Entropic uncertainty relations for anti-commuting observables Stephanie Wehner Joint work with Andreas Winter

Outline Uncertainty Relations A meta-uncertainty relation What are they? What is known? A meta-uncertainty relation Clifford algebra Characterizing quantum states Strong uncertainty relations .. for the collision entropy …for the Shannon entropy Relation to BB84 Open questions

The setting ½ X=[k] Y=[k] 1 PX(1) 2 ½ PX(2) : X=[k] PX(k) k Y=[k] In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more "uncertain" we might say that that characteristic is for the system. The Heisenberg uncertainty principle gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously. More precisely, the product of the standard deviations , where is the reduced Planck constant. The principle generalizes to many other pairs of quantities besides position and momentum (for example, angular momentum about two different axes), and can be derived directly from the axioms of quantum mechanics. It has often been regarded as the most distinctive feature in which quantum mechanics differs from classical theories of the physical world. Roughly speaking, the uncertainty principle (for position and momentum) states that one cannot assign exact simultaneous values to the position and momentum of a physical system. Rather, these quantities can only be determined with some characteristic ‘uncertainties’ that cannot become arbitrarily small simultaneously. But what is the exact meaning of this principle, and indeed, is it really a principle of quantum mechanics Shannon entropy H(PX) = - j2 X PX(j) log PX(j) Collision entropy H2 (PX) = - log  j2 X PX (j)2

What are uncertainty relations? Each measurement Set of possible outcomes X Measuring a state: distribution PX over X Historic uncertainty relations bound variance And depend on state to be measured! Entropic uncertainty relations (Byalynicki-Birula, Mycielski CMP’75, Deutsch PRL’83) Only depend on chosen properties! Goal: for any state, give a lower bound for ?

Entropic uncertainty relations Example for two outcomes X = {0,1} and Y = {0,1} No entropy for one, means maximal entropy for the other! Maximally incompatible! But, what characterizes incompatible measurements? Find such measurements Minimize over states

So, why they are interesting? Physics Quantum cryptography Quantum key distribution Quantum cryptography in the bounded storage model: security based on uncertainty relations! Non-local games (interactive proof systems with entanglement)

Outline Uncertainty Relations A meta-uncertainty relation What are they? What is known? A meta-uncertainty relation Clifford algebra Characterizing quantum states Strong uncertainty relations .. for the collision entropy …for the Shannon entropy Relation to BB84 Open questions

Known results: 2 measurements Two measurements (Maassen&Uffink, PRL’89) A = {|a1>,…,|ad>} and B = {|b1>,…,|bd>} ½ (H(A|½) + H(B|½)) ¸ - log c(A,B) c(A,B) = max{|<a|b>| | |a> 2 A, |b> 2 B} H(A|½) = - Tr(|aj><aj|½) log Tr(|aj><aj|½) Maximal if overlaps are all equal and

Known results: 2 measurements Maximized if the two bases are mutually unbiased Intuition: If we measure any element of one basis in the other, all outcomes are equally likely.

Known results: many measurements All mutually unbiased bases (Sanchez,PLA ‘93) Random (Hayden,Leung,Shor,Winter,CMP’04) How about more than two measurements? Maybe chosing them to be mutually unbiased is good enough? NO! (Ballester, W, PRA’07) For d=2: … ¸ 2/3

Goal What characterizes maximally incompatible measurements? Find measurements for which we get strong uncertainty relations Here: for two outcome measurements Need anti-commutation!

Outline Uncertainty Relations A meta-uncertainty relation What are they? What is known? A meta-uncertainty relation Clifford algebra Characterizing quantum states Strong uncertainty relations .. for the collision entropy …for the Shannon entropy Relation to BB84 Open questions

What is Clifford algebra? Algebra generated by Generators ¡1,…,¡N For all pairs: {¡i,¡j} = ¡i ¡j + ¡j ¡i = 0 For each: ¡i 2 = I View ¡1,..,¡N as basis vectors for an N-dimensional real vector space. Vector a = (a1,…,aN) = j aj ¡j

Matrix representation For j = 1,…,n ¡2j = Y­ (j-1)­ Z ­ I­ (n-j) ¡2j-1 = Y­ (j-1) ­ X ­ I­ (n-j) For =2 (d=2) ¡1 = X, ¡2 = Z Represented N=2n basis vectors with 2n £ 2n matrices To recall: a 2 R2n is a vector: a =  aj ¡j a is also a 2n £ 2n matrix Two matrices anti-commute iff vectors are orthogonal!

Rotating the basis If the original vectors generated a Clifford algebra, so do the new ones: inner products are preserved ..and thus all anti-commutation relations!

Outline Uncertainty Relations A meta-uncertainty relation What are they? What is known? A meta-uncertainty relation Clifford algebra Characterizing quantum states Strong uncertainty relations .. for the collision entropy …for the Shannon entropy Relation to BB84 Open questions

Basis for dxd matrices Generators also give a basis for matrices with

Basis rotations on matrices Given a matrix .. then basis elements transform as vectors: Transform 1st grade as vector Can implement it as a unitary Sketch of argument Can extend this to include the final term! 2n+1 dimensional real vector space!

Tool 1: Rotation Given a matrix Can rotate  gj ¡j onto ¡1 R½Ry = (1/d)(I + √(i gi 2i) ¡1 + j < k g’jk ¡ j ¡k+….) Transform 1st grade as vector Can implement it as a unitary Sketch of argument

Tool 2: Sign flips Given a matrix ½= (1/d)(I + √(i gi 2i) ¡1 + g’jk ¡ j ¡k+ ….) Can find unitary Fj which maps ¡j ! -¡j ¡k ! ¡k for k  j Fj ½ Fj y = (1/d)(I + √(i gi 2i) ¡1 - g’jk ¡ j ¡k+….) Transform 1st grade as vector Can implement it as a unitary Sketch of argument

When is a matrix ρ a valid state? To minimize over states ρ, we often want to parametrize them But what conditions do the coefficients need to satisfy such that ρ is a state? Minimize over states! So how do we parametrize them? ??? OK!

The Bloch sphere: d=2 Difficult for d > 2! When is ½ a quantum state? (½ ¸ 0) Difficult for d > 2!

Generalized Bloch sphere Given state ½ = (1/d) (I + i gi ¡i + i,k gik ¡i¡k + …) Rotate onto ¡1: R ½ Ry = (1/d) (I + √(i gi 2i) ¡1 + i,k g’ik ¡i ¡k + …) Dephase: Fj ¡j Fjy = -¡j and Fj ¡k Fjy = ¡k Dj = (1/2) (½ + Fj ½ Fjy) Iterate for all j 2 {2,…,N} Left with ½’= (1/d) (I + √(i gi 2) ¡1) Must have I ¸ (√i gi 2) ¡1 ¸ -I, hence i gi 2 · 1

Generalized Bloch sphere For any quantum state ½ in d=2n and K· 2n+1 observables ¡j

Outline Uncertainty Relations A meta-uncertainty relation What are they? What is known? A meta-uncertainty relation Clifford algebra Characterizing quantum states Strong uncertainty relations .. for the collision entropy …for the Shannon entropy Relation to BB84 Open questions

Matrix representation Measurements we will consider! Algebra generated by Generators ¡1,…,¡N For all pairs: {¡i,¡j} = ¡i ¡j + ¡j ¡i = 0 For each: ¡i 2 = I Two eigenvalues +/- 1: ¡j = ¡j 0 - ¡j 1

Why may this be a good choice? Similarly ‘’unbiased’’: Observing ‘’0’’ or ‘’1’’ has equal probability when measuring with a different observable.

Collision entropy For d=2n and K · 2n+1 observables ¡j:

Outline Uncertainty Relations A meta-uncertainty relation What are they? What is known? A meta-uncertainty relation Clifford algebra Characterizing quantum states Strong uncertainty relations .. for the collision entropy …for the Shannon entropy Relation to BB84 Open questions

Shannon entropy For d=2n and K· 2n+1 observables ¡j: Rewrite as before and use ‘’meta’’-uncertainty relation Maximally strong: Maximum uncertainty for all measurements except one!

Outline Uncertainty Relations A meta-uncertainty relation What are they? What is known? A meta-uncertainty relation Clifford algebra Characterizing quantum states Strong uncertainty relations .. for the collision entropy …for the Shannon entropy Relation to BB84 Open questions

Relation to BB84 Encode a bit into the positive or negative eigenspace of these operators. Choose operator at random Task: determine bit without knowing operator “BB84 scenario”: Alice picks x 2 R {0,1} and b 2 R {+,£} Sends |x>b to Bob Extension: Alice picks x 2 R {0,1}, and b 2 R {¡1,…¡N} Sends ¡b x to Bob Previous uncertainty relations for MUBs (BB84 states) in d=2 follow as a special case!

Applications Could implement 1-k oblivious transfer protocols in the bounded storage model. (Damgard, Fehr, Renner, Salvail, Schaffner, CRYPTO 07) Same for the noisy storage model (but no direct proof) Relatively easy to implement in practice.

Open questions For two-outcome measurements, anti-commutation is enough! Optimal uncertainty relations for Shannon entropy Near optimal uncertainty relations for collision entropy Strong uncertainty relations for more then two outcome measurements? Other applications? Small state within a large state! Non-local games: need the same operators to implement quantum XOR-games