Some Problems in Quantum Information Theory

Complex MUBs SIC-POVMs Problems over the Real Numbers

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 Find d2, if possible. d I Find, as many as you can, equiangular lines in R (unit vectors whose inner products have constant abs value). d+1 Find 2 , if possible. d I Find, as many as you can, orthonormal bases in C where unit vectors from distinct bases have inner products with constant modulus. Find d + 1, if possible. d I Find, as many as you can, orthonormal bases in R where unit vectors from distinct bases have inner products with constant d absolute value. Find 2 + 1, if possible.

Complex MUBs SIC-POVMs Problems over the Real Numbers

The Whole Talk in One Slide

You were born to solve these problems :

d I Find, as many as you can, equiangular lines in C (unit vectors whose inner products have constant modulus).

William J. Martin Quantum Information Theory d I Find, as many as you can, equiangular lines in R (unit vectors whose inner products have constant abs value). d+1 Find 2 , if possible. d I Find, as many as you can, orthonormal bases in C where unit vectors from distinct bases have inner products with constant modulus. Find d + 1, if possible. d I Find, as many as you can, orthonormal bases in R where unit vectors from distinct bases have inner products with constant d absolute value. Find 2 + 1, if possible.

Complex MUBs SIC-POVMs Problems over the Real Numbers

The Whole Talk in One Slide

You were born to solve these problems :

d I Find, as many as you can, equiangular lines in C (unit vectors whose inner products have constant modulus). Find d2, if possible.

William J. Martin Quantum Information Theory d+1 Find 2 , if possible. d I Find, as many as you can, orthonormal bases in C where unit vectors from distinct bases have inner products with constant modulus. Find d + 1, if possible. d I Find, as many as you can, orthonormal bases in R where unit vectors from distinct bases have inner products with constant d absolute value. Find 2 + 1, if possible.

Complex MUBs SIC-POVMs Problems over the Real Numbers

The Whole Talk in One Slide

You were born to solve these problems :

William J. Martin Quantum Information Theory d I Find, as many as you can, orthonormal bases in C where unit vectors from distinct bases have inner products with constant modulus. Find d + 1, if possible. d I Find, as many as you can, orthonormal bases in R where unit vectors from distinct bases have inner products with constant d absolute value. Find 2 + 1, if possible.

Complex MUBs SIC-POVMs Problems over the Real Numbers

The Whole Talk in One Slide

You were born to solve these problems :

William J. Martin Quantum Information Theory d I Find, as many as you can, orthonormal bases in R where unit vectors from distinct bases have inner products with constant d absolute value. Find 2 + 1, if possible.

Complex MUBs SIC-POVMs Problems over the Real Numbers

The Whole Talk in One Slide

You were born to solve these problems :

d I Find, as many as you can, equiangular lines in C (unit vectors whose inner products have constant modulus). Find d2, if possible. d I Find, as many as you can, equiangular lines in R (unit vectors whose inner products have constant abs value). d+1 Find 2 , if possible. d I Find, as many as you can, orthonormal bases in C 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 :

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Uncertainty Principle

Exact knowledge of one diminishes knowledge of the other

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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”.

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 appendix

William J. Martin Quantum Information Theory A mixed state is encoded as a positive semideﬁnite Hermitian matrix with trace one.

k X ∗ ρ = pj vj vj j=1

(speciﬁed by d2 − 1 real parameters)

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

I basis states 1, 2,..., d d I pure state (superposition) v ∈ C 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 Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

I basis states 1, 2,..., d d I pure state (superposition) v ∈ C 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.

k X ∗ ρ = pj vj vj j=1

(speciﬁed by d2 − 1 real parameters)

William J. Martin Quantum Information Theory (1) the state is expressed as a linear combination

ψ = p1v1 + p2v2 + ··· + pnvn

P 2 we assume ψ is a unit vector so that j |pj | = 1. 2 (2) an index j is chosen at random with probability |pj | (3) our measurement returns only the index j

(4) and the state collapses to vj .

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

In the simplest model, a measurement is an orthogonal n decomposition of C . 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 P 2 we assume ψ is a unit vector so that j |pj | = 1. 2 (2) an index j is chosen at random with probability |pj | (3) our measurement returns only the index j

(4) and the state collapses to vj .

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

In the simplest model, a measurement is an orthogonal n decomposition of C . 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 = p1v1 + p2v2 + ··· + pnvn

William J. Martin Quantum Information Theory 2 (2) an index j is chosen at random with probability |pj | (3) our measurement returns only the index j

(4) and the state collapses to vj .

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

In the simplest model, a measurement is an orthogonal n decomposition of C . 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 = p1v1 + p2v2 + ··· + pnvn

P 2 we assume ψ is a unit vector so that j |pj | = 1.

William J. Martin Quantum Information Theory (3) our measurement returns only the index j

(4) and the state collapses to vj .

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

In the simplest model, a measurement is an orthogonal n decomposition of C . 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 = p1v1 + p2v2 + ··· + pnvn

P 2 we assume ψ is a unit vector so that j |pj | = 1. 2 (2) an index j is chosen at random with probability |pj |

William J. Martin Quantum Information Theory (4) and the state collapses to vj .

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

In the simplest model, a measurement is an orthogonal n decomposition of C . 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 = p1v1 + p2v2 + ··· + pnvn

P 2 we assume ψ is a unit vector so that j |pj | = 1. 2 (2) an index j is chosen at random with probability |pj | (3) our measurement returns only the index j

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges States and Measurements

In the simplest model, a measurement is an orthogonal n decomposition of C . 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 = p1v1 + p2v2 + ··· + pnvn

P 2 we assume ψ is a unit vector so that j |pj | = 1. 2 (2) an index j is chosen at random with probability |pj | (3) our measurement returns only the index j

(4) and the state collapses to vj .

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges The cost of clarity

”Truth and clarity are complementary.” - Niels Bohr

William J. Martin Quantum Information Theory √ 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)

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complex MUBs

0 d Two orthonormal bases B and B for C are unbiased if

|hu, vi| = α

for some constant α, for every u ∈ B and every v ∈ B0.

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complex MUBs

0 d Two orthonormal bases B and B for C 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 d Let M(d) denote the maximum size of a set of MUBs in C .

Theorem[Delsarte, Goethals, Seidel (1978)] (proven in terms of line systems)

M(d) ≤ d + 1

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complete sets of MUBs

d A collection {B1,..., BM } of orthonormal bases for C 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 Theorem[Delsarte, Goethals, Seidel (1978)] (proven in terms of line systems)

M(d) ≤ d + 1

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complete sets of MUBs

d A collection {B1,..., BM } of orthonormal bases for C is a collection/set of mutually unbiased bases if each pair is unbiased. We say we have a set of “k MUBs”.

d Let M(d) denote the maximum size of a set of MUBs in C .

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complete sets of MUBs

d A collection {B1,..., BM } of orthonormal bases for C is a collection/set of mutually unbiased bases if each pair is unbiased. We say we have a set of “k MUBs”.

d Let M(d) denote the maximum size of a set of MUBs in C .

Theorem[Delsarte, Goethals, Seidel (1978)] (proven in terms of line systems)

M(d) ≤ d + 1

William J. Martin Quantum Information Theory 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 √ ∗ I Since these are unitary, Hj Hj = I and each dHj is a complex Hadamard matrix

I so we seek mutually unbiased Hadamard matrices (“MUCH”, ∗ Best and Kharaghani, 2010): Hj Hk ﬂat ∀ j 6= k

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complex Hadamard matrices

Say a matrix is ﬂat if all entries have the same modulus.

William J. Martin Quantum Information Theory 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 √ ∗ I Since these are unitary, Hj Hj = I and each dHj is a complex Hadamard matrix

I so we seek mutually unbiased Hadamard matrices (“MUCH”, ∗ Best and Kharaghani, 2010): Hj Hk ﬂat ∀ j 6= k

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 I in this case, all the other√ matrices (bases) in a set of MUBs are ﬂat, with modulus 1/ d √ ∗ I Since these are unitary, Hj Hj = I and each dHj is a complex Hadamard matrix

I so we seek mutually unbiased Hadamard matrices (“MUCH”, ∗ Best and Kharaghani, 2010): Hj Hk ﬂat ∀ j 6= k

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 √ ∗ I Since these are unitary, Hj Hj = I and each dHj is a complex Hadamard matrix

I so we seek mutually unbiased Hadamard matrices (“MUCH”, ∗ Best and Kharaghani, 2010): Hj Hk ﬂat ∀ j 6= k

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 I so we seek mutually unbiased Hadamard matrices (“MUCH”, ∗ Best and Kharaghani, 2010): Hj Hk ﬂat ∀ j 6= k

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complex Hadamard matrices

Say a matrix is ﬂat if all entries have the same modulus.

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complex Hadamard matrices

Say a matrix is ﬂat if all entries have the same modulus.

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 Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 d2 − 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 Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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)

2 is a bijection. (E.g., for q odd f : Fq → Fq via f (x) = x .)

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 r planar. For q = p , Fq is a vector space over Zp (p prime); let ζ th Pp−1 j be a primitive complex p root of unity so that j=0 ζ = 0.

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Complete Sets of MUBs in Prime Power Dimensions

Roy-Scott (2007), Godsil-Roy (2009) d Write the standard basis for C (d prime) as {eb | b ∈ Fq}. Let B∞ = {eb | b ∈ F }. For f ∈ S, let ( ) 1 X a·b+f (b) Bf = √ ζ eb | a ∈ F . d b∈F

Then the collection {Bf : f ∈ S} ∪ {B∞} is a set of |S| + 1 d MUBs in C . 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 Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 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.

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges MUBs and planes

William J. Martin Quantum Information Theory 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.

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges MUBs and planes

I All known constructions of complete sets are somehow related to aﬃne planes

William J. Martin Quantum Information Theory I Yet this does not mean that all complete sets of MUBs must come from planes.

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 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?

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges What is needed?

William J. Martin Quantum Information Theory 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?

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges What is needed?

I Ideas!

William J. Martin Quantum Information Theory 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?

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges What is needed?

I Ideas!

I New constructions — particularly when d is not a prime power

William J. Martin Quantum Information Theory 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?

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 I Is there any combinatorics in the inner products? A coherent conﬁguration?

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers 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 2 I Beth-Wocjan construction (2004): If d = s and there exist k MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d, we obtain M(d) ≥ d1/29.6 (Beats MacNeish lower bound inﬁnitely often)

I Kharaghani and co-authors have found mutually unbiased Hadamard matrices in various small dimensions

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Examples of non-complete sets

I Analogue of MacNeish construction: if d has prime r1 r2 rk factorization d = p1 p2 ··· pk , then

r1 r2 rk M(d) ≥ min{M(p1 ), M(p2 ),..., M(pk )}. via taking tensor products of this many MUBs in each prime-power dimension

William J. Martin Quantum Information Theory (Beats MacNeish lower bound inﬁnitely often)

I Kharaghani and co-authors have found mutually unbiased Hadamard matrices in various small dimensions

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Examples of non-complete sets

I Analogue of MacNeish construction: if d has prime r1 r2 rk factorization d = p1 p2 ··· pk , then

r1 r2 rk M(d) ≥ min{M(p1 ), M(p2 ),..., M(pk )}. via taking tensor products of this many MUBs in each prime-power dimension 2 I Beth-Wocjan construction (2004): If d = s and there exist k MOLS(s) then M(d) ≥ k + 2. So, for square dimensions d, we obtain M(d) ≥ d1/29.6

William J. Martin Quantum Information Theory I Kharaghani and co-authors have found mutually unbiased Hadamard matrices in various small dimensions

Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Examples of non-complete sets

I Analogue of MacNeish construction: if d has prime r1 r2 rk factorization d = p1 p2 ··· pk , then

William J. Martin Quantum Information Theory Just a bit of physics Complex MUBs Basics of MUBs SIC-POVMs MUB Constructions Problems over the Real Numbers MUB Challenges Examples of non-complete sets

I Analogue of MacNeish construction: if d has prime r1 r2 rk factorization d = p1 p2 ··· pk , then

I Kharaghani and co-authors have found mutually unbiased Hadamard matrices in various small dimensions

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

SIC-POVMs

A set of d + 1 MUBs give us d + 1 measurements that together estimate d2 − 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Complex Equiangular Lines

d In each complex vector space C , 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.

2 Example: In C , the lines spanned by

(τ, i), (τ, −i), (1, iτ), (1, −iτ) √ 1+ 3 are equiangular for τ = 2 (1 + i).

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

SIC-POVMs

Physicists are interested in equiangular line sets of size exactly d2: these are called symmetric informationally complete positive operator-valued measures (“SIC-POVMs”)

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Why are these measurements?

Why are physicists interested? In the simplest model, a measurement is an orthogonal n decomposition of C . We identify the measurement with an orthonormal basis {v1,..., vn}. As we said before, the state is expressed as a linear combination

ψ = p1v1 + p2v2 + ··· + pnvn

P 2 where j |pj | = 1. When we perform this measurement, an index 2 j is chosen at random with probability |pj | and the state collapses to vj .

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Von Neumann Measurements

This measurement is represented by the Hermitian matrix P ∗ 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Special POVMs

∗ Symmetric: Constant inner product Tr(EhEj ) over all h 6= j. ∗ ∗ But Eh = vhvh, Ej = vj vj gives

∗ ∗ ∗ ∗ ∗ 2 Tr(EhEj ) = Tr(vhvhvj vj ) = Tr(vj vhvhvj ) = |hvh, vj i| .

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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 d2 − 1 real parameters. Our POVM has probabilities summing to one, so we need d2 outcomes in order to be “informationally complete”.

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Some Nice Constructions

d Let N(d) denote the max size of a set of equiangular lines in C . Theorem[Delsarte, Goethals, Seidel (1978)]

N(d) ≤ d2

2 d Here are the general techniques to obtain exactly d lines in C :

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Three Constructions of Exactly d 2 Lines

8 I S. Hoggar (1981,1998): 64 lines in C 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Some Nice Constructions of Less Than d 2 Lines

I K¨onig(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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Sets of Quadratic Size

Recall that the upper bound is N(d) ≤ d2

I The K¨onigconstruction gives

N(d) ≥ d2 − d + 1

for d = pr + 1 (p prime)

I Converting MUBs to equiangular lines (Greaves, et al., Jedwab & Wiebe) gives

N(d) = Θ(d2)

for many other dimensions d.

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 ab χa(b) = ω 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, 2−4−5, 4−1 = 3, 4−2 = 2.

William J. Martin Quantum Information Theory 0 1 2 3 4 5 6 2 3 4 5 6 χ1 = [ 1 ω ω ω ω ω ω ] 2 4 6 3 5 χ2 = [ 1 ω ω ω ω ω ω ] 4 5 2 6 3 χ4 = [ 1 ω ω ω ω ω ω ] 3 This gives us a conﬁguration of 7 vectors in C

X = (1, 1, 1), (ω, ω2, ω4), (ω2, ω4, ω), (ω3, ω6, ω5), (ω4, ω, ω2), (ω5, ω3, ω6), (ω6, ω5, ω3) which span 7 equiangular lines.

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 = e2πi/7

William J. Martin Quantum Information Theory 3 This gives us a conﬁguration of 7 vectors in C

X = (1, 1, 1), (ω, ω2, ω4), (ω2, ω4, ω), (ω3, ω6, ω5), (ω4, ω, ω2), (ω5, ω3, ω6), (ω6, ω5, ω3) which span 7 equiangular lines.

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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:

0 1 2 3 4 5 6 2 3 4 5 6 2πi/7 χ1 = [ 1 ω ω ω ω ω ω ] With ω = e 2 4 6 3 5 χ2 = [ 1 ω ω ω ω ω ω ] 4 5 2 6 3 χ4 = [ 1 ω ω ω ω ω ω ]

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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:

0 1 2 3 4 5 6 2 3 4 5 6 2πi/7 χ1 = [ 1 ω ω ω ω ω ω ] With ω = e 2 4 6 3 5 χ2 = [ 1 ω ω ω ω ω ω ] 4 5 2 6 3 χ4 = [ 1 ω ω ω ω ω ω ] 3 This gives us a conﬁguration of 7 vectors in C

X = (1, 1, 1), (ω, ω2, ω4), (ω2, ω4, ω), (ω3, ω6, ω5), (ω4, ω, ω2), (ω5, ω3, ω6), (ω6, ω5, ω3) which span 7 equiangular lines.

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 d2 and forms a SIC-POVM.

William J. Martin Quantum Information Theory So what’s holding us up?

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 Gr¨obner basis techniques) in the following dimensions

d = 2, 3,..., 16, 19, 24, 28, 35, 48

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 Gr¨obner 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 (with a 206-page supplement).

10 Fiducial vector for numerical example in C .

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Zauner’s Conjecture

Grassl and Scott summarize these solutions in a 16-page paper

William J. Martin Quantum Information Theory 10 Fiducial vector for numerical example in C .

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Zauner’s Conjecture

Grassl and Scott summarize these solutions in a 16-page paper (with a 206-page supplement).

10 Fiducial vector for numerical example in C .

William J. Martin Quantum Information Theory 10 Fiducial vector for analytic example in C requires some preliminary deﬁnitions of constants.

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Zauner’s Conjecture

Second part of Grassl-Scott supplement gives exact solutions.

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Zauner’s Conjecture

Second part of Grassl-Scott supplement gives exact solutions.

10 Fiducial vector for analytic example in C requires some preliminary deﬁnitions of constants.

William J. Martin Quantum Information Theory 10 Here is the ﬁrst entry of the ﬁducial vector for the example in C .

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Zauner’s Conjecture Second part of Grassl-Scott supplement gives exact solutions.

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Zauner’s Conjecture Second part of Grassl-Scott supplement gives exact solutions.

10 Here is the ﬁrst entry of the ﬁducial vector for the example in C .

William J. Martin Quantum Information Theory 3 Then the 9 rows of the following matrices form a SIC-POVM in C .  −2 1 1   1 −2 1   1 1 −2   −2 ω ω2  ,  1 −2ω ω2  ,  1 ω −2ω2  −2ω 1 ω2 ω −2 ω2 ω 1 −2ω2

Good News! This works again in dimensions 2 and 8. Bad News. That’s it for this technique, in this form at least.

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e2πi/3, consider the complex Hadamard matrix

 1 1 1  H =  1 ω ω2  . ω 1 ω2

William J. Martin Quantum Information Theory Good News! This works again in dimensions 2 and 8. Bad News. That’s it for this technique, in this form at least.

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e2πi/3, consider the complex Hadamard matrix

 1 1 1  H =  1 ω ω2  . ω 1 ω2

3 Then the 9 rows of the following matrices form a SIC-POVM in C .  −2 1 1   1 −2 1   1 1 −2   −2 ω ω2  ,  1 −2ω ω2  ,  1 ω −2ω2  −2ω 1 ω2 ω −2 ω2 ω 1 −2ω2

William J. Martin Quantum Information Theory Bad News. That’s it for this technique, in this form at least.

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e2πi/3, consider the complex Hadamard matrix

 1 1 1  H =  1 ω ω2  . ω 1 ω2

Good News! This works again in dimensions 2 and 8.

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers Equiangular Tight Frames

Nine Lines in C3 Jedwab & Wiebe (2014): With ω = e2πi/3, consider the complex Hadamard matrix

 1 1 1  H =  1 ω ω2  . ω 1 ω2

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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 d2 lines. But even for d2 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 d precisely the problem of finding equiangular lines in C ; 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 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”)

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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.

d If we have a set X of N unit vectors in C with maximum norm on all pairwise inner products, then s N − d ≥ . d(N − 1)

William J. Martin Quantum Information Theory Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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.

d If we have a set X of N unit vectors in C 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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.

d If we have a set X of N unit vectors in C 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 Note also: • orthonormal basis has N = d • regular simplex has N = d + 1

Complex MUBs Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 = ps + 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 Deﬁnition and physics SIC-POVMs Constructions Problems over the Real Numbers 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 = ps + 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) d d t I known extremal examples of 2 + 1 MUBs in R (d = 4 ) come from Kerdock sets d+1 d I examples of 2 equiangular lines in R only known for d = 2, 3, 7, 23

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! :

William J. Martin Quantum Information Theory Complex MUBs SIC-POVMs Problems over the Real Numbers

Quantum States

pure state (superposition) √ √ √ 1 2 ψ = 0.40 + 0.61 = √ √ 5 3 2 2 with amplitudes a1, a2 satisfying |a1| + |a2| = 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 = a1v1 + a2v2 + ··· + ad vd

2 2 2 with |a1| + |a2| + ··· + |ad | = 1. 2 I an oracle chooses j, 1 ≤ j ≤ d with probability |aj | 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 =   , 01 =   , 10 =   , 11 =   .  0   0   1   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 φj 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

k k ∗ X ∗ ∗ X 2 pb = b ρb = pj b φj φj b = pj |hb, φj i| j=1 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 ] is distinct from the pure state P p φ . For example, the √ j j j√ density matrix for pure state 0.40 + 0.61 is √ √2/5 6/5 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. Back to the introduction William J. Martin Quantum Information Theory