Bounds on Bipartite Entanglement from Marginal Measures Giuseppe - - PowerPoint PPT Presentation

Bounds on Bipartite Entanglement from Marginal Measures

Supervisors: Giuseppe Baio Antonino Messina (Università degli studi di Palermo, Italy) Dariusz Chruściński (Nicolaus Copernicus University in Toruń, Poland)

49th Symposium on Mathematical Physics Toruń, Poland

Introduction

 Achieving robust manipulation of quantum dynamics is still a far reaching challenge which

• ffers a remarkable amount of applications.

 Entanglement as a Resource Multipartite Quantum Systems Superposition of states Entanglement Suitable criteria to check Entanglement Measures to characterize the amount of entanglement

Introduction

 Concurrence (Ent. of Formation) :

• Two–Qubit Case:

 Negativity (PPT criterion):

• PPT criterion necessary and sufficient for

 Problem: Suppose to prefix the reduced DM (Marginals). There exists restrictions on such measures stemming from the assigned marginals?

W.K. Wootters, Phys. Rev. Lett. 80, 2245, 1998.

• G. Vidal & R.F. Werner, Phys. Rev. A. 65, 032314, 2002.

Sommario

 Our Problem and Its Motivation:

1. The Quantum Marginal Problem 2. Parametrization of States with constrained marginals 3. Maximally Entangled Mixed states

 Results:

1. Two Qubits case (X-states) 2. Higher Dimensions (Circulant States)

 Conclusive Remarks

Outline

The Quantum Marginal Problem

 Given a multipartite quantum system and a set of marginals:  The answer is always positive:  The restrictions on the class of possible joint states can be described efficiently in terms of spectral inequalities.

e.g.

?

Are they compatible with a compound state?

The Quantum Marginal Problem: Simple cases

 Example: Pure states of a bipartite system.  Mixed Two-Qubit QMP (S. Bravyi, 2004)  Marginal constraints implies restrictions on the purity of the joint state.

Schmidt decomp.: Reduced density Matrices They always have the same spectrum:

The Quantum Marginal Problem: Known Results

 For any Bipartite QMP the constraints can be derived algorithmically (Klyachko, 2004).  E.g. N-qubit pure QMP – «Polygonal Inequalities»:  The set of possible marginal spectra of a pure state always form a convex polytope.

• A. A. Klyachko, "Quantum marginal problem and N-representability", Journ. of Phys: Confer. Ser., 36: 72–86, 2006.

Entanglement Class Spectral Polytope

Parametrization of States

 Bloch Vector Parametrization – Simple geometrical interpretation, Easy control of composite systems.  Determining the boundary of the allowed values for the set {βi} is a complex problem in general :

Positive definite Convex subset Bloch/ Coherence vector Decartes’ rule of signs Characteristic Polynomial

• E. Brüning, H. Mäkelä, A. Messina & F. Petruccione, Journal of Modern Optics 59.1 (2012): 1-20.

Parametrization of States with constrained marginals

 Parametrization of joint states suitable for two-qubits and two qutrits case:  Positive semidefinit. conditions can be derived using Cholesky factorization algorithm.  The result is valid for arbitrary dimension.  This allows us to construct varieties of states according to the constrained marginals.

Correlation Matrix positive semidefinite iff Lower triangular with non negative diagonal entries n-1 inequalities : Conditions on the entries of Δcorr

Parametrization of States with constrained marginals

 E.g. two qubits case:

• Valid if all Lii are stricly positive!
• We can also obtain similar results

for the semidefinite case.

Maximally Entangled Mixed States

 MEMS : States such that, for a fixed purity, their EoF cannot be increased by any global unitary transformation

W.J. Munro, D.F.V. James, A.G. White & P.G. Kwiat, Phys. Rev. A, 56, 4452, 1991.

Werner States MEMS

Linear Entropy Tangle (C2)

Maximally Entangled Mixed States

 MEMS : States such that, for a fixed purity, their EoF (Concurrence) cannot be increased by any global unitary transformation  Theorem: Given a state, the unitary transformation maximizing the EoF is of the following form (Verstraete, 2001):  MEMS are within the class of X-states : we can restrict ourselves to this class for two- qubits.

• S. Ishizaka & T. Hiroshima, Phys, Rev. A, 62, 22310, 2000.

W.J. Munro, D.F.V. James, A.G. White & P.G. Kwiat, Phys. Rev. A, 56, 4452, 2001.

Eigenvalue dec.

• F. Verstraete, K. Audenart, T.D. Bie & B.D. Moor, Phys. Rev. A, 56, 030302, 2001.

Two-Qubit Case

 We compare such inequalities with the study of concurrence. Class of X-States Given Marginals Positivity Conditions

Parametrized in terms of ε :

Two Qubit Case

 Concurrence of an X-state:  The maximum of concurrence is obtained when s=1 ρAB is entangled iff

Two Qubit Case

 Concurrence of an X-state:  This value of Concurrence represents then the upper bound on the entanglement with fixed marginals (See also Adesso, Illuminati, De Siena, 2003).  How can we generalise this result to higher dimensions? ρAB is entangled iff

Generalizing X-States

See also S.R. Hedemann, "Evidence that all states are unitarily equivalent to X states of the same entanglement." arXiv preprint arXiv:1310.7038, 2013.

 Consider the transformation by Verstraete et. Al :  Let us generalize naively this transformation to 3x3 case :

Basis with 2 Maximally entangled States (Bell States) States with Cyclic Structure (Circulant)

Circulant States

 Ciclic property of two qubit X-states:  Let us construct 3x3 states with such a property:

• D. Chruściński & A. Kossakowski, Phys. Rev. A 76.3, 032308, 2007.

Circulant States

 3x3 Circulant states :  Partial Transpose is again Circulant: Positive iff PPT iff

Circulant States

 Class of 3x3 Circulant states compatible with Marginals:

Circulant States

 Class of 3x3 Circulant states compatible with Marginals:  Negativity:

Circulant States

 Class of 3x3 Circulant states compatible with Marginals:  Maximization of purity: We conjecture that, in our problem, maximizing purity is the same as maximizing Entanglement. (In the qubit case the two coincide).

Circulant States

 Possible candidates within the class:

Conclusions

 A certain Marginal measure implies restrictions of the Purity of the joint States (Marginal Constraints). Therefore, it also implies upper bounds on the Entanglement measures.  We try to develop some tools to estimate such bounds.  At this stage we need refined numerical methods that could confirm (or contradict)

• ur intuition.

 Convex Optimization methods (Semi-Definite Programming) might tell us if our candidates are true MEMS with fixed marginal within our simple 3x3 class. (See Mendonça et Al., 2017)  Next Step: Analyze nonclassical correlations by means of Quantum Discord in the same situation of fixed marginals.

Thank you for your attention