Bounding the set of finite dimensional
correlations
Miguel Navascués1 and Tamás Vértesi2
1Institute
for Quantum Optics and Quantum Information (IQOQI), Vienna.
2Institute for Nuclear Research, Hungarian Academy of Sciences.
MN and T. Vértesi, Phys. Rev. Lett. 115, 020501 (2015) .
MN, A. Feix, M. Araújo and T. Vértesi, Phys. Rev. A 92, 042117 (2015).
Optimization theory
max
∈
Variational methods
max
∈
Variational methods
max
∈
ansatz
Variational methods
max
∈
ansatz
Variational methods
max
∈
ansatz
‐usually, pretty straightforward
E.g.: Newton's method
‐usually, universal
‐sometimes arise as intuitions gathered via numerical experiments. E.g.: MPS.
Relaxations
max
∈
Relaxations
max
∈
Relaxations
max
∈
Relaxations
max
∈
‐Depend on the structure of the problem
‐"Aha!" idea
E.g.: the NPA hierarchy
MN, S. Pironio and A. Acín, Phys. Rev. Lett. 98, 010401 (2007).
MN, S. Pironio and A. Acín, New J. Phys. 10, 073013 (2008).
Alice
Bob
x
y
a
b
Alice
Bob
x
y
a
b
Are the ouputs of this experiment compatible with
quantum mechanics?
"Easy"
Variational methods
"Difficult"
Mathematical coincidences
"Difficult"
"Easy"
Mathematical coincidences
Variational methods
relaxation?
is not quantum
The problem resists brute force approach!!
is not quantum
The problem resists brute force approach!!
What is a Hilbert space?
visible averages
invisible averages
Moment matrix
...
1
, | ,
′| ′
′, | ′,
,
Moment matrix
...
1
, | ,
′| ′
′, | ′,
Moment matrix
...
1
Hilbert space
, | ,
′| ′
⋮
′, | ′,
, | , admits a positive semidefinite moment matrix for the operators , ,
Q1
1st‐order moment matrix
, | , admits a positive semidefinite moment matrix for the operators 2nd‐order , , ,
,
moment ,
Q2
Q3
3rd‐order moment matrix
Q
matrix
MN, S. Pironio and A. Acín, Phys. Rev. Lett. 98, 010401 (2007).
MN, S. Pironio and A. Acín, New J. Phys. 10, 073013 (2008).
, | , admits a positive semidefinite moment matrix for the operators , , ,
,
,
, ,
, etc.
NPA
Variational methods
max
,
,
Bounding the set of finite dimensional
quantum correlations
MN and T. Vértesi, Phys. Rev. Lett. 115, 020501 (2015) .
Alice
Bob
x
y
dimension dimension ,
a
,
,
|
∈
⨂
|
,
b
"Easy"
Variational methods
"Difficult"
Mathematical coincidences
relaxation?
"Easy"
Variational methods
"Difficult"
Mathematical coincidences
Quantum communication complexity
Alice
Bob
H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Rev. Mod. Phys. 82, 665 (2010).
Quantum communication complexity
Alice
Bob
H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Rev. Mod. Phys. 82, 665 (2010).
Quantum communication complexity
Alice
Bob
H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Rev. Mod. Phys. 82, 665 (2010).
Quantum communication complexity
Dimension D
Alice
Bob
H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Rev. Mod. Phys. 82, 665 (2010).
Quantum communication complexity
Dimension D
Alice
Bob
Maximum probability that H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Rev. Mod. Phys. 82, 665 (2010).
Quantum communication complexity
Dimension D
Alice
Variational methods
Bob
Relaxation?
H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Rev. Mod. Phys. 82, 665 (2010).
max
1
max
2
∗
s.t.
,
∈
s.t. ∈
,
⨂
, ,
0,
⨂
, , ,
,
,
,
,
,
1
,
,
∈B
1
,projectors,
1,
dim
Brute force theoretically posible, but impractical
What is a D-dimensional Hilbert space?
MN and T. Vértesi, Phys. Rev. Lett. 115, 020501 (2015) .
D=# of entries
D=# of columns
Moment matrix
...
1
, | ,
′| ′
′, | ′,
,
How to incorporate dimension constraints?
Matrix polynomial identities
Matrix polynomial identities
E.g.:
1
Matrix polynomial identities
E.g.:
1
2
Matrix polynomial identities
E.g.:
1
2
Central polynomial (commutes with everything)
Standard Identity
matrices
C. Procesi. Rings with polynomial identities. Marcel Dekker, New York, 1973
Moment matrix
...
1
, | ,
′| ′
′, | ′,
Moment matrix
...
1
, | ,
′| ′
∈
0, ∀
,
, … ,
′, | ′,
,
0
,
,
,
0
Moment matrix
...
1
, | ,
′| ′
′, | ′,
Moment matrices of D‐level quantum systems are subject to non‐trivial linear constraints!!!
D-dimensional
Hilbert space
D-dimensional
Hilbert space
?
Toy problem
max
1
,
s.t. ,
,
1
Temporal quantum correlations
projective measurement
1,‐1
C. Budroni, T. Moroder, M. Kleinmann, O. Gühne, Phys. Rev. Lett. 111, 020403 (2013) Temporal quantum correlations
00
1
1
‐1
C. Budroni, T. Moroder, M. Kleinmann, O. Gühne, Phys. Rev. Lett. 111, 020403 (2013) Toy problem
∗
max
|
1
,
,
,
1,
s.t. 1
|
Toy problem
∗
max
|
1
,
,
|
,
1,
s.t. 1
In the dimension‐free case, this kind of problems can be solved with a single SDP
S. Pironio, M. Navascués and A. Acín, SIAM J. Optim. 20, 5, 2157‐2180 (2010).
C. Budroni, T. Moroder, M. Kleinmann, O. Gühne, Phys. Rev. Lett. 111, 020403 (2013) ∗
Temporal quantum correlations
projective measurement
two‐dimensional system
1,‐1
C. Budroni, T. Moroder, M. Kleinmann, O. Gühne, Phys. Rev. Lett. 111, 020403 (2013) Toy problem
∗
max
|
1
,
,
|
,
1,
s.t. 1,
|
∈
,
,
∈
Solving the toy problem (I): divide the problem into classes
1
1
1
2
Trivial
1
2
|
1
|
1
A random instance can be generated easily
0
1
1
non‐trivial
Solving the toy problem (I): divide the problem into classes
729 classes
Each class labeled by a vector ∈ 0,1,2
Classed toy problem
∗
max
|
1
,
,
|
,
1,
s.t. 1,
|
∈
,
,
1
1
,
∈
,
max
1
,
,
,
s.t. Γ ,
Γ
Γ∈
1
0,
,
Γ
Solving the toy problem (II): Identify max
1
,
,
,
s.t. Γ ,
Γ
Γ∈
1
0,
,
,
Γ
High level description
1
High level description
1
Generate random dichotomic operators
, ∈B
(with app. ranks) and vector ∈
High level description
1
Generate random dichotomic operators
, ∈B
(with app. ranks) and vector ∈
Build moment matrix Γ
Γ , = Γ
,
1,
= ,
,
High level description
1
Generate random dichotomic operators
, ∈B
(with app. ranks) and vector ∈
Build moment matrix Γ
Γ , = Γ
,
1,
,
= 1
,
Repeat until the is moment matrix Γ
a linear combination of Γ ,…,Γ
High level description
1
Generate random dichotomic operators
, ∈B
(with app. ranks) and vector ∈
Build moment matrix Γ
Γ , = Γ
,
1,
,
= ,
At that point, any
feasible moment matrix Γ must be a linear combination of Γ ,…,Γ
Γ
1
,…,Γ
)
SDP relaxation
max
1
,
,
,
s.t. Γ ,
Γ
Γ
Γ
1
0,
Γ
SDP relaxation
max
1
,
,
,
s.t. Γ ,
Γ
Problems with SDP solvers
Γ
Γ
1
0,
Lack of strict feasibility, i.e., there are no Γ 0
Γ
| |≫1
Implementation tips
1
Why?
Isometry to the support of Γ
Strict positivity
Implementation tips
2
Use (modified) Gram‐Schmidt as you generate the matrices Γ , Γ …
Sequence of orthogonal matrices Γ , Γ …
at s=N+1, Γ
Γ
0, up to computer precision
Γ
Γ
Γ
Γ
SDP relaxation
,
,
min
Γ
,
1,
s.t. Γ ,
Γ
Γ
,
, , ,
, , ,
Γ
,
,
min
1,
s.t. Γ ,
Γ
0,
Γ
Γ
0,
Γ
Free dimensionality
∗
Second relaxation, D=2
Generalization
NPO problem with dimension constraints
∗
, ,
s.t. MN, A. Feix, M. Araújo and T. Vértesi, Phys. Rev. A 92, 042117 (2015).
∗
min
1,
s.t. 0,
0,
/
∑
∗
,
∗
∈ .
∗
min
1,
s.t. 0,
0,
/
∑
,
∈ .
Archimedean
condition
,
,
→
∗
∗
,
Related hierarchies
SDP hierarchy for quantum non‐locality under dimension constraints
Alice
x
Bob
,
,
max
x
, | ,
, , ,
dimension dimension ,
a
,
,
|
∈
⨂
|
,
y
s.t. ∈
, ⨂
,
∈B
,
1
,projectors,
1,
dim
s.t. ∈
, ⨂
,
,
∈B
1
,projectors,
1,
dim
rank
,
, rank
High level description
1
,
Generate random projectors ,
, ∈B
(with app. ranks) and vector ∈
Repeat until the is moment matrix Γ
a linear combination of Γ ,…,Γ
Build moment matrix Γ
Γ , = Γ
,
⨂
= 1,
,
⨂
1
,
,
SDP hierarchy
s.t. Γ ,
Γ
1
0, Γ
Γ
Convergence is guaranteed
MN, A. Feix, M. Araújo and T. Vértesi, Phys. Rev. A 92, 042117 (2015).
NPA
NPA
Pironio‐Bell
0.2532
0.250875
Pironio‐Bell
0.2071
0.25
D=2
D=2, 3
5.9907
quantum mechanics
5.8310
D=2, 3
dimension MN, A. Feix, M. Araújo and T. Vértesi, Phys. Rev. A 92, 042117 (2015).
max
|
1
3
?
| |
High level description
1
,
Generate random projectors ,
, ∈B
(with app. ranks)
Repeat until the is moment matrix Γ
a linear combination of Γ ,…,Γ
Build moment matrix Γ
Γ , = Γ
,
⨂
= 1,
,
⨂
1
,
,
max
|
0.229771
1
3
| |
SDP hierarchy for quantum communication complexity
Dimension D
Alice
Bob
Maximum probability that ∗
,
,
such that
,
∈
,
,
0,
1
High level description
1
Generate random
,
∈B
projectors ,
app. ranks)
(with Build moment matrix Γ
Γ , = Γ
,
,
,
= 1
),
Repeat until the is moment matrix Γ
a linear combination of Γ ,…,Γ
SDP hierarchy
1
max
2
s.t. Γ ,
Γ
Γ
, ,
,
,
1
0, Γ
Γ
No proof of convergence!!!
MN, A. Feix, M. Araújo and T. Vértesi, Phys. Rev. A 92, 042117 (2015).
Random Access Codes (RAC)
Alice
Dimension D
Bob
Random Access Codes (RAC)
buttons
d levels
R. Gallego, N. Brunner, C. Hadley and A. Acín, Phys. Rev. Lett. 105, 230501 (2010). Prepare‐and‐measure dimension witnesses
Witness
Your quantum technology allows you to manipulate systems of dimensión at least D+1
R. Gallego, N. Brunner, C. Hadley and A. Acín, Phys. Rev. Lett. 105, 230501 (2010). R. Gallego, N. Brunner, C. Hadley and A. Acín, Phys. Rev. Lett. 105, 230501 (2010). In the lab
Witness?
Variational methods
hope: the inequality is satisfied by the bigger set
R. Gallego, N. Brunner, C. Hadley and A. Acín, Phys. Rev. Lett. 105, 230501 (2010). In the lab
Witness?
Variational methods
hope: the inequality is satisfied by the bigger set
R. Gallego, N. Brunner, C. Hadley and A. Acín, Phys. Rev. Lett. 105, 230501 (2010). We proved that all these witnesses were sound, hence validating of the conclusions of the experimental papers
M. Hendrych, R. Gallego, M. Micuda, N. Brunner, A. Acín and J. P. Torres, Nat. Phys. 8, 588 (2012).
J. Ahrens, P. Badziag, A. Cabello, M. Bourennane, Nat. Phys. 8, 592 (2012). Hierarchy for real quantum systems
1
Generate real random ,
∈B
(with projectors ,
app. ranks)
Build moment matrix Γ
Γ , = Γ
,
1,
,
= 1
),
Repeat until the is moment matrix Γ
a linear combination of Γ ,…,Γ
Real vs. Complex quantum mechanics
Your technology allows you to go beyond complex quantum mechanics in dimension d
Your technology allows you to go beyond real quantum mechanics in dimension d
max ∑
, ,
,
| ,
Random Access Codes (RAC)
Alice
Dimension D
real qubits
complex qubits
3→1
Bob
0.788675
3→1
0.7696723
The structure of Matrix Product States
MN and T. Vértesi, arXiv:1509.04507.
s‐local term
(acts on particles j,...,j+s‐1)
k particles
physical dimension ground state
Impossible to even store for high k
Matrix product states
…
∈
) physical dimension boundary condition
bond dimension ,…,
D. Perez‐Garcia, F. Verstraete, M.M. Wolf and J.I. Cirac, Quantum Inf. Comput. 7, 401 (2007). Matrix product states
Features:
a) Efficient computation of expectation values
b) Calculations in the thermodynamical limit → ∞
D. Perez‐Garcia, F. Verstraete, M.M. Wolf and J.I. Cirac, Quantum Inf. Comput. 7, 401 (2007). Correspondence between polynomials and states
,…,
,
homogeneous polynomial of degree m
…
MN and T. Vértesi, arXiv:1509.04507.
|
∗
| ,…,
Overlap between a polynomial and a MPS
k particles
|
…
|
⨂ … ⨂|
,…,
| m particles
…
…
,…, ,
,…,
"defect"
MN and T. Vértesi, arXiv:1509.04507.
| ,…, ,
,…
Existence of annihilation operators
…
…
| ,…, ,
,…
,…, ,
,…,
, MPI for dimension D
0, for all MPS with bond dimension D!!
There exist local operators which annihilate all MPS of bond dimension D or smaller
MN and T. Vértesi, arXiv:1509.04507.
Actually, for high m, almost all m‐local operators are annihilation operators, because
Local dimension of m particles
Local dimension of MPS subspace
Existence of cut‐and‐glue operators
…
…
| ,…, ,
,…
,…, ,
,…,
scalar
p A | ′
, central polynomial for dimension D
| ′
…
…
| ,…, ,
,…
,…, ,
,…,
MN and T. Vértesi, arXiv:1509.04507.
MPS with m particles less
, basis of homogeneous central polynomials for dimension D of degree m
MN and T. Vértesi, arXiv:1509.04507.
bond dimension D
,…,
C, cut‐and‐glue operator
m particles projected onto a pure state (cut form the chain)
Two ends glued back
C, cut‐and‐glue operator
|
…
,…,
|
⨂ … ⨂|
C, cut‐and‐glue operator for dimension D
⊗1
⊗ 1 , separable
, k‐site MPS of bond dimension D
|
…
,…,
|
⨂ … ⨂|
Theoretical complexity of the SDP hierarchy and connection with MPS
MN and T. Vértesi, arXiv:1509.04507.
Complexity of implementing the kth relaxation
SDP constraint
free variables
Free‐dimensional problems have the same complexity as dimension‐constrained ones!?
Connection with MPSs
Feasible moment matrices are extendible MPSs of bond dimension D!!! MN and T. Vértesi, arXiv:1509.04507.
Complexity of implementing the kth relaxation
SDP constraint
Supp∈span{poly(k) k‐site MPS of bond dimension D}
free variables
∈span{poly(k) 2k‐site MPS of bond dimension D}
Complexity of implementing the kth relaxation
MN and T. Vértesi, arXiv:1509.04507.
Finite dimensions are exponentially easier to characterize than infinite dimensions!!!
Extra PSD conditions to boost speed of convergence
Γ
MN and T. Vértesi, arXiv:1509.04507.
Extra PSD conditions to boost speed of convergence
Γ
cut‐and‐glue‐operator
C
MN and T. Vértesi, arXiv:1509.04507.
separable operator
Extra PSD conditions to boost speed of convergence
Γ
cut‐and‐glue‐operator
C
New PSD conditions
separable operator
C
MN and T. Vértesi, arXiv:1509.04507.
PPT
Conclusions
‐Simple, easy‐to‐program technique to enforce dimension constraints in noncommutative polynomial optimization.
‐Plenty of numerical evidence suggests that it is effective.
‐It can be combined with other hierarchies, such as MLHG or BFS.
T. Moroder, J.‐D. Bancal, Y.‐C. Liang, M. Hofmann and O. Gühne, Phys. Rev. Lett. 111, 030501 (2013).
M. Berta, O. Fawzi and V. B. Scholz, arXiv:1506.08810.