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Complex MUBs SIC-POVMs Problems over the Real Numbers Some Problems in Quantum Information Theory William J. Martin Department of Mathematical Sciences and Department of Computer Science Worcester Polytechnic Institute BIRS Workshop on Mathematics of Communications Banff, Alberta January 29, 2015 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Thanks Thanks to the organizers! Yesterday, along the Bow River In preparing this talk, I benefited from conversations with Bill Kantor. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers The Whole Talk in One Slide You were born to solve these problems : I Find, as many as you can, equiangular lines in Cd (unit vectors whose inner products have constant modulus). William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers The Whole Talk in One Slide You were born to solve these problems : I Find, as many as you can, equiangular lines in Cd (unit vectors whose inner products have constant modulus). Find d 2 , if possible. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers The Whole Talk in One Slide You were born to solve these problems : I I Find, as many as you can, equiangular lines vectors whose inner products have constant Find d 2 , if possible. Find, as many as you can, equiangular lines vectors whose inner products have constant William J. Martin in Cd (unit modulus). in Rd (unit abs value). Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers The Whole Talk in One Slide You were born to solve these problems : I I Find, as many as you can, equiangular lines vectors whose inner products have constant Find d 2 , if possible. Find, as many as you can, equiangular lines vectors whose inner products have constant d+1 Find 2 , if possible. William J. Martin in Cd (unit modulus). in Rd (unit abs value). Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers The Whole Talk in One Slide You were born to solve these problems : I I I Find, as many as you can, equiangular lines in Cd (unit vectors whose inner products have constant modulus). Find d 2 , if possible. Find, as many as you can, equiangular lines in Rd (unit vectors whose inner products have constant abs value). d+1 Find 2 , if possible. Find, as many as you can, orthonormal bases in Cd where unit vectors from distinct bases have inner products with constant modulus. Find d + 1, if possible. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers The Whole Talk in One Slide You were born to solve these problems : I I I I Find, as many as you can, equiangular lines in Cd (unit vectors whose inner products have constant modulus). Find d 2 , if possible. Find, as many as you can, equiangular lines in Rd (unit vectors whose inner products have constant abs value). d+1 Find 2 , if possible. Find, as many as you can, orthonormal bases in Cd where unit vectors from distinct bases have inner products with constant modulus. Find d + 1, if possible. Find, as many as you can, orthonormal bases in Rd where unit vectors from distinct bases have inner products with constant absolute value. Find d2 + 1, if possible. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Uncertainty Principle Exact knowledge of one diminishes knowledge of the other William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges A Cartoon Version of Quantum Physics Kraus: “Two observables A and B of an n-level system (i.e., a quantum system with n-dimensional state space) are called complementary, if knowledge of the measured value of A implies maximal uncertainty of the measured value of B, and vice versa.” Also called “maximally incompatible”. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges A Cartoon Version of Quantum Physics Complementary Observables: If we measure one exactly, a subsequent measurement of the other produces all outcomes with equal probability. I have more to say about measurements in an William J. Martin appendix Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements I basis states 1, 2, . . . , d I pure state (superposition) v ∈ Cd I A mixed state is a probability distribution on pure states state v1 v2 · · · vk probability p1 p2 · · · pk William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements I basis states 1, 2, . . . , d I pure state (superposition) v ∈ Cd I A mixed state is a probability distribution on pure states state v1 v2 · · · vk probability p1 p2 · · · pk A mixed state is encoded as a positive semidefinite Hermitian matrix with trace one. ρ= k X pj vj vj∗ j=1 (specified by d 2 − 1 real parameters) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements In the simplest model, a measurement is an orthogonal decomposition of Cn . We identify the measurement with an orthonormal basis {v1 , . . . , vn }. Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }: William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements In the simplest model, a measurement is an orthogonal decomposition of Cn . We identify the measurement with an orthonormal basis {v1 , . . . , vn }. Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }: (1) the state is expressed as a linear combination ψ = p1 v1 + p2 v2 + · · · + pn vn William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements In the simplest model, a measurement is an orthogonal decomposition of Cn . We identify the measurement with an orthonormal basis {v1 , . . . , vn }. Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }: (1) the state is expressed as a linear combination ψ = p1 v1 + p2 v2 + · · · + pn vn P we assume ψ is a unit vector so that j |pj |2 = 1. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements In the simplest model, a measurement is an orthogonal decomposition of Cn . We identify the measurement with an orthonormal basis {v1 , . . . , vn }. Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }: (1) the state is expressed as a linear combination ψ = p1 v1 + p2 v2 + · · · + pn vn P we assume ψ is a unit vector so that j |pj |2 = 1. (2) an index j is chosen at random with probability |pj |2 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements In the simplest model, a measurement is an orthogonal decomposition of Cn . We identify the measurement with an orthonormal basis {v1 , . . . , vn }. Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }: (1) the state is expressed as a linear combination ψ = p1 v1 + p2 v2 + · · · + pn vn P we assume ψ is a unit vector so that j |pj |2 = 1. (2) an index j is chosen at random with probability |pj |2 (3) our measurement returns only the index j William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges States and Measurements In the simplest model, a measurement is an orthogonal decomposition of Cn . We identify the measurement with an orthonormal basis {v1 , . . . , vn }. Measurement of quantum state ψ w.r.t. basis {v1 , . . . , vn }: (1) the state is expressed as a linear combination ψ = p1 v1 + p2 v2 + · · · + pn vn P we assume ψ is a unit vector so that j |pj |2 = 1. (2) an index j is chosen at random with probability |pj |2 (3) our measurement returns only the index j (4) and the state collapses to vj . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complementary Observables — I repeat myself: Given measurements (“observables”) A and B on an n-dimensional quantum system, we say A and B are complementary if whenever we measure one exactly, a subsequent measurement of the other produces all outcomes with equal probability. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges The cost of clarity ”Truth and clarity are complementary.” - Niels Bohr William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex MUBs Two orthonormal bases B and B 0 for Cd are unbiased if |hu, vi| = α for some constant α, for every u ∈ B and every v ∈ B 0 . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex MUBs Two orthonormal bases B and B 0 for Cd are unbiased if |hu, vi| = α 0 for some constant α, for every √ u ∈ B and every v ∈ B . Easy to check that α = 1/ d. Schwinger (1960), Ivanovic (1981) - measurements Wootters & Fields (1989) - “MUB” Alltop (1980) - autocorrelation Seidel, et al. (1970s) - line systems various physicists (1980s) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complete sets of MUBs A collection {B1 , . . . , BM } of orthonormal bases for Cd is a collection/set of mutually unbiased bases if each pair is unbiased. We say we have a set of “k MUBs”. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complete sets of MUBs A collection {B1 , . . . , BM } of orthonormal bases for Cd is a collection/set of mutually unbiased bases if each pair is unbiased. We say we have a set of “k MUBs”. Let M(d) denote the maximum size of a set of MUBs in Cd . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complete sets of MUBs A collection {B1 , . . . , BM } of orthonormal bases for Cd is a collection/set of mutually unbiased bases if each pair is unbiased. We say we have a set of “k MUBs”. Let M(d) denote the maximum size of a set of MUBs in Cd . Theorem[Delsarte, Goethals, Seidel (1978)] (proven in terms of line systems) M(d) ≤ d + 1 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex Hadamard matrices Say a matrix is flat if all entries have the same modulus. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex Hadamard matrices Say a matrix is flat if all entries have the same modulus. I We represent each basis Bj as the set of columns of some unitary matrix Hj William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex Hadamard matrices Say a matrix is flat if all entries have the same modulus. I We represent each basis Bj as the set of columns of some unitary matrix Hj I applying a unitary transformation, we may assume the first matrix is the identity William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex Hadamard matrices Say a matrix is flat if all entries have the same modulus. I We represent each basis Bj as the set of columns of some unitary matrix Hj I applying a unitary transformation, we may assume the first matrix is the identity I in this case, all the other √ matrices (bases) in a set of MUBs are flat, with modulus 1/ d William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex Hadamard matrices Say a matrix is flat if all entries have the same modulus. I We represent each basis Bj as the set of columns of some unitary matrix Hj I applying a unitary transformation, we may assume the first matrix is the identity I in this case, all the other √ matrices (bases) in a set of MUBs are flat, with modulus 1/ d √ Since these are unitary, Hj∗ Hj = I and each dHj is a complex Hadamard matrix I William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complex Hadamard matrices Say a matrix is flat if all entries have the same modulus. I We represent each basis Bj as the set of columns of some unitary matrix Hj I applying a unitary transformation, we may assume the first matrix is the identity I in this case, all the other √ matrices (bases) in a set of MUBs are flat, with modulus 1/ d √ Since these are unitary, Hj∗ Hj = I and each dHj is a complex Hadamard matrix I I so we seek mutually unbiased Hadamard matrices (“MUCH”, Best and Kharaghani, 2010): Hj∗ Hk flat ∀ j 6= k William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Retrieving Complete Information about a State Note that each basis corresponds to a measurement which, statistically, reveals d − 1 pieces of information (probabilities sum to one). So d + 1 measurements represent d 2 − 1 real parameters, which is the amount of information needed to represent a general mixed state k X ρ= pj vj vj∗ j=1 we have d real values along the diagonal summing to one and 2 × d(d − 1)/2 real parameters to specify the complex entries above the diagonal William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Planar Functions Let F be an additive group (e.g., F = Fq ). A mapping f : F → F is a planar function if, for every non-zero a ∈ F the map x 7→ f (x + a) − f (x) is a bijection. (E.g., for q odd f : Fq → Fq via f (x) = x 2 .) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Planar Functions Hypothesis: We suppose that S is a set of functions Fq → Fq , including the zero function, such that the difference of any two is planar. For q = p r , Fq is a vector space over Zp (p prime); let ζ P j be a primitive complex p th root of unity so that p−1 j=0 ζ = 0. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Complete Sets of MUBs in Prime Power Dimensions Roy-Scott (2007), Godsil-Roy (2009) Write the standard basis for Cd (d prime) as {eb | b ∈ Fq }. Let B∞ = {eb | b ∈ F }. For f ∈ S, let ) ( 1 X a·b+f (b) ζ eb | a ∈ F . Bf = √ d b∈F Then the collection {Bf : f ∈ S} ∪ {B∞ } is a set of |S| + 1 MUBs in Cd . Here d can also be a prime power and we can find S of size q using symplectic spreads (CCKS, K). William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Bent Functions Another flavour of construction arises from a bent function on a vector space over Fp . These two techniques, using bent functions f : V → F and planar functions f : F → F cover all the constructions I know of complete sets of complex MUBs. Can we do something in between? f : V → W for a non-trivial subspace W of V ? (Or modules over rings?) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges MUBs and planes William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges MUBs and planes I All known constructions of complete sets are somehow related to affine planes William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges MUBs and planes I All known constructions of complete sets are somehow related to affine planes I Kantor (2012): symplectic spreads yield complete sets of MUBs and inequivalent spreads give rise to inequivalent MUBs, so there are a lot of them William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges MUBs and planes I All known constructions of complete sets are somehow related to affine planes I Kantor (2012): symplectic spreads yield complete sets of MUBs and inequivalent spreads give rise to inequivalent MUBs, so there are a lot of them I Yet this does not mean that all complete sets of MUBs must come from planes. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges What is needed? William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges What is needed? I Ideas! William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges What is needed? I Ideas! I New constructions — particularly when d is not a prime power William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges What is needed? I Ideas! I New constructions — particularly when d is not a prime power I large, but not complete, sets for non-prime power orders William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges What is needed? I Ideas! I New constructions — particularly when d is not a prime power I large, but not complete, sets for non-prime power orders I M(d) = Ω(d) (or even a lower bound that grows with d) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges What is needed? I Ideas! I New constructions — particularly when d is not a prime power I large, but not complete, sets for non-prime power orders I M(d) = Ω(d) (or even a lower bound that grows with d) I Is there any combinatorics in the inner products? A coherent configuration? William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Examples of non-complete sets I Analogue of MacNeish construction: if d has prime factorization d = p1r1 p2r2 · · · pkrk , then M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}. via taking tensor products of this many MUBs in each prime-power dimension William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Examples of non-complete sets I Analogue of MacNeish construction: if d has prime factorization d = p1r1 p2r2 · · · pkrk , then M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}. via taking tensor products of this many MUBs in each prime-power dimension I Beth-Wocjan construction (2004): If d = s 2 and there exist k MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d, we obtain M(d) ≥ d 1/29.6 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Examples of non-complete sets I Analogue of MacNeish construction: if d has prime factorization d = p1r1 p2r2 · · · pkrk , then M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}. via taking tensor products of this many MUBs in each prime-power dimension I Beth-Wocjan construction (2004): If d = s 2 and there exist k MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d, we obtain M(d) ≥ d 1/29.6 (Beats MacNeish lower bound infinitely often) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Just a bit of physics Basics of MUBs MUB Constructions MUB Challenges Examples of non-complete sets I Analogue of MacNeish construction: if d has prime factorization d = p1r1 p2r2 · · · pkrk , then M(d) ≥ min{M(p1r1 ), M(p2r2 ), . . . , M(pkrk )}. via taking tensor products of this many MUBs in each prime-power dimension I Beth-Wocjan construction (2004): If d = s 2 and there exist k MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d, we obtain M(d) ≥ d 1/29.6 (Beats MacNeish lower bound infinitely often) I Kharaghani and co-authors have found mutually unbiased Hadamard matrices in various small dimensions William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames SIC-POVMs A set of d + 1 MUBs give us d + 1 measurements that together estimate d 2 − 1 real parameters. Can we do this with one measurement? Whereas MUBs give physicists a way to perform quantum state tomography on a single mixed state (prepared repeatedly) by performing a series of measurements (one for each basis), SIC-POVMs achieve the same thing with a single measurement (repeatedly applied). But this is not our usual sort of measurement: the outcome states are not pairwise orthogonal. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Complex Equiangular Lines In each complex vector space Cd , we seek the maximum size of a set L of equiangular lines. These were studied by Seidel and co-authors in the 1960s and 1970s. Example: In C2 , the lines spanned by (τ, i), (τ, −i), are equiangular for τ = √ 1+ 3 2 (1 William J. Martin (1, iτ ), (1, −iτ ) + i). Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames SIC-POVMs Physicists are interested in equiangular line sets of size exactly d 2 : these are called symmetric informationally complete positive operator-valued measures (“SIC-POVMs”) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Why are these measurements? Why are physicists interested? In the simplest model, a measurement is an orthogonal decomposition of Cn . We identify the measurement with an orthonormal basis {v1 , . . . , vn }. As we said before, the state is expressed as a linear combination ψ = p1 v1 + p2 v2 + · · · + pn vn P where j |pj |2 = 1. When we perform this measurement, an index j is chosen at random with probability |pj |2 and the state collapses to vj . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Von Neumann Measurements This P measurement is represented by the Hermitian matrix ψ = j pj vj vj∗ , a sum of n rank-one projections, or by the ensemble {vj vj∗ } of n pairwise orthogonal rank-one Hermitian matrices summing to I . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Measure the whole system to determine state of a subsystem Now suppose we are interested in a quantum subsystem of some system. Let matrix P denote orthogonal projection onto this subspace. A measurement on the overall system is an ensemble of rank-one projections. Restricted to this subspace, these correspond to projections of rank one (or zero) Pvj vj∗ P. These are not necessarily pairwise orthogonal, but they still sum to the identity. A POVM is an ensemble {Ej } of positive semidefinite Hermitian matrices which sum to I . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Special POVMs Symmetric: Constant inner product Tr(Eh Ej∗ ) over all h 6= j. But Eh = vh vh∗ , Ej = vj vj∗ gives Tr(Eh Ej∗ ) = Tr(vh vh∗ vj vj∗ ) = Tr(vj∗ vh vh∗ vj ) = |hvh , vj i|2 . So, for rank one projectors, “symmetry” means “equiangular”. By expanding I 2 inside Tr(I 2 ) = d, we find the common modulus of inner products must be 1/(d + 1). William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Special POVMs Informationally Complete: Now we can get more than d − 1 real values from the same measurement. So we consider repeatedly applying the same POVM. As before, the general mixed state is described by d 2 − 1 real parameters. Our POVM has probabilities summing to one, so we need d 2 outcomes in order to be “informationally complete”. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames SIC-POVMs Physicists are interested in these for both practical and theoretical connections: I quantum state tomography I quantum cryptography I evaluating “hidden variable” theories (Kochen-Specker Theorem) (Some classical applications are mentioned as well.) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Some Nice Constructions Let N(d) denote the max size of a set of equiangular lines in Cd . Theorem[Delsarte, Goethals, Seidel (1978)] N(d) ≤ d 2 Here are the general techniques to obtain exactly d 2 lines in Cd : I I I William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Three Constructions of Exactly d 2 Lines I S. Hoggar (1981,1998): 64 lines in C8 from a 4-dimensional quaternionic polytope I Jedwab & Wiebe (2014): Elegant solutions in dimensions d = 2, 3, 8 from Hadamard matrices I Various authors: “Zauner method” William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Some Nice Constructions of Less Than d 2 Lines I König (1995,1999): Characters of abelian groups restricted to (v , k, 1)-difference sets (rediscovered several times) I Godsil & Roy (2012): line systems from relative difference sets I Greaves, et al. (2014)/Jedwab & Wiebe (2014): equiangular lines from MUBs I equiangular tight frame constructions William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Sets of Quadratic Size Recall that the upper bound is N(d) ≤ d 2 I The König construction gives N(d) ≥ d 2 − d + 1 for d = p r + 1 (p prime) I Converting MUBs to equiangular lines (Greaves, et al., Jedwab & Wiebe) gives N(d) = Θ(d 2 ) for many other dimensions d. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Equiangular Lines from Difference Sets Let G be a finite abelian group. A character of G is a group homomorphism χ : G → C∗ Example: For G = Z7 we have seven distinct characters, χa (0 ≤ a < 7) given by χa (b) = ω ab where ω is any primitive complex 7th root of unity. The quadratic residues in Z7 form a difference set D = {1, 2, 4}: 1−2 = 6, 1−4 = 4, 2−1 = 1, William J. Martin 2−4−5, 4−1 = 3, Quantum Information Theory 4−2 = 2. Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Lines from Difference sets The quadratic residues in Z7 form a difference set D = {1, 2, 4} with parameters (7, 3, 1). The corresponding characters χ1 , χ2 , χ4 give us 7 vectors by arranging them in a 3 × 7 array and transposing: With ω = e 2πi/7 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Lines from Difference sets The quadratic residues in Z7 form a difference set D = {1, 2, 4} with parameters (7, 3, 1). The corresponding characters χ1 , χ2 , χ4 give us 7 vectors by arranging them in a 3 × 7 array and transposing: With ω = e 2πi/7 χ1 χ2 χ4 = = = 0 [1 [1 [1 William J. Martin 1 ω ω2 ω4 2 ω2 ω4 ω 3 ω3 ω6 ω5 4 ω4 ω ω2 Quantum Information Theory 5 ω5 ω3 ω6 6 ω6] ω5] ω3] Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Lines from Difference sets The quadratic residues in Z7 form a difference set D = {1, 2, 4} with parameters (7, 3, 1). The corresponding characters χ1 , χ2 , χ4 give us 7 vectors by arranging them in a 3 × 7 array and transposing: χ1 = [ With ω = e 2πi/7 χ2 = [ χ4 = [ This gives us a configuration 0 1 1 1 of 1 2 3 ω ω2 ω3 ω2 ω4 ω6 ω4 ω ω5 7 vectors in C3 4 ω4 ω ω2 5 ω5 ω3 ω6 X = (1, 1, 1), (ω, ω 2 , ω 4 ), (ω 2 , ω 4 , ω), (ω 3 , ω 6 , ω 5 ), (ω 4 , ω, ω 2 ), (ω 5 , ω 3 , ω 6 ), (ω 6 , ω 5 , ω 3 ) which span 7 equiangular lines. William J. Martin Quantum Information Theory 6 ω6] ω5] ω3] Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Conjecture (G. Zauner, 1999) SIC-POVMs exist in all dimensions d. Moreover, in each dimension d, there is a “fiducial” vector (a common eigenvector of all operators in some specific abelian subgroup of the Heisenberg-Weyl group) whose orbit under the Heisenberg-Weyl group has size d 2 and forms a SIC-POVM. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture From a computational viewpoint, the conjecture seems to work! “Numerical” examples of SIC-POVMs have been found to high precision in all dimensions d ≤ 67. Exact analytic solutions have been found (mostly by Gröbner basis techniques) in the following dimensions d = 2, 3, . . . , 16, 19, 24, 28, 35, 48 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture From a computational viewpoint, the conjecture seems to work! “Numerical” examples of SIC-POVMs have been found to high precision in all dimensions d ≤ 67. Exact analytic solutions have been found (mostly by Gröbner basis techniques) in the following dimensions d = 2, 3, . . . , 16, 19, 24, 28, 35, 48 So what’s holding us up? William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Grassl and Scott summarize these solutions in a 16-page paper William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Grassl and Scott summarize these solutions in a 16-page paper (with a 206-page supplement). William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Grassl and Scott summarize these solutions in a 16-page paper (with a 206-page supplement). Fiducial vector for numerical example in C10 . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Second part of Grassl-Scott supplement gives exact solutions. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Second part of Grassl-Scott supplement gives exact solutions. Fiducial vector for analytic example in C10 requires some preliminary definitions of constants. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Second part of Grassl-Scott supplement gives exact solutions. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Zauner’s Conjecture Second part of Grassl-Scott supplement gives exact solutions. Here is the first entry of the fiducial vector for the example in C10 . William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e 2πi/3 , consider the complex Hadamard matrix 1 1 1 H = 1 ω ω2 . ω 1 ω2 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e 2πi/3 , consider the complex Hadamard matrix 1 1 1 H = 1 ω ω2 . ω 1 ω2 Then the 9 rows of the following matrices form a SIC-POVM in C3 . −2 1 1 1 −2 1 1 1 −2 −2 1 −2ω ω 2 , 1 ω −2ω 2 ω ω2 , −2ω 1 ω 2 ω −2 ω2 ω 1 −2ω 2 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e 2πi/3 , consider the complex Hadamard matrix 1 1 1 H = 1 ω ω2 . ω 1 ω2 Then the 9 rows of the following matrices form a SIC-POVM in C3 . −2 1 1 1 −2 1 1 1 −2 −2 1 −2ω ω 2 , 1 ω −2ω 2 ω ω2 , −2ω 1 ω 2 ω −2 ω2 ω 1 −2ω 2 Good News! This works again in dimensions 2 and 8. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e 2πi/3 , consider the complex Hadamard matrix 1 1 1 H = 1 ω ω2 . ω 1 ω2 Then the 9 rows of the following matrices form a SIC-POVM in C3 . −2 1 1 1 −2 1 1 1 −2 −2 1 −2ω ω 2 , 1 ω −2ω 2 ω ω2 , −2ω 1 ω 2 ω −2 ω2 ω 1 −2ω 2 Good News! This works again in dimensions 2 and 8. Bad News. That’s it for this technique, in this form at least. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames We Need More Ideas Like This Even if Zauner’s conjecture is true, it will be helpful to find more constructions — efficiently computable, reasonably verifiable examples. Clearly there is better chance for progress if we are happy with less than d 2 lines. But even for d 2 lines, we must be careful to avoid assumptions that apparent connections to finite design theory are in fact constraints. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Some nonsense on Wikipedia “This seems similar in nature to the symmetric property of SIC-POVMs. In fact, the problem of finding a SIC-POVM is precisely the problem of finding equiangular lines in Cd ; whereas mutually unbiased bases are analogous to affine spaces. In fact it can be shown that the geometric analogy of finding a ‘complete set of N+1 mutually unbiased bases is identical to the geometric structure analogous to a SIC-POVM’. It is important to note that the equivalence of these problems is in the strict sense of an abstract geometry, and since the space on which each of these geometric analogues differs, there’s no guarantee that a solution on one space will directly correlate with the other. An example of where this analogous relation has yet to necessarily produce results is the case of 6-dimensional Hilbert space, in which a SIC-POVM has been analytically computed using mathematical software, but no complete mutually unbiased bases has yet been discovered.” William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames The Welch Bound In telecommunications, it is beneficial to spread n unit vectors as far apart as possible on the unit sphere. If we have a set X of N unit vectors in Cd with maximum norm on all pairwise inner products, then s N −d ≥ . d(N − 1) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames The Welch Bound In telecommunications, it is beneficial to spread n unit vectors as far apart as possible on the unit sphere. If we have a set X of N unit vectors in Cd with maximum norm on all pairwise inner products, then s N −d ≥ . d(N − 1) If equality holds, then the Gram matrix G = [x · y]x,y has only two eigenvalues and all off-diagonal entries equal to ±ε. (“ETF”) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames The Welch Bound In telecommunications, it is beneficial to spread n unit vectors as far apart as possible on the unit sphere. If we have a set X of N unit vectors in Cd with maximum norm on all pairwise inner products, then s N −d ≥ . d(N − 1) If equality holds, then the Gram matrix G = [x · y]x,y has only two eigenvalues and all off-diagonal entries equal to ±ε. (“ETF”) William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Equiangular Tight Frames (ETFs) Only four non-trivial general constructions known: I N = 2d where ∃ conference matrix (d = 2s+1 or d = p s + 1, s ≥ 1, p odd prime) I characters from difference sets I Incidence matrices of Steiner systems 2-(v , k, 1) I eigenspace projections of strongly regular graphs with special parameters William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Definition and physics Constructions Equiangular Tight Frames Equiangular Tight Frames (ETFs) Only four non-trivial general constructions known: I N = 2d where ∃ conference matrix (d = 2s+1 or d = p s + 1, s ≥ 1, p odd prime) I characters from difference sets I Incidence matrices of Steiner systems 2-(v , k, 1) I eigenspace projections of strongly regular graphs with special parameters Note also: • orthonormal basis has N = d • regular simplex has N = d + 1 William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Real MUBs and Equiangular Lines A few brief remarks about the real case: I Upper bounds are roughly half the corresponding LP bounds for MUBs and equiangular lines I more combinatorics and number-theoretic restrictions (|x| = α for only two values of x) I known extremal examples of d2 + 1 MUBs in Rd (d = 4t ) come from Kerdock sets examples of d+1 equiangular lines in Rd only known for 2 d = 2, 3, 7, 23 I I many examples yield or come from association schemes (regular two-graphs, all sets of real MUBs, de Caen’s set of 2 2 9 d lines) And we can ask both questions over the quaternions, as well! William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers The End. Thank You! You are the optimal people to solve these challenges! : I I I I Find, as many as you can, equiangular lines in Cd (unit vectors whose inner products have constant modulus). Find d 2 , if possible. Find, as many as you can, equiangular lines in Rd (unit vectors whose inner products have constant abs value). d+1 Find 2 , if possible. Find, as many as you can, orthonormal bases in Cd where unit vectors from distinct bases have inner products with constant modulus. Find d + 1, if possible. Find, as many as you can, orthonormal bases in Rd where unit vectors from distinct bases have inner products with constant absolute value. Find d2 + 1, if possible. William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Quantum States pure state (superposition) ψ= √ 0.40 + √ 1 0.61 = √ 5 √ √2 3 with amplitudes a1 , a2 satisfying |a1 |2 + |a2 |2 = 1. If we measure state ψ in the standard basis, we will see 0 with probability 0.4 and 1 with probability 0.6. Back to the introduction William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Measurements A quantum measurement of a d-dimensional quantum system is specified by an orthonormal basis {w1 , w2 , . . . , wd } of the space. What happens: I the current state is expressed as a linear combination ψ = a1 v1 + a2 v2 + · · · + ad vd with |a1 |2 + |a2 |2 + · · · + |ad |2 = 1. I an oracle chooses j, 1 ≤ j ≤ d with probability |aj |2 I only the integer j is returned by the measurement I the state ψ collapses to vj Back to the introduction William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers “Collapse of the wave function” William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers States as Rank One Trace One Hermitian Matrices We may represent this as a rank one Hermitian matrix √ 2/5 6/5 ∗ √ Ψ = ψψ = 6/5 3/5 with trace one. Then, the probability of a measurement yielding basis state b is pb = b∗ Ψb. Back to the introduction William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Entanglement of Qubits entanglement (superposition of basis states in a multiparticle system) Suppose the basis states are 1 0 0 0 0 1 0 0 00 = 0 , 01 = 0 , 10 = 1 , 11 = 0 . 0 0 0 1 Then the collection of pure states includes, for example, 1 1 ψ = √ 00 − √ 11 2 2 in which the two qubits are “entangled”. (ψ is not expressible as a tensor product of two single-qubit state vectors). Back to the introduction William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Mixed States Physicists also allow probability distributions on pure states. Suppose the system is in state φ1 with probability p1 , in state φ2 with probability p2 , etc. Such a mixed state is prescribed by a pair of tuples [φ1 , φ2 , . . . , φk ] , [p1 , p2 , . . . , pk ]. Here, each pj is non-negative and p1 + p2 + · · · + pk = 1. A general mixed state is represented by its corresponding density matrix k X ρ= pj φj φ∗j j=1 a positive semidefinite Hermitian matrix of trace one. Back to the introduction William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Measuring Mixed States A measurement with respect to an orthonormal basis containing basis vector b returns b with probability pb = b∗ ρb = k X pj b∗ φj φ∗j b = j=1 k X pj |hb, φj i|2 j=1 which, intuitively, can be thought of as a two-stage probabilistic event. Back to the introduction William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers Two notes about mixed states I The mixed state described by [φ1 , φ2 , . . . , φk ] , [p1 , p2 , . . . , pk ] P is distinct from the pure state j pj φj . For example, the √ √ density matrix for pure state 0.40 + 0.61 is √ 6/5 √2/5 6/5 3/5 while the mixed state which is equal to 0 with probability 0.4 and equal to 1 with probability 0.6 has a density matrix with zeros off the diagonal. I Two mixed states are indistinguishable if and only if they have the same density matrix. So we may as well identify (mixed) quantum states with positive semidefinite Hermitian matrices with trace one. Quantum Information Theory Back to the introduction William J. Martin