Dvoretzky's Theorem and the Complexity of Entanglement Detection

The well-known Horodecki criterion asserts that a state $\rho$ on $\mathbf{C}^d \otimes \mathbf{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathrm{Id})(\rho)$ is not positive semi-definite. We show that the number of such maps needed to detect all the robustly entangled states (i.e., states $\rho$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on the 1977 inequality of Figiel--Lindenstrauss--Milman, which ultimately relies on Dvoretzky's theorem about almost spherical sections of convex bodies. We interpret that inequality as a statement about approximability of convex bodies by polytopes with few vertices or with few faces and apply it to the study of fine properties of the set of quantum states and that of separable states. Our results can be thought of as geometrical manifestations of the complexity of entanglement detection.

We present a lower bound for complexity of entanglement detection which (ultimately) relies on the 1961 Dvoretzky's theorem, a fundamental result from Asymptotic Geometric Analysis asserting that high-dimensional convex sets typically look round when we observe only their section with a randomly chosen subspaces of smaller dimension. The dichotomy between entanglement vs. separability is fundamental in quantum theory. Entanglement is indispensable for most protocols of quantum information theory/cryptography/computing/teleportation. . .
Since entanglement is defined as non-membership in a (closed) convex set, it follows from the Hahn-Banach separation theorem that for every entangled state ρ, there is a linear form f such that f a on Sep and f (ρ) > a. Such f certifies, or witnesses, the entanglement of ρ.
Since entanglement is defined as non-membership in a (closed) convex set, it follows from the Hahn-Banach separation theorem that for every entangled state ρ, there is a linear form f such that f a on Sep and f (ρ) > a. Such f certifies, or witnesses, the entanglement of ρ.
Equivalently, Sep is the intersection of half-spaces, which leads to a natural scheme for approximating Sep by polytopes.
Since entanglement is defined as non-membership in a (closed) convex set, it follows from the Hahn-Banach separation theorem that for every entangled state ρ, there is a linear form f such that f a on Sep and f (ρ) > a. Such f certifies, or witnesses, the entanglement of ρ.
Equivalently, Sep is the intersection of half-spaces, which leads to a natural scheme for approximating Sep by polytopes.
A naive -but meaningful -way of measuring complexity of entanglement detection would be determining how well Sep can be approximated by a polytope with N faces.
Since entanglement is defined as non-membership in a (closed) convex set, it follows from the Hahn-Banach separation theorem that for every entangled state ρ, there is a linear form f such that f a on Sep and f (ρ) > a. Such f certifies, or witnesses, the entanglement of ρ.
Equivalently, Sep is the intersection of half-spaces, which leads to a natural scheme for approximating Sep by polytopes.
A naive -but meaningful -way of measuring complexity of entanglement detection would be determining how well Sep can be approximated by a polytope with N faces.
More generally, it is known (Gurvits 2003) that deciding whether a state is entangled or separable is, in general, NP-hard. Later refinements have been due to Ioannou (2007), Gharibian (2010) and others; in particular some upper bounds were supplied by Brandão et al (2011).
A more structured scheme of witnessing entanglement is given by the following (M d stands for the space of n × n complex matrices).

Theorem (The Horodecki criterion 1996)
is not positive semi-definite (one says that Φ witnesses the entanglement of ρ).
A more structured scheme of witnessing entanglement is given by the following (M d stands for the space of n × n complex matrices).

Theorem (The Horodecki criterion 1996)
is not positive semi-definite (one says that Φ witnesses the entanglement of ρ).
A linear map Φ is positive if Φ(PSD) ⊂ PSD where PSD denotes the positive semi-definite cone, more naturally contained in M sa d .
A more structured scheme of witnessing entanglement is given by the following (M d stands for the space of n × n complex matrices). Denote by • homotheties with respect to ρ * := I H dim H :

Theorem (The Horodecki criterion 1996)
Say that a state ρ is robustly entangled if 1 2 • ρ is entangled. Robustly entangled states remain entangled in the presence of randomizing noise.

Theorem (Aubrun-Szarek)
Suppose that Φ 1 , . . . , Φ N are positive maps on M d such that, for any robustly entangled state ρ on C d ⊗ C d , there is an index i such that This shows that the set of separable states is complex, and not because of some fine features of its boundary.
Denote by • homotheties with respect to ρ * := I H dim H : Say that a state ρ is robustly entangled if 1 2 • ρ is entangled. Robustly entangled states remain entangled in the presence of randomizing noise.

Theorem (Aubrun-Szarek)
Suppose that Φ 1 , . . . , Φ N are positive maps on M d such that, for any robustly entangled state ρ on This shows that the set of separable states is complex, and not because of some fine features of its boundary. Results about NP-hardness of entanglement detection have usually focused on boundary effects.
Let K ⊂ R n be a convex compact set with 0 in the interior. Define the verticial and facial dimensions of K as where the infima run over all polytopes P ⊂ R n . The affine invariants dim F and dim V are measures of complexity.
Let K ⊂ R n be a convex compact set with 0 in the interior. Define the verticial and facial dimensions of K as where the infima run over all polytopes P ⊂ R n . The affine invariants dim F and dim V are measures of complexity.
These are dual concepts since dim Let K ⊂ R n be a convex compact set with 0 in the interior. Define the verticial and facial dimensions of K as where the infima run over all polytopes P ⊂ R n . The affine invariants dim F and dim V are measures of complexity.
These are dual concepts since dim Let K ⊂ R n be a convex compact set with 0 in the interior. Define the verticial and facial dimensions of K as where the infima run over all polytopes P ⊂ R n . The affine invariants dim F and dim V are measures of complexity.
These are dual concepts since dim Let K ⊂ R n be a convex compact set with 0 in the interior. Define the verticial and facial dimensions of K as where the infima run over all polytopes P ⊂ R n . The affine invariants dim F and dim V are measures of complexity.
These are dual concepts since dim x, y 1, ∀y ∈ K } is the polar of K .
One has dim F (K ) = O(n) and dim V (K ) = O(n) if (say) the origin is the center of mass of K .
We have dim F (B n 2 ) = dim V (B n 2 ) = Θ(n) (B n 2 is the Euclidean ball). Another parameter is the asphericity of K defined as a(K ) := inf {R/r : rB n 2 ⊂ K ⊂ RB n 2 } .
A fundamental property ("complexity must lie somewhere") of convex sets is the following.

Theorem (Figiel-Lindenstrauss-Milman 1977)
For any convex body K ⊂ R n containing the origin in the interior we have A fundamental property ("complexity must lie somewhere") of convex sets is the following.

Theorem (Figiel-Lindenstrauss-Milman 1977)
For any convex body K ⊂ R n containing the origin in the interior we have This result is a consequence of the tangible version of Dvoretzky's theorem due to Milman, which gives a sharp formula for the dimension of almost Euclidean sections of convex bodies.
If K ⊂ R n is a convex body such that rB n 2 ⊂ K and M = M(K ) denotes the average of the "norm" · K over the sphere, then K has lots of almost Euclidean sections of dimension k = Ω(nr 2 M 2 ).
A fundamental property ("complexity must lie somewhere") of convex sets is the following.

Theorem (Figiel-Lindenstrauss-Milman 1977)
For any convex body K ⊂ R n containing the origin in the interior we have This result is a consequence of the tangible version of Dvoretzky's theorem due to Milman, which gives a sharp formula for the dimension of almost Euclidean sections of convex bodies.
If K ⊂ R n is a convex body such that rB n 2 ⊂ K and M = M(K ) denotes the average of the "norm" · K over the sphere, then K has lots of almost Euclidean sections of dimension k = Ω(nr 2 M 2 ).
Thus the facial dimension of K exceeds ck.
A fundamental property ("complexity must lie somewhere") of convex sets is the following.

Theorem (Figiel-Lindenstrauss-Milman 1977)
For any convex body K ⊂ R n containing the origin in the interior we have This result is a consequence of the tangible version of Dvoretzky's theorem due to Milman, which gives a sharp formula for the dimension of almost Euclidean sections of convex bodies.
If K ⊂ R n is a convex body such that rB n 2 ⊂ K and M = M(K ) denotes the average of the "norm" · K over the sphere, then K has lots of almost Euclidean sections of dimension k = Ω(nr 2 M 2 ).
Thus the facial dimension of K exceeds ck. Applying the same argument to K • and using the inequality M(K )M(K • ) 1 yields the FLM bound.

The FLM bound -examples
We illustrate the FLM bound on some examples where it is sharp up to polylog factors Recall that B n 2 is the n-dimensional Euclidean ball, while ∆ n is the n-dimensional simplex.
And here are some more examples related to entanglement detection.
The value of a(D) is elementary to compute; the value of a(Sep) is due to Gurvits-Barnum (2002).
The verticial dimensions of D and Sep are easier to compute since these sets are defined by convex hulls. However, there are some surprises.
And here are some more examples related to entanglement detection.
The value of a(D) is elementary to compute; the value of a(Sep) is due to Gurvits-Barnum (2002).
The verticial dimensions of D and Sep are easier to compute since these sets are defined by convex hulls. However, there are some surprises.

Quantum-related examples, II
We now complete the table.
The set D is self-dual (or, more precisely, D • = (−m) • D), so its facial dimension equals its verticial dimension.
We now complete the table.
The set D is self-dual (or, more precisely, D • = (−m) • D), so its facial dimension equals its verticial dimension.
The set D is self-dual (or, more precisely, D • = (−m) • D), so its facial dimension equals its verticial dimension.
The lower bound on the facial dimension of Sep follows from the Figiel-Lindenstrauss-Milman inequality However, it is conceivable that we actually have dim F (Sep) = Θ(d 4 ). Let Φ 1 , . . . , Φ N be N positive maps on M d with the property that for every robustly entangled state ρ, there exists an index i such that (Φ i ⊗ Id)(ρ) is not positive. This hypothesis is equivalent to the following inclusion Aubrun & Szarek (Lyon & CWRU/Paris 6) Dvoretzky's theorem and entanglement Banff, January 11, 2016 16 / 18 Let Φ 1 , . . . , Φ N be N positive maps on M d with the property that for every robustly entangled state ρ, there exists an index i such that (Φ i ⊗ Id)(ρ) is not positive. This hypothesis is equivalent to the following inclusion By considering X → Φ i (I) −1/2 Φ i (X )Φ i (I) −1/2 , we may assume that Φ i (I) = I for all i = 1, . . . , N.
Let Φ 1 , . . . , Φ N be N positive maps on M d with the property that for every robustly entangled state ρ, there exists an index i such that (Φ i ⊗ Id)(ρ) is not positive. This hypothesis is equivalent to the following inclusion By considering X → Φ i (I) −1/2 Φ i (X )Φ i (I) −1/2 , we may assume that Φ i (I) = I for all i = 1, . . . , N.
Next, for simplicity, let us assume first that each Φ i is trace-preserving, i.e., Φ i (ρ * ) = ρ * . Consider the convex body Let Φ 1 , . . . , Φ N be N positive maps on M d with the property that for every robustly entangled state ρ, there exists an index i such that (Φ i ⊗ Id)(ρ) is not positive. This hypothesis is equivalent to the following inclusion By considering X → Φ i (I) −1/2 Φ i (X )Φ i (I) −1/2 , we may assume that Φ i (I) = I for all i = 1, . . . , N.
Next, for simplicity, let us assume first that each Φ i is trace-preserving, i.e., Φ i (ρ * ) = ρ * . Consider the convex body which satisfies Sep ⊂ K ⊂ 2 • Sep. Note the trace-preserving condition assures that for all i's we are •-dilating with respect to the same point.
Since the facial dimension of D(C d ⊗ C d ) is of order d 2 , there exists a polytope P with at most exp(Cd 2 ) facets such that 1 2 • D ⊂ P ⊂ D. Then the polytope Since #facets(P 1 ∩ P 2 ) #facets(P 1 ) + #facets(P 2 ), the polytope Q has at most (N + 1) exp(Cd 2 ) facets .
Since the facial dimension of D(C d ⊗ C d ) is of order d 2 , there exists a polytope P with at most exp(Cd 2 ) facets such that 1 2 • D ⊂ P ⊂ D. Then the polytope Since #facets(P 1 ∩ P 2 ) #facets(P 1 ) + #facets(P 2 ), the polytope Q has at most (N + 1) exp(Cd 2 ) facets .
Since the facial dimension of Sep is Ω(d 3 / log d), it follows that log((N + 1) exp(Cd 2 )) cd 3 / log d so that N exp(cd 3 / log d) as claimed.
Since the facial dimension of D(C d ⊗ C d ) is of order d 2 , there exists a polytope P with at most exp(Cd 2 ) facets such that 1 2 • D ⊂ P ⊂ D. Then the polytope Since #facets(P 1 ∩ P 2 ) #facets(P 1 ) + #facets(P 2 ), the polytope Q has at most (N + 1) exp(Cd 2 ) facets .
Since the facial dimension of Sep is Ω(d 3 / log d), it follows that log((N + 1) exp(Cd 2 )) cd 3 / log d so that N exp(cd 3 / log d) as claimed.
The general situation (without the trace-preserving restriction) is handled similarly starting with the assumption that (1 − 1 2d ) • D ⊂ P ⊂ D.
We illustrated the complexity of robust entanglement by showing that super-exponentially many positive maps are needed to detect it -at least if used non-adaptively/without reflection.
The proof is via a facet-counting argument (even if the set of separable states is not a polytope itself) and ultimately relies on the bound due to Figiel-Lindenstrauss-Milman which asserts that -between (i) the number of vertices, (ii) the number of facets, and (iii) asphericity -complexity must lie somewhere.
We illustrated the complexity of robust entanglement by showing that super-exponentially many positive maps are needed to detect it -at least if used non-adaptively/without reflection.
The proof is via a facet-counting argument (even if the set of separable states is not a polytope itself) and ultimately relies on the bound due to Figiel-Lindenstrauss-Milman which asserts that -between (i) the number of vertices, (ii) the number of facets, and (iii) asphericity -complexity must lie somewhere.
Can this approach be used to handle other problems in complexity theory?

Conclusion
We illustrated the complexity of robust entanglement by showing that super-exponentially many positive maps are needed to detect it -at least if used non-adaptively/without reflection.
The proof is via a facet-counting argument (even if the set of separable states is not a polytope itself) and ultimately relies on the bound due to Figiel-Lindenstrauss-Milman which asserts that -between (i) the number of vertices, (ii) the number of facets, and (iii) asphericity -complexity must lie somewhere.
Can this approach be used to handle other problems in complexity theory? Some other directions in which this work can be continued are: • Upper bounds; in particular, what is the order of d F (Sep)?
• The multipartite or "unbalanced" (H = C d ⊗ C m ) setting