Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics

Correlations in Measurement-Based Quantum Computing and Bell Inequalities Browne, Daniel 2010

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Bell inequalities  for (Measurement Based Quantum) Computation s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  Dan Browne  Dept. of Physics and Astronomy University College London Sunday, 1 August 2010  (MeasurementBased) Quantum Computation  Quantum Foundations  Sunday, 1 August 2010  Measurement-based quantum computation  Sunday, 1 August 2010  Measurement-based quantum computation  • is a model of universal quantum computation.  R. Raussendorf and H.J. Briegel, “A one-way quantum computer”, PRL 2000. Sunday, 1 August 2010  Measurement-based quantum computation  • is a model of universal quantum computation. • 1. Create a special multi-qubit entangled state  R. Raussendorf and H.J. Briegel, “A one-way quantum computer”, PRL 2000. Sunday, 1 August 2010  Measurement-based quantum computation  • is a model of universal quantum computation. • 1. Create a special multi-qubit entangled state • e.g. A cluster state  R. Raussendorf and H.J. Briegel, “A one-way quantum computer”, PRL 2000. Sunday, 1 August 2010  Measurement-based quantum computation  • is a model of universal quantum computation. • 1. Create a special multi-qubit entangled state • e.g. A cluster state • 2. Then measure qubits individually, in certain (adaptive) bases, and post-process outcomes.  R. Raussendorf and H.J. Briegel, “A one-way quantum computer”, PRL 2000. Sunday, 1 August 2010  Measurement-based quantum computation  • is a model of universal quantum computation. • 1. Create a special multi-qubit entangled state • e.g. A cluster state • 2. Then measure qubits individually, in certain (adaptive) bases, and post-process outcomes.  • Via a suitable choice of measurements, any quantum computation can be acheived.  R. Raussendorf and H.J. Briegel, “A one-way quantum computer”, PRL 2000. Sunday, 1 August 2010  Measurement-based quantum computation Prepare entangled resource state  Measure a sub-set of qubits  e.g. cluster state Process measurement results  Choose bases for next subset of measurements  Computational Output  Requires classical “side-computation” Sunday, 1 August 2010  Measurement-based quantum computation sj ∈ {0, 1} cos θj X + (−1)sj sin θj Y +1 → 0  −1→1  mj ∈ {0, 1}  R. Raussendorf, D. E. Browne and H.J. Briegel, PRA (2003). Sunday, 1 August 2010  Measurement-based quantum computation  •  For a cluster state resource it suffices for side-computation to be linear.  •  Each measurement is of the form  sj ∈ {0, 1}  cos θj X + (−1)sj sin θj Y and meas. outcomes are relabelled  +1 → 0  −1→1  mj ∈ {0, 1}  R. Raussendorf, D. E. Browne and H.J. Briegel, PRA (2003). Sunday, 1 August 2010  Measurement-based quantum computation  •  For a cluster state resource it suffices for side-computation to be linear.  •  Each measurement is of the form  cos θj X + (−1)sj sin θj Y and meas. outcomes are relabelled  •  sj ∈ {0, 1}  +1 → 0  −1→1  mj ∈ {0, 1}  The angle θj is pre-set and differs for each measurement.  R. Raussendorf, D. E. Browne and H.J. Briegel, PRA (2003). Sunday, 1 August 2010  Measurement-based quantum computation  •  For a cluster state resource it suffices for side-computation to be linear.  •  Each measurement is of the form  cos θj X + (−1)sj sin θj Y and meas. outcomes are relabelled  • •  sj ∈ {0, 1}  +1 → 0  −1→1  mj ∈ {0, 1}  The angle θj is pre-set and differs for each measurement. But bit-value sj is calculated on the fly - and set equal to the parity of a sub-set of previous measurement outcomes.  R. Raussendorf, D. E. Browne and H.J. Briegel, PRA (2003). Sunday, 1 August 2010  Measurement-based quantum computation  •  For a cluster state resource it suffices for side-computation to be linear.  •  Each measurement is of the form  sj ∈ {0, 1}  cos θj X + (−1)sj sin θj Y and meas. outcomes are relabelled  +1 → 0  −1→1  mj ∈ {0, 1}  • •  The angle θj is pre-set and differs for each measurement.  •  Final output bits are encoded in the parity of a sub-set of the measurement outcomes.  But bit-value sj is calculated on the fly - and set equal to the parity of a sub-set of previous measurement outcomes.  R. Raussendorf, D. E. Browne and H.J. Briegel, PRA (2003). Sunday, 1 August 2010  Measurement-based quantum computation Prepare entangled resource state  Measure a sub-set of qubits  e.g. cluster state Process measurement results  Choose bases for next subset of measurements  Computational Output  Requires (linear / XOR) classical “side-computation” Sunday, 1 August 2010  Bell inequalities  Sunday, 1 August 2010  Bell inequalities  Sunday, 1 August 2010  Bell inequalities  • Bell inequalities (BIs) express restrictions on the joint  probability distributions for spatially separated measurements in local hidden variable (LHV) theories.  Sunday, 1 August 2010  CHSH inequality Setting:  s1 ∈ {0, 1}  s2 ∈ {0, 1}  Outcome:  m1 ∈ {0, 1}  m2 ∈ {0, 1}  • We focus on the parity of the outcomes and define: Es1 ,s2 = p(m1 ⊕ m2 = 0|s1 , s2 ) − p(m1 ⊕ m2 = 1|s1 , s2 )  Sunday, 1 August 2010  CHSH inequality Setting:  s1 ∈ {0, 1}  s2 ∈ {0, 1}  Outcome:  m1 ∈ {0, 1}  m2 ∈ {0, 1}  • We focus on the parity of the outcomes and define: Es1 ,s2 = p(m1 ⊕ m2 = 0|s1 , s2 ) − p(m1 ⊕ m2 = 1|s1 , s2 )  • and show that for correlations in any LHV theory: Sunday, 1 August 2010  E0,0 + E0,1 + E1,0 − E1,1 ≤ 2  CHSH inequality • With entangled quantum state, Alice and Bob can  violate this inequality, although not exceeding Tsirelson’s bound: √  E0,0 + E0,1 + E1,0 − E1,1 ≤ 2 2  Sunday, 1 August 2010  CHSH inequality • With entangled quantum state, Alice and Bob can  violate this inequality, although not exceeding Tsirelson’s bound: √  E0,0 + E0,1 + E1,0 − E1,1 ≤ 2 2  • A (loophole-free) demonstration of a Bell Inequality  violation would refute local hidden variable theories.  Sunday, 1 August 2010  CHSH inequality • With entangled quantum state, Alice and Bob can  violate this inequality, although not exceeding Tsirelson’s bound: √  E0,0 + E0,1 + E1,0 − E1,1 ≤ 2 2  • A (loophole-free) demonstration of a Bell Inequality  violation would refute local hidden variable theories.  • The maximal violation (stronger than QM) is achieved by the Popescu-Rohrlich (PR) Box, which acheives E0,0 + E0,1 + E1,0 − E1,1 = 4 B. S. Tsirelson, Lett. Math. Phys. (1980). S. Popescu and D. Rohrlich, Found. Phys. (1994) Sunday, 1 August 2010  CHSH inequality E0,0 + E0,1 + E1,0 − E1,1 ≤ 2  • It is useful to re-express the CHSH Inequality directly in terms of conditional probabilities.  Es1 ,s2 = p(m1 ⊕ m2 = 0|s1 , s2 ) − p(m1 ⊕ m2 = 1|s1 , s2 )  Sunday, 1 August 2010  CHSH inequality E0,0 + E0,1 + E1,0 − E1,1 ≤ 2  • It is useful to re-express the CHSH Inequality directly in terms of conditional probabilities.  Es1 ,s2 = p(m1 ⊕ m2 = 0|s1 , s2 ) − p(m1 ⊕ m2 = 1|s1 , s2 )  • Some simple algebra gives us a very neat representation [1] of the CHSH inequality:  3 1 � p(m1 ⊕ m2 = s1 s2 |s1 s2 ) ≤ 4 s ,s 4 1  2  [1] QIP Folklore: earliest reference I know: Wim Van Dam, PHD Thesis (2000) Sunday, 1 August 2010  CHSH inequality E0,0 + E0,1 + E1,0 − E1,1 ≤ 2  • It is useful to re-express the CHSH Inequality directly in terms of conditional probabilities.  Es1 ,s2 = p(m1 ⊕ m2 = 0|s1 , s2 ) − p(m1 ⊕ m2 = 1|s1 , s2 )  • Some simple algebra gives us a very neat representation [1] of the CHSH inequality:  3 1 � p(m1 ⊕ m2 = s1 s2 |s1 s2 ) ≤ 4 s ,s 4 1  2  • Thus we can phrase the CHSH inequality in terms of a game. [1] QIP Folklore: earliest reference I know: Wim Van Dam, PHD Thesis (2000) Sunday, 1 August 2010  CHSH game  • Rules: • Alice, Bob are given independent bits s , s 1  2  from a  uniform distribution.  • They may not communicate during the game.  • Aim: • They should each produce a bit m , m 1  2, such  that  m1 ⊕ m2 = s1 s2 m1 XOR m2 = s1 AND s2  • We call games where the desired data is encoded in the XOR of measurement outcomes XOR-games.  Sunday, 1 August 2010  CHSH game  1 � p(m1 + m2 = s1 s2 ) 4 s ,s 1  2  3 ≤ 4 √ 2+ 2 ≤ ≈ 0.85 4 =1  Sunday, 1 August 2010  LHV Quantum PR Box  CHSH game • CHSH inequalities bound the mean success probability of the game.  1 � p(m1 + m2 = s1 s2 ) 4 s ,s 1  2  3 ≤ 4 √ 2+ 2 ≤ ≈ 0.85 4 =1  Sunday, 1 August 2010  LHV Quantum PR Box  CHSH game • CHSH inequalities bound the mean success probability of the game.  1 � p(m1 + m2 = s1 s2 ) 4 s ,s 1  2  3 ≤ 4 √ 2+ 2 ≤ ≈ 0.85 4 =1  LHV Quantum PR Box  • The GHZ paradox can also be related to a very similar game.  Sunday, 1 August 2010  GHZ correlation |ψ� = |001� + |110�  (uniquely) satisfies:  X ⊗ X ⊗ X|ψ� = |ψ� X ⊗ Y ⊗ Y |ψ� = |ψ� Y ⊗ X ⊗ Y |ψ� = |ψ�  which also imply:  Y ⊗ Y ⊗ X|ψ� = −|ψ�  }  N. D. Mermin (1990), building on Greenberger, et al. (1989) Sunday, 1 August 2010  Correlations in outcomes of local measurements  GHZ correlation |ψ� = |001� + |110�  (uniquely) satisfies:  X ⊗ X ⊗ X|ψ� = |ψ� X ⊗ Y ⊗ Y |ψ� = |ψ� Y ⊗ X ⊗ Y |ψ� = |ψ�  which also imply:  Y ⊗ Y ⊗ X|ψ� = −|ψ�  }  N. D. Mermin (1990), building on Greenberger, et al. (1989) Sunday, 1 August 2010  Correlations in outcomes of local measurements  GHZ correlation |ψ� = |001� + |110�  (uniquely) satisfies:  X ⊗ X ⊗ X|ψ� = |ψ� X ⊗ Y ⊗ Y |ψ� = |ψ� Y ⊗ X ⊗ Y |ψ� = |ψ�  which also imply:  Y ⊗ Y ⊗ X|ψ� = −|ψ�  }  Correlations in outcomes of local measurements  GHZ “Paradox”: No (non-contextual) real number assignment of X and Y can satisfy all of these. N. D. Mermin (1990), building on Greenberger, et al. (1989) Sunday, 1 August 2010  GHZ “paradox”  • A very clean way to express this correlation is to use the binary notation introduced above.  s1  s2  s1 + s2  m1  m2  m3  m1 ⊕ m2 ⊕ m3 = s1 s2  • I.e. the correlations “win” an XOR-game, nearly identical to the CHSH game.  J. Anders and D. E. Browne PRL (2009). Sunday, 1 August 2010  Geometric approach to Bell inequalities  Sunday, 1 August 2010  Geometric interpretation of BIs • Another useful representation of Bell inequalities is to form a vector from the conditional probabilities.  p(s1 , s2 ) ≡ p(m1 ⊕ m2 = 1|s1 , s2 )    p(0, 0) p(0, 1)  p� =  p(1, 0) p(1, 1)  •  1,1,1,1  conditional probability space  0,0,0,0 Each possible set of conditional probabilities is represented a point in a unit hypercube.  Sunday, 1 August 2010  Geometric interpretation of BIs  We can thus classify the regions of conditional probability space (2-D Schematic). PR Box Quantum region LHV region: “Bell polytope” Bell inequalities define facets of polytope. Marcel Froissart: Nouvo Cimento (1981), B.S. Tsirelson, J. Sov. Math. (1987) Sunday, 1 August 2010  Convex polytopes • The convex hull of a set of vectors (vertices) in R^d. Vertex  Facet inequality Facet  • Generalisation of polygons, polyhedra to higher d. • May be defined in terms of its vertices or its facets (as a set of inequalities defining half planes.)  Sunday, 1 August 2010  Geometric interpretation of BIs • The “Bell polytope” represents the region of  conditional probabilities allowed in LHV theories.  • It is a hyper-octahedron. The facets represent the  CHSH inequalities (and normalisation conditions).  Marcel Froissart: Nouvo Cimento (1981). Sunday, 1 August 2010  Many-party Bell inequalities  Sunday, 1 August 2010  Many-party Bell-inequalities •  Werner and Wolf (2001) generalised the CHSH setting to n-parties.  •  We still keep 2-settings, 2-outputs per meas and consider conditional probs for the XOR of all outputs.  s1 M=  �  mj  j  Sunday, 1 August 2010  m1  sn  s2  m2  ···  mn  Many-party Bell-inequalities •  Werner and Wolf (2001) generalised the CHSH setting to n-parties.  •  We still keep 2-settings, 2-outputs per meas and consider conditional probs for the XOR of all outputs.  s1 M=  �  mj  j  Sunday, 1 August 2010  m1  sn  s2  m2  ···  mn  Many-party Bell-inequalities •  Werner and Wolf (2001) generalised the CHSH setting to n-parties.  •  We still keep 2-settings, 2-outputs per meas and consider conditional probs for the XOR of all outputs.  s1 M=  �  mj  j  m1  • Sunday, 1 August 2010  sn  s2  m2  ···  mn  W & W derived the full n-party Bell polytope - and found that, for any n, it is a hyper-octahedron in 2^n dimensions.  Loopholes in Bell inequality Experiments  Sunday, 1 August 2010  Loopholes in Bell Inequality experiments  Sunday, 1 August 2010  Loopholes in Bell Inequality experiments  • The beauty of Bell inequalities is that they are experimentally testable.  Sunday, 1 August 2010  Loopholes in Bell Inequality experiments  • The beauty of Bell inequalities is that they are experimentally testable.  • However, Bell’s assumptions are strict. • Space-like separated measurements • Perfect detection efficiency • Measurement settings chosen at random (free-will).  Sunday, 1 August 2010  Loopholes in Bell Inequality experiments  • The beauty of Bell inequalities is that they are experimentally testable.  • However, Bell’s assumptions are strict. • Space-like separated measurements • Perfect detection efficiency • Measurement settings chosen at random (free-will).  • If these do not hold, then the BIs may not hold - there may be loopholes.  Sunday, 1 August 2010  Loopholes in Bell Inequality experiments Loopholes make the LHV region larger.  Allowed LHV correlations under Bell’s assumptions  Sunday, 1 August 2010  Loopholes in Bell Inequality experiments Loopholes make the LHV region larger.  Allowed LHV correlations under Bell’s assumptions  Sunday, 1 August 2010  Allowed LHV correlations under actual experimental conditions  Bell inequalities vs Measurement-Based Quantum Computation  Sunday, 1 August 2010  MBQC vs BIs s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  Sunday, 1 August 2010  MBQC vs BIs s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  • Both • Require (only) XOR side-processing to perform computational game or task.  • This task is impossible (non-linear) with XOR gates alone (linear gates).  Sunday, 1 August 2010  MBQC vs BIs s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  Sunday, 1 August 2010  MBQC vs BIs s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  • But • MBQC requires adaptive measurements, and thus measurements cannot be space-like separated.  • Spatial separation is one of the most important assumptions in deriving the Bell inequalities.  Sunday, 1 August 2010  This talk s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  •  •  Main result:  •  We will derive an equivalent of the Bell polytope for MBQC.  •  I.e. Within LHV theories, which measurement-based computations can be achieved?  Main tool:  •  Sunday, 1 August 2010  Polytopes over Boolean functions.  Boolean Functions  Sunday, 1 August 2010  Boolean functions • A Boolean function is a map from n-bits to a single bit. • Every such function can be represented by a 2 bit n  vector listing the outputs for each of the 2n inputs.  •    E.g. f (0 . . . 00) f (0 . . . 01)   f (0 . . . 10) � f =    ..   . f (1 . . . 11)  to enumerate the Boolean functions • Itas is convenient where j =1, ..., 2^(2^n). Sunday, 1 August 2010  Boolean functions • A Boolean function is a map from n-bits to a single bit. • Every such function can be represented by a 2 bit n  vector listing the outputs for each of the 2n inputs.  •    E.g. f (0 . . . 00) f (0 . . . 01)   f (0 . . . 10) � f =    ..   .  • I.e. There 2^(2^n)  Boolean functions.  f (1 . . . 11)  to enumerate the Boolean functions • Itas is convenient where j =1, ..., 2^(2^n). Sunday, 1 August 2010  Boolean functions • A Boolean function is a map from n-bits to a single bit. • Every such function can be represented by a 2 bit n  vector listing the outputs for each of the 2n inputs.  •    E.g. f (0 . . . 00) f (0 . . . 01)   f (0 . . . 10) � f =    ..   .  • I.e. There 2^(2^n)  Boolean functions.  f (1 . . . 11)  to enumerate the Boolean functions • Itas isfconvenient (�s) where j =1, ..., 2^(2^n). j  Sunday, 1 August 2010  Boolean functions • Every Boolean function may be expressed as a polynomial (modulo 2) (known as “algebraic normal form”).  • E.g. if input is the bitstring  �s = s1 s2 . . . sn  f (s) = a0 + a1 s1 + a2 s2 + · · · + a12 s1 s2 + · · ·  Sunday, 1 August 2010  Boolean functions • Every Boolean function may be expressed as a polynomial (modulo 2) (known as “algebraic normal form”).  • E.g. if input is the bitstring  �s = s1 s2 . . . sn  f (s) = a0 + a1 s1 + a2 s2 + · · · + a12 s1 s2 + · · ·  Sunday, 1 August 2010  Boolean functions • Every Boolean function may be expressed as a polynomial (modulo 2) (known as “algebraic normal form”).  • E.g. if input is the bitstring  �s = s1 s2 . . . sn  f (s) = a0 + a1 s1 + a2 s2 + · · · + a12 s1 s2 + · · ·  • In other words, any Boolean function can be expressed as a sequence of XOR (add mod 2) and AND (multiply) gates.  Sunday, 1 August 2010  Boolean functions • Every Boolean function may be expressed as a polynomial (modulo 2) (known as “algebraic normal form”).  • E.g. if input is the bitstring  �s = s1 s2 . . . sn  f (s) = a0 + a1 s1 + a2 s2 + · · · + a12 s1 s2 + · · ·  • In other words, any Boolean function can be expressed as a sequence of XOR (add mod 2) and AND (multiply) gates.  • The degree of the polynomial is a useful way of classifying Boolean functions.  Sunday, 1 August 2010  Linear Boolean functions • Linear Boolean functions are degree 1 polynomials, and can be most generally written l(s) = a0 +  n � j=1  Sunday, 1 August 2010  aj sj  ak ∈ {0, 1}  Linear Boolean functions • Linear Boolean functions are degree 1 polynomials, and can be most generally written l(s) = a0 +  n � j=1  Sunday, 1 August 2010  aj sj  ak ∈ {0, 1}  Linear Boolean functions • Linear Boolean functions are degree 1 polynomials, and can be most generally written l(s) = a0 +  n �  aj sj  j=1  ak ∈ {0, 1}  2^(n+1) such functions, which we shall • There are thus l (�s) enumerate  Sunday, 1 August 2010  k  where k =1, ..., 2^(n+1).  Linear Boolean functions • Linear Boolean functions are degree 1 polynomials, and can be most generally written l(s) = a0 +  n �  aj sj  j=1  ak ∈ {0, 1}  2^(n+1) such functions, which we shall • There are thus l (�s) enumerate  k  where k =1, ..., 2^(n+1).  • All linear functions may be generated via XOR gates alone.  Sunday, 1 August 2010  Linear Boolean functions • Linear Boolean functions are degree 1 polynomials, and can be most generally written l(s) = a0 +  n �  aj sj  j=1  ak ∈ {0, 1}  2^(n+1) such functions, which we shall • There are thus l (�s) enumerate  k  where k =1, ..., 2^(n+1).  • All linear functions may be generated via XOR gates alone. • Linear functions are closed under composition.  Sunday, 1 August 2010  Linear Boolean functions • Linear Boolean functions are degree 1 polynomials, and can be most generally written l(s) = a0 +  n �  aj sj  j=1  ak ∈ {0, 1}  2^(n+1) such functions, which we shall • There are thus l (�s) enumerate  k  where k =1, ..., 2^(n+1).  • All linear functions may be generated via XOR gates alone. • Linear functions are closed under composition. • In contrast, by composing a quadratic gate (e.g. NAND) one can generate all Booleans.  Sunday, 1 August 2010  Stochastic Boolean maps given input string • Consider a stochastic machine which, λ s, outputs fj (�s) with probability  �s  Sunday, 1 August 2010  j.  prob: λj choice: j  fj (�s)  Stochastic Boolean maps given input string • Consider a stochastic machine which, λ s, outputs fj (�s) with probability  �s  j.  prob: λj choice: j  fj (�s)  • The probability that the output bit of the machine is 1, conditional upon the input s, is given by: � � p(1|�s) = λj p(fj (�s) = 1) λj = 1 j  Sunday, 1 August 2010  j  Stochastic Boolean maps  Sunday, 1 August 2010  Stochastic Boolean maps • We can simplify this expression, due to a very convenient identity: p(fj (�s) = 1) = fj (�s)  Sunday, 1 August 2010  Stochastic Boolean maps • We can simplify this expression, due to a very convenient identity: p(fj (�s) = 1) = fj (�s)  •  Sunday, 1 August 2010  and thus obtain  p(1|�s) =  � j  λj fj (�s)  Stochastic Boolean maps • We can simplify this expression, due to a very convenient identity: p(fj (�s) = 1) = fj (�s)  •  and thus obtain  • or equivalently  p(1|�s) =  � j  p� =  � j  Sunday, 1 August 2010  λj fj (�s)  λj f�j  Stochastic Boolean maps • We can simplify this expression, due to a very convenient identity: p(fj (�s) = 1) = fj (�s)  •  and thus obtain  p(1|�s) =  �  λj fj (�s)  j  • or equivalently  p� =  �  λj f�j  j  • This is a convex combination of real-space vectors and thus a polytope, we call it the Boolean polytope.  Sunday, 1 August 2010  Stochastic Boolean maps • We can simplify this expression, due to a very convenient identity: p(fj (�s) = 1) = fj (�s)  •  and thus obtain  p(1|�s) =  �  λj fj (�s)  j  • or equivalently  p� =  �  λj f�j  j  • This is a convex combination of real-space vectors and thus a polytope, we call it the Boolean polytope.  • Geometrically, it is a unit 2^n d. hypercube. Sunday, 1 August 2010  The Linear Polytope  �s  Sunday, 1 August 2010  prob: λj choice: j  lj (�s)  The Linear Polytope • Consider now a stochastic machine which can only implement linear functions: �s  Sunday, 1 August 2010  prob: λj choice: j  lj (�s)  The Linear Polytope • Consider now a stochastic machine which can only implement linear functions: prob: λj  �s  choice: j  lj (�s)  • The output probability of a linear stochastic  machine lies in the 2^(n+1) vertex polytope: p� =  n+1 2�  j=1  Sunday, 1 August 2010  λj l�j  The Linear Polytope • Consider now a stochastic machine which can only implement linear functions: prob: λj  �s  choice: j  lj (�s)  • The output probability of a linear stochastic  machine lies in the 2^(n+1) vertex polytope: p� =  n+1 2�  λj l�j  j=1  • We shall call this the (n-bit) linear polytope. Sunday, 1 August 2010  The linear polytope • We can classify the linear polytope using standard techniques.  • It has 2^(n+1) vertices in a 2^n dimensional space. • Can show that it is a hyper-octahedron.  Sunday, 1 August 2010  Bell inequalities for MBQC  Sunday, 1 August 2010  Bell inequalities for MBQC s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  • Let us ask a “Bell inequality” type question for MBQC:  • Using the correlations from a LHV theory, within an MBQC setting what computations can we achieve?  Sunday, 1 August 2010  Bell inequalities for MBQC s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  Sunday, 1 August 2010  Bell inequalities for MBQC s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  • If we allow side computation to be universal, we could access the full Boolean polytope with side computation alone.  Sunday, 1 August 2010  Bell inequalities for MBQC s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  • If we allow side computation to be universal, we could access the full Boolean polytope with side computation alone.  • For our question to make sense we make the following key assumption:  Sunday, 1 August 2010  Bell inequalities for MBQC s1 ∈ {0, 1}  s2 ∈ {0, 1}  m1 ∈ {0, 1}  m2 ∈ {0, 1}  vs  • If we allow side computation to be universal, we could access the full Boolean polytope with side computation alone.  • For our question to make sense we make the following key assumption:  • Side-computation will be solely linear. Sunday, 1 August 2010  Bell inequalities for MBQC • To simplify further, let’s initially adopt the precise CHSH (Werner-Wolf) setup.  • Spatially separated parties with independent inputs. • Output of computation encoded in the XOR of all outcome bits. s1  m1  s2  m2 M=  sn  ··· � j  Sunday, 1 August 2010  mj  mn  Bell inequalities for MBQC • What computations M (�s) can this setup perform? • LHV model may be stochastic and thus the computations may include stochastic maps allowed computations will form a region of the Boolean polytope.  Sunday, 1 August 2010  LHV?  Bell inequalities for MBQC • What computations M (�s) can this setup perform? • LHV model may be stochastic and thus the computations may include stochastic maps allowed computations will form a region of the Boolean polytope.  Sunday, 1 August 2010  LHV?  Bell inequalities for MBQC • What computations M (�s) can this setup perform? • LHV model may be stochastic and thus the computations may include stochastic maps allowed computations will form a region of the Boolean polytope.  LHV?  • Randomness arises in LHV theories by random assignment of the hidden variables.  Sunday, 1 August 2010  Bell inequalities for MBQC • What computations M (�s) can this setup perform? • LHV model may be stochastic and thus the computations may include stochastic maps allowed computations will form a region of the Boolean polytope.  LHV?  • Randomness arises in LHV theories by random assignment of the hidden variables.  • Hence, this region will be a convex polytope, whose  vertices correspond to the deterministic computations.  Sunday, 1 August 2010  Bell inequalities for MBQC  • First let us consider a single box. sj  mj  • The most general deterministic relationships between output and input can be written: mj = aj + bj mj  aj ∈ {0, 1}  bj ∈ {0, 1}  • I.e. there are only 4 1-bit to 1-bit functions. Sunday, 1 August 2010  Bell inequalities for MBQC • If we associates bit a and b with each box, the most j  j  general expression for the output XOR bit M (�s) is s1  s2  m1 M=  ···  m2 �  mj =  j  M =a+  � j  Sunday, 1 August 2010  sn  �  aj +  j  bj sj  � j  mn bj sj  Bell inequalities for MBQC M (�s) = a +  � j  Sunday, 1 August 2010  bj sj  Bell inequalities for MBQC M (�s) = a +  •  Sunday, 1 August 2010  We see  � j  bj sj  Bell inequalities for MBQC M (�s) = a +  •  Sunday, 1 August 2010  We see  •  this is linear in s,  � j  bj sj  Bell inequalities for MBQC M (�s) = a +  •  Sunday, 1 August 2010  We see  • •  �  bj sj  j  this is linear in s, by suitable choice of aj and bj all 2^(n+1) linear functions can be achieved.  Bell inequalities for MBQC M (�s) = a +  •  •  Sunday, 1 August 2010  We see  • •  �  bj sj  j  this is linear in s, by suitable choice of aj and bj all 2^(n+1) linear functions can be achieved.  Hence the full range of computations achievable by LHV correlations with XOR on the outcomes  Bell inequalities for MBQC M (�s) = a +  •  •  We see  • •  bj sj  j  this is linear in s, by suitable choice of aj and bj all 2^(n+1) linear functions can be achieved.  Hence the full range of computations achievable by LHV correlations with XOR on the outcomes  •  Sunday, 1 August 2010  �  is the Linear Polytope.  The linear polytope vs the Werner Wolf polytope  Sunday, 1 August 2010  The linear polytope vs the Werner Wolf polytope  • •  The linear polytope is a hyper-octahedron with 2^(n+1) vertices.  •  Are they the same polytope?  Sunday, 1 August 2010  The Werner-Wolf polytope is a hyper-octahedron with 2^(n+1) vertices.  The linear polytope vs the Werner Wolf polytope  • •  The linear polytope is a hyper-octahedron with 2^(n+1) vertices.  • •  Are they the same polytope?  Sunday, 1 August 2010  The Werner-Wolf polytope is a hyper-octahedron with 2^(n+1) vertices.  Yes! In fact, the derivation above is essentially a line-by-line recasting of Werner and Wolf (2001).  The linear polytope vs the Werner Wolf polytope  • •  The linear polytope is a hyper-octahedron with 2^(n+1) vertices.  • •  Are they the same polytope?  •  Our derivation is a computational reformulation of the traditional Bell inequalities.  Sunday, 1 August 2010  The Werner-Wolf polytope is a hyper-octahedron with 2^(n+1) vertices.  Yes! In fact, the derivation above is essentially a line-by-line recasting of Werner and Wolf (2001).  Bell inequalities for MBQC  • So is this a failure?  Sunday, 1 August 2010  Bell inequalities for MBQC  • So is this a failure? • We have reproduced a 10-year old result.  Sunday, 1 August 2010  Bell inequalities for MBQC  • So is this a failure? • We have reproduced a 10-year old result. • We have failed to derive Bell inequalities relevant to  standard MBQC since that utilises adaptive measurements.  Sunday, 1 August 2010  Bell inequalities for MBQC  • So is this a failure? • We have reproduced a 10-year old result. • We have failed to derive Bell inequalities relevant to  standard MBQC since that utilises adaptive measurements.  • However, sometimes new representations give new insights.... Sunday, 1 August 2010  Loopholes and linearity • Our Bell polytope is the polytope of linear functions.  Sunday, 1 August 2010  Loopholes and linearity • Our Bell polytope is the polytope of linear functions. • Recall that a “loophole” is any experimental imperfection which enlarges the LHV region.  Sunday, 1 August 2010  Loopholes and linearity • Our Bell polytope is the polytope of linear functions. • Recall that a “loophole” is any experimental imperfection which enlarges the LHV region.  • Since the LHV region is the linear polytope, and since linear functions are closed under composition.  Sunday, 1 August 2010  Loopholes and linearity • Our Bell polytope is the polytope of linear functions. • Recall that a “loophole” is any experimental imperfection which enlarges the LHV region.  • Since the LHV region is the linear polytope, and since linear functions are closed under composition.  • Introducing extra linear computation will never introduce a loophole!  Sunday, 1 August 2010  Loopholes and linearity • Our Bell polytope is the polytope of linear functions. • Recall that a “loophole” is any experimental imperfection which enlarges the LHV region.  • Since the LHV region is the linear polytope, and since linear functions are closed under composition.  • Introducing extra linear computation will never introduce a loophole!  • All known loopholes must be associated with some non-linear computation.  Sunday, 1 August 2010  Loopholes and linearity • Our Bell polytope is the polytope of linear functions. • Recall that a “loophole” is any experimental imperfection which enlarges the LHV region.  • Since the LHV region is the linear polytope, and since linear functions are closed under composition.  • Introducing extra linear computation will never introduce a loophole!  • All known loopholes must be associated with some non-linear computation.  • This allows us to weaken our assumptions and derive the same Bell inequalities.  Sunday, 1 August 2010  An example: pre-processing in GHZ  • Notice the third input depends upon the other two! s1  s2  s1 + s2  m1  m2  m3  m1 ⊕ m2 ⊕ m3 = s1 s2  Sunday, 1 August 2010  An example: pre-processing in GHZ  • Notice the third input depends upon the other two! s1  s2  s1 + s2  m1  m2  m3  m1 ⊕ m2 ⊕ m3 = s1 s2  • Why does this not induce a loophole? Sunday, 1 August 2010  An example: pre-processing in GHZ  • Notice the third input depends upon the other two! s1  s2  s1 + s2  m1  m2  m3  m1 ⊕ m2 ⊕ m3 = s1 s2  • Why does this not induce a loophole? • In an experiment, we simulate the dependency using post-selection.  Sunday, 1 August 2010  An example: GHZ • In an experimental test of GHZ, the third input is set at random. s1  s2  r  m1  m2  m3  our data, and only keep • We then post-select r =s +s data where  Sunday, 1 August 2010  1  .  2  An example: GHZ • In an experimental test of GHZ, the third input is set at random. s1  s2  r  m1  m2  m3  our data, and only keep • We then post-select r =s +s data where  1  .  2  • Does this induce a detection loop-hole? No, it doesn’t...  Sunday, 1 August 2010  Linear measurement post-selection  • Consider a more general setting. Let us set all inputs to n measurements as n uniformly random bits.  r1 M =a+  �  r2  bj rj  j  m1  m2  rn  ···  mn  • Let s be the input bit-string (provided by a referee, say). • We can post-select data such that each r is a linear j  function of the bits in s.  Sunday, 1 August 2010  Linear measurement post-selection  • Observation: This post-selection does not induce a loophole.  • Reason: closure of linear maps under composition.  Sunday, 1 August 2010  Linear measurement post-selection  • Observation: This post-selection does not induce a loophole.  • Reason: closure of linear maps under composition. M =a+  � j  Sunday, 1 August 2010  bj rj  Linear measurement post-selection  • Observation: This post-selection does not induce a loophole.  • Reason: closure of linear maps under composition. M =a+  �  bj rj  j  • M is linear in r . The r ’s are linear in s. Hence M is j  linear in s.  Sunday, 1 August 2010  j  Linear measurement post-selection  • Observation: This post-selection does not induce a loophole.  • Reason: closure of linear maps under composition. M =a+  �  bj rj  j  • M is linear in r . The r ’s are linear in s. Hence M is j  j  linear in s.  • M remains inside the linear polytope and no  loopholes are induced - the LHV region is no larger than before.  Sunday, 1 August 2010  Linear measurement post-selection •  Sunday, 1 August 2010  But we can go further.  r1  r2  m1  m2  Linear measurement post-selection • •  Sunday, 1 August 2010  But we can go further. By the same reasoning, we can post-select r to be a linear function in both s and the measurement output bits mj.  r1  r2  m1  m2  Linear measurement post-selection • •  Sunday, 1 August 2010  But we can go further. By the same reasoning, we can post-select r to be a linear function in both s and the measurement output bits mj.  s1  m1 + s2  m1  m2  Linear measurement post-selection • •  But we can go further.  •  Since mj are linear in r, M will remain within the linear polytope!  Sunday, 1 August 2010  By the same reasoning, we can post-select r to be a linear function in both s and the measurement output bits mj.  s1  m1 + s2  m1  m2  Linear measurement post-selection • •  But we can go further.  • •  Since mj are linear in r, M will remain within the linear polytope!  Sunday, 1 August 2010  By the same reasoning, we can post-select r to be a linear function in both s and the measurement output bits mj. This allows us to simulate linearly adaptive measurements,  s1  m1 + s2  m1  m2  Linear measurement post-selection • •  But we can go further.  • •  Since mj are linear in r, M will remain within the linear polytope!  • Sunday, 1 August 2010  By the same reasoning, we can post-select r to be a linear function in both s and the measurement output bits mj. This allows us to simulate linearly adaptive measurements,  s1  m1 + s2  m1  m2  And the linear polytope will still describe all LHV correlations.  Computational Bell inequalities  Sunday, 1 August 2010  Computational Bell Inequalities •  Sunday, 1 August 2010  We can use these observations to generalise the traditional Bell inequalities.  Computational Bell Inequalities •  We can use these observations to generalise the traditional Bell inequalities.  •  A computational Bell inequality is a facet of the polytope of Boolean stochastic maps achieved in any LHV theory  Sunday, 1 August 2010  • •  On a random m-bit input string s  •  With post-selection of measurement settings which are a linear function of input data and other measurement outcomes (simulated linear adaptivity)  •  And arbitrary linear pre- and post-computation.  Given n 2-setting, 2-outcome space-like separated measurements  Computational Bell Inequalities  • The “Computational Bell inequalities” are easy to characterise.  • For n ≥ m they are facets of the m-bit linear polytope.  • Setting n = m and forbidding post-selection, we  recover the traditional definition of CHSH inequalities.  Sunday, 1 August 2010  From LHV to Quantum Correlations  • A significant strand of Bell inequality research has been to characterise the set of conditional probabilities / stochastic maps allowed within Quantum Mechanics.  [1] Known as Tsirelson - Landau - Masanes region. Sunday, 1 August 2010  From LHV to Quantum Correlations  • A significant strand of Bell inequality research has been to characterise the set of conditional probabilities / stochastic maps allowed within Quantum Mechanics.  • Usually only non-  adaptive measurements are considered.  [1] Known as Tsirelson - Landau - Masanes region. Sunday, 1 August 2010  From LHV to Quantum Correlations  • A significant strand of Bell inequality research has been to characterise the set of conditional probabilities / stochastic maps allowed within Quantum Mechanics.  • Usually only non-  adaptive measurements are considered.  • But in our new framework, we can also allow extra parties, and simulated linear adaptive measurement.  [1] Known as Tsirelson - Landau - Masanes region. Sunday, 1 August 2010  From LHV to Quantum Correlations  • A significant strand of Bell inequality research has been to characterise the set of conditional probabilities / stochastic maps allowed within Quantum Mechanics.  • Usually only non-  adaptive measurements are considered.  • But in our new framework, we can also allow extra parties, and simulated linear adaptive measurement.  • Do these enlarge the quantum region? [1] Known as Tsirelson - Landau - Masanes region. Sunday, 1 August 2010  A bigger quantum set?  From LHV to Quantum Correlations  • Yes! • For a fixed n, • Allowing extra parties • Allowing linear post-selection (simulated adaptivity).  • can increase the quantum region.  Sunday, 1 August 2010  From LHV to Quantum Correlations  • Yes! • For a fixed n, • Allowing extra parties • Allowing linear post-selection (simulated adaptivity).  • can increase the quantum region. • This can provide a greater degree of violation of Bell Inequalities than in the standard setting.  Sunday, 1 August 2010  Quantum correlations with adaptive measurements  • For example, if we limit the number of parties to 6. • With linear adaptive measurement we can access the deterministic 3-bit AND function.  • The triple product s s s  1 2 3  linear  linear  of input bits s1, s2, s3.  linear  • With no adaptivity, deterministic computation of this function is impossible with only 6 measurements [1].  [1] In Hoban et al (2010 - arxiv next week) we show that 2^(n-1) qubits are required to acheive an n-bit AND deterministically with no adaptivity. Sunday, 1 August 2010  Summary • We derived Bell inequalities from the perspective of measurement-based quantum computation.  Sunday, 1 August 2010  Summary • We derived Bell inequalities from the perspective of measurement-based quantum computation.  • This allowed us to characterise all 2-setting, 2-output Bell inequalities in terms of the linear Boolean polytope.  Sunday, 1 August 2010  Summary • We derived Bell inequalities from the perspective of measurement-based quantum computation.  • This allowed us to characterise all 2-setting, 2-output Bell inequalities in terms of the linear Boolean polytope.  • And made it clear that extra linear computations  (including simulated adaptivity) do not lead to loopholes.  Sunday, 1 August 2010  Summary • We derived Bell inequalities from the perspective of measurement-based quantum computation.  • This allowed us to characterise all 2-setting, 2-output Bell inequalities in terms of the linear Boolean polytope.  • And made it clear that extra linear computations  (including simulated adaptivity) do not lead to loopholes.  • This enables one to consider a richer structure of  quantum correlations in the context of BIs (including the adaptive measurements arising in MBQC) which we are only starting to investigate.  Sunday, 1 August 2010  Summary • We derived Bell inequalities from the perspective of measurement-based quantum computation.  • This allowed us to characterise all 2-setting, 2-output Bell inequalities in terms of the linear Boolean polytope.  • And made it clear that extra linear computations  (including simulated adaptivity) do not lead to loopholes.  • This enables one to consider a richer structure of  quantum correlations in the context of BIs (including the adaptive measurements arising in MBQC) which we are only starting to investigate.  • Are Bell inequality violations more about non-linearity than non-locality...?  Sunday, 1 August 2010  Acknowledgements • • •  Sunday, 1 August 2010  Joint work with members of my UCL group  •  Matty Hoban, Janet Anders, Earl Campbell and Klearchos Loukopoulos.  These ideas benefitted from discussions with Robert Raussendorf, Robert Spekkens, Miguel Navascues, Akimasa Miyake, Robin Blume-Kohout Debbie Leung and many others... References  • • •  R. Werner and M. Wolf, Phys. Rev. A 64 32112 (2001)  •  M. J. Hoban and D. E. Browne, Bell inequalities and adaptive measurements, arxiv soon.  J. Anders and D. E. Browne, Phys. Rev. Lett. (2009) M. J. Hoban, E.T. Campbell, K. Loukopoulos, D.E. Browne, "Non-adaptive measurement-based quantum computation and multi-party Bell inequalities", arxiv very very soon.  

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