Bounding the Elliptope of Quantum Correlations Proving Separability in Mixed States

1. Make sure you can read this text. Notes will show up in the margins. BoundingtheElliptopeofQuantum Correlations Proving SeparabilityinMixed States & Elie Wolfe 11/26/2013

2. What is the linear quantum bound? Probability of correlated output “Expected Values” More familiar form Most general CHSH linear inequality Well known One can easily switch between forms Preference explainedon slide 7 We want to find quantum limit for all weights e.g. Tsirelson Bound DESIDERATUM: To determine Quantum Bound for any weighted marginal-involving linear inequality

3. Motivation: Quantum Elliptope The TLM criterion (in a form given by Masanes) Linear bounds = facets. Can “fake” a curved boundary. Therefore this is an interesting problem! Necessary and also sufficient, assuming marginals are 100% random See Supporting Hyperplane Theorem NPA has given a result. We’ll come back to this point. • 4-d quantum elliptopesolved(TLM: ’87, ’88, ‘03) • 8-d quantum elliptopeopen problem (NPA hierarchy) PROPOSAL: Generate very many Tsirelsoninequaltiesto circumscribe the quantum elliptope with a collection of facets

4. Why an 8 dimensional space? Every corner square sums to 1. Here’s a general formula for the dimension of the elliptope Drop four more by equivalent ways to get the marginals. Derived yesterday. Ask me later for a proof.

5. Primary Source Recall that this is what we are tying to solve Citations imply an existing result in the literature …but I couldn’t get it to work. For now we are only interesting in the QM limit. “Theorem 2” has a formula for the general CHSH quantum bound So I asked for help. 1980 B.S. Cirel'sonCited by 753… “Quantum generalizations of Bell's inequality.” Letters in Mathematical Physics 4, 93-100. ALREADY SOLVED? From: Elie Wolfe<elupus@gmail.com>To: Tsirelson Boris<tsirel@post.tau.ac.il>Feb 22 2011 Dr. Tsirelson, …I am having difficulty understanding and implementing the condition that you expressed… Could point me to other papers discussing this work, in particular the paper which contains your proof of this theorem? ~Elie

6. Tsirelson’s Story From: Tsirelson Boris<tsirel@post.tau.ac.il>Feb 23 2011 to: Elie Wolfe<elupus@gmail.com> Dear Elie, …my formulas appear to be wrong. It means, some computational error was made by me many years ago. Now, if you want to, you can try to retrace the calculations, find the error and fix it.Unfortunately, the proof was never published (by me), and the calculations are lost. Why did not I publish it? Well, I was very discouraged that no one was interestedat all in these results.  Nowadays it seems strange, but check the literature of 1980-1990: quite few related works (only L. Landau, and Summers, Werner). And in addition, that time I was in Alia refusal in Soviet Union; it was difficult to me to publish anything at all, the more so, something "strange, neither mathematics nor physics, of very little interest". Best wishes, --Boris

7. Measures=Observables, Limit=Eigenvalue This is the recipe for determine linear quantum bounds. The use of tensor product Hilbert spaces may not always be ok. See: Tsirelson’s Problem We use X and Y bases b/c empty along the diagonals This is why we use expected values. Variation between the two measurements is all that matters. Reflection symmetry makes is easier, but is not needed. • Define only local quantum measurements with 1 eigenvalues. • The joint observations are composed operators. Hence the joy of • For qubits we draw from the Pauli Matrices • Measurements for different parties must commute. • For dichotomicscenarios, one degree of freedom is sufficient.

8. Find the Largest Eigenvalue Key point: Translate the linear inequality into a quantum operator Roots are just the eigenvals We must maximize this operator over all possible quantum states Wait, solve for the roots?? Right… Express the matrix →obtain characteristic polynomial→ solve for the roots → find largest root → vary over θA& θB(to span all polynomial coefficients). How large can we get the largest root to be? This is the Quantum Bound

9. A Better Way / An Analytic Approach Solving for the many (#marginal-observables=4) roots of the polynomial is HARD. Simultaneous maximization over multiple degrees of freedom (θA,θB…=2) is HARD. Advantage: Hyperbolic Polynomial (all roots real) Use Intermediate Value Theorem to tightly define the region larger than the largest root.

10. Intermediate Value Theorem Just a random quartic with four real roots. All sign changes in the function and in its derivatives happen BEFORE largest root. From maximize over multiple… We can always set the leading coefficient to one. Look at behavior near +infinity. Must be the same!! …to MINimizeover just one! Holds for any HermitianEigenvalue Problem!

11. “ForAll” Constrained Optimization We want to maximize the right hand side And hence we have derived the quantum bound! We drop weak conditions Back-subsitute Easier to maximize of Cosine squared than θ First step: convert to an operator. Trivial algebra Note how this gives weights to a marginal Constrain: The C.P. and all derivatives must be positive. This is a non-trivial bound! First condition is stronger (given non-negative) Which condition is stronger? Right hand side is at-most 9. Easy! Only these two matter now This is what is important! Specific form is not important. We’ll work this out together now. We want worst-case scenario, needing largest m 

12. Results: New Quantum Bounds This top inequality is the one we just derived. TAKE HOME MESSAGE: Simple marginal-involving bounds

13. Upgrade: Nonlinear Quantum Bounds SURPRISE! Analytic technique work for non-numbers as well!

14. Sample Contrast with NPA Here is a lightly-biased scenario, a test point in 8-space Question is, is it quantum? Could this ever happen? A linear quantum bound, deemed QB3 NPA1, criterion for first level of hierarchy NPA1 says it’s ok… but QB3 rejects it!

15. EXTENSION #1 The authors sought to compare their IC bound to the true QM bound. They used NPA1. But QB3 is tighter across this facet. “Recovering part of the quantum boundary from information causality“ PRA 80, 040103 (2009) Jonathan AllcockNicolas BrunnerMarcinPawlowskiValerio Scarani

16. EXTENSION #2 Cabello solved for the fraction of statistics compatible with QM in the 4-dimensional correlations subspace But the 8-dimensional volume for the total quantum elliptope… “How much larger are quantum correlations than classical ones?“ PRA 72, 012113 (2005) Adán Cabello

17. SummarySlide Quantum Linear Inequalities are (again) State of the Art Quantum Linear Inequalities are Readily Derived Hermitian-Matrix Eigenvalue-Maximization can always be reformulated as single-variable constrained-optimization due to the Intermediate Value Theorem

18. Certifying Separability of Mixed States Two Easy Ideas

19. Appearances can be deceiving Here are some highly entangled Bell states… they can be expressed as product states… and when mixed are seen to be separable!

20. We want to certify full separability PPT PPT along a bipartition WEAK Separable over that bipartition MEDIUM Separable over all bipartitions STRONG Completely Separable Last resort: Try to find an explicitly separable decomposition

21. Easy Idea #1: Build & Check Entangled States All Seperable States Seperable States of a Given Form Does the mixed state fit the form of some family of separable states?

22. Diagonally Symmetric States GDS are (usually) entangled. SDS are made to be separable GDS are (usually) entangled. The entangled states might fit the separable form! Does it fit? Just an example. Convexity conditions are important. a tailor-made sufficient separability criterion:

23. Four-Qubit Explicit Example For diagonally symmetric states, this system of equations checks fit Five equations for a system of four qubits. Our form gives x & y in pairs. Derived on previous slide. Always N+1 equations. Should be same # of degrees of freedom! So when odd # of equations we set last y to zero. 5th parameter

24. Dicke Model Superradiance These equations govern superrad evolution Starting with just separable, becomes a mixture over entangled. Does superradiance time evolution lead to entangled states ?

25. Easy Idea #2: Equality by Integration Did we do a good job? Incredible! ALL separable states certified by build-&-check! Trying to ballpark the fraction of separable states it certifies… Two birds one stone: PPT also proven sufficient on this family of states. So we have a sufficient separability criterion… How “good” is it? Could it be tight? Compare to PPT INTEGRATE THE VOLUME OF STATES PER CRITERION PPT: We define a 0/1 indicator function on the state SDS: We use a Jacobian transform to determine a volume element Volume SDS = Volume PPT  Necessary & Sufficient!

26. SummarySlide Generate Sufficient Separability Criteria by Build & Check Rank Criteria by their Volume

27. Detailed Nonlinear Calculation

28. Dicke States Four qubitDicke state basis The Dicke states are sums-over-permutations of computational basis states, and therefore generally highly entangled.

29. Hardy Nonlocality?

30. Upper bound according to LHVM & NOSIG • NOSIG 8 constraints • LHVM 8 additional Bell inequalities Alternative: Maximize over convex combinations of the 16 deterministic possibilities. The space of this set, however, is exactly the Bell inequalities.

31. NOSIG constraint derivation -1 +1 Discrepancy in the marginal expectations limits the allowed correlations

32. Original Manuscript