of basis vectors • 3. All the other MUBs are related to curve bundles • in the finite phase-space completely separable Inseparable completely or partially 4. Curves in the phase space offer a regular way to classify MUBs and determine an optimal MUB

  23. Three qubit MUB Standard table of MUB operators: 3 factorizable + 6 non factorizable 63 operators arranged in 7 9 table separability J. Lawrence, C. Brukner, and A. Zeilinger, Phys. Rev. A 65, 032320 (2002) A.B. Klimov, L. L. Sánchez-Soto, and H. de Guise, J. Phys. A 38, 2747 (2005) J. Romero, G.Bjork, A.B.Klimov, L.L. Sanchez-Soto, Phys.Rev.A 72 062310 (2005)

  24. Non standard table of MUB operators There are no factorizable nor 3-entangled states separability

  25. What kind of mathematics do we need to work with n qubits Consider a single qubit The Hilbert space: Operators acting in this space: operations by mod 2 So that: {0,1} form a group with respect to multiplication and 1 with respect to summation There is a single set of MUB: eigenstates of

  26. Elements of the finite field theory Finite field is an Abelian group with respect to sum and multiplication Number of elements of is Trivial example: General case: and - primitive element: root of a minimal (n - th degree) polynomial Example: - primitive polynomial - primitive polynomial

  27. Finite field and quibits Mapping Trace operation: can be considered as an n-dim linear space Basis in the field: So that for any Orthonormal basis (self-dual basis): Representation of n – qubit state

  28. Main idea: label states and operators of d - dim QS with elements ofGF(d) - eigenstates of - operator, • root of unity Is conjugated to operator elements of the Pauli group: are related through the operator of the finite Fourier transform

  29. Relation to one- particle operators Expansion of field element in a self-dual basis: Thus Factorization of and in one – particle operators and powers

  30. Standard construction of MUB basis index - operator of discrete rotations where stabilizer state for Different MUBs are eigenstates of sets of commuting operators There are such sets operators Commutativity: A.B.Klimov, C. Munoz, JL Romero J. Phys. A 39 14471 (2006)

  31. Discrete phase space for qubits Wootters W K Found. Phys. 36 112 (2006) Phase space is a grid endowed with a finite geometry structure Finite geometry: there exists a concept of straight lines which intersect only at a single point or parallel - fixed elements of Equation of a straight line: line-ray: parallel line: some other line: There are rays. For each ray there are parallel lines There are lines. Each line contains points.

  32. Phase space points and discrete operators Consider a monomial: Such monomial is labeled by points of a discrete phase space GF(4) (2 qubits)

  33. Commuting operators rays in the discrete phase space Set Set ray ray ray - fixed element of Set Example:GF(4):5 rays, each ray contains 3 points 5 sets of commuting operators Each set contains 3 operators

  34. The simplest example: GF(4) - two qubits Abstract operators: FB FB EB EB FB where

  35. Abelian structures in the discrete phase space General equation for a ray in the parametric form: - fixed elements of - parameter General Abelian structure in the phase-space Abelian curves G. Björk, J.Romero, A. B. Klimov, LL. Sánchez-Soto JOSA B 24. 371 (2007) A. B. Klimov, J.Romero, G. Björk, LL. Sánchez-Soto J.Phys.A (2007)

  36. Commutative curve: operators labeled with points of a curve commute and are points of the same curve Rays and are commutative curves We look for a set of commuting operators Each curve should contain operators Nonsingular curves: curves with no self intersections self intersection Rays and are nonsingular curves

  37. Types of curves or 1. Non degenerated curves: 2. - degenerated: but not 3. - degenerated: but not 4. Exceptional curves: - degenerated curve: Non degenerated curve: Exceptional curve: - degenerated curve:

  38. Types of curves: Example GF(4) - degenerated curve: Exceptional curve: Non degenerated curve: or - degenerated curve:

  39. Single non-singular commutative curve Set of commuting operators Non intersecting curves disjoint set of commuting operators Curve bundle: set of non-intersecting commutative curves Problem: find all Abelian commutative nonsingular curves and separate in bundles of mutually non intersecting (except at the origin) curves Operators labeled with points of curves in each bundle form tables of operators which eigenstates form MUBs

  40. Central problem: Factorization of MUB operators Given a basis for any So that Given a commutative curve What is the factorization of the string?? Such factorization is invariant under local transformations!

  41. Example: Total number of Abelian structures: 15 5 rays + 10 curves organized in 6 bundles One particular bundle EB FB FB FB EB

  42. Local transformations Rotations by radians around the z-, x- or y- axes applied to a single particle z - transformation: x - transformation y - transformation Effect of local transformations: nonlinear transformation curve to curve Example: GF(4) curve ray We have to factor out all the curves equivalent under local transformations!

  43. There is a regular way to classify all non equivalent curves! • All the curves can be obtained from two rays • by applying local transformations and All the bundles of curves are equivalent – can be obtained by the bundle of rays by local transformations, so that there is a unique factorization structure Not all the bundles of curves are equivalent There are 4 different factorization structures primitive polynomial Example: Corresponding set of commuting operators Corresponding basis is bi-separable:

  44. Factorization structure of the MUBs for 8 dim space (m,n,k) inseparable rays exceptional curves separable bi-separable

  45. Factorization structures existing in 8 dim space (3 qubits) 9 rays 2 rays + 7 curves 1 ray + 8 curves 9 curves (3,0,6) (2,3,4) (1,6,2) (0,9,0) (m,n,k) separable bi-separable non-separable

  46. Basis complexity and optimal MUBs N qubits: possible MUB factorizations: Basis complexity : P – fidelity of the CNOT gate, n – number of CNOT gates to generate corresponding basis total number of CNOT gates

  47. Examples 1. N=3 (0,9,0) is the optimum basis for quantum tomography

  48. Conclusions • MUBs are related to geometric structures in the • finite phase-space • 2. Rays give just one MUB set related to • the factorization of basis vectors • 3. All the other MUBs are related to curve bundles • in the finite phase-space completely separable Inseparable completely or partially 4. Curves in the phase space offer a regular way to classify MUBs and determine an optimal MUB

  49. MUBs for qubits can be defined using the Pauli operators and the identity in ordered sets (the operator tables). To each different MUB-structure, there is an associated phase-space construction of lines or curves. The standard MUB construction is based on ”straight lines” in phase-space [the (2,3,4) MUB]. We show that there exist a finite set of different possible constructions, but that these are based on curves in phase-space, and not lines. The different MUB structures can be used to define a quantitative (un)certainty relation between the individual and the joint properties of the qubits.

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