Entropic uncertainty relations for anticommuting observables

1
Entropic uncertainty relations for anti-commuting
observables

• Stephanie Wehner
• Joint work with Andreas Winter

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Outline

• Uncertainty Relations
• 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

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The setting
1
PX(1)
2
½
PX(2)

Xk
PX(k)
k
Yk
Shannon entropy H(PX) - ?j2 X PX(j) log PX(j)
Collision entropy H2 (PX) - log ? j2 X PX (j)2
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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
  CMP75, Deutsch PRL83)
• Only depend on chosen properties!
• Goal for any state, give a lower bound for

?
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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

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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)

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Outline

• Uncertainty Relations
• 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

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Known results 2 measurements

• Two measurements (MaassenUffink, PRL89)
• A a1gt,,adgt and B b1gt,,bdgt
• ½ (H(A½) H(B½)) - log c(A,B)
• c(A,B) maxltabgt agt 2 A, bgt 2 B
• H(A½) -? Tr(ajgtltaj½) log Tr(ajgtltaj½)
• Maximal if overlaps are all equal and

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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.

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Known results many measurements

• All mutually unbiased bases (Sanchez,PLA 93)
• Random (Hayden,Leung,Shor,Winter,CMP04)
• How about more than two measurements?
• Maybe chosing them to be mutually unbiased is
  good enough?
• NO! (Ballester, W, PRA07)

For d2 2/3
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Goal

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

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Outline

• Uncertainty Relations
• 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

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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

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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 (d2)
• 1 X, 2 Z
• Represented N2n 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!

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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!

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Outline

• Uncertainty Relations
• 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

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Basis for dxd matrices

• Generators also give a basis for matrices with

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Basis rotations on matrices

• Given a matrix
• .. then basis elements transform as vectors

Can extend this to include the final term! 2n1
dimensional real vector space!
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Tool 1 Rotation

• Given a matrix
• Can rotate ? gj j onto 1
• R½Ry (1/d)(I v(?i gi 2i) 1 ?j lt k gjk j
  k.)

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Tool 2 Sign flips

• Given a matrix
• ½ (1/d)(I v(?i gi 2i) 1 gjk j
  k .)
• Can find unitary Fj which maps
• j ! -j
• k ! k for k ? j
• Fj ½ Fj y (1/d)(I v(?i gi 2i) 1 - gjk j
  k.)

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When is a matrix ? a valid state?

• To minimize over states ?, we often want to
  parametrize them
• But what conditions do the coefficients need to
  satisfy such that ? is a state?

OK!
???
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The Bloch sphere d2

When is ½ a quantum state? (½ 0)
Difficult for d gt 2!
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Generalized Bloch sphere

• Given state ½ (1/d) (I ?i gi i ?i,k gik
  ik )
• Rotate onto 1
• R ½ Ry (1/d) (I v(?i gi 2i) 1 ?i,k gik 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 v(?i gi 2) 1)
• Must have I (v?i gi 2) 1 -I, hence ?i gi 2
  1

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Generalized Bloch sphere

• For any quantum state ½ in d2n and K 2n1
  observables j

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Outline

• Uncertainty Relations
• 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

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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

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Why may this be a good choice?

• Similarly unbiased
• Observing 0 or 1 has equal probability
  when measuring with a different observable.

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Collision entropy

• For d2n and K 2n1 observables j

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Outline

• Uncertainty Relations
• 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

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Shannon entropy

• For d2n and K 2n1 observables j
• Rewrite as before and use meta-uncertainty
  relation
• Maximally strong Maximum uncertainty for all
  measurements except one!

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Outline

• Uncertainty Relations
• 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

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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 xgtb 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 d2 follow as a special case!
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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.

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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