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

Bounds on Bipartite Entanglement from Marginal Measures Giuseppe Baio Antonino Messina (Università degli studi di Palermo, Italy ) Supervisors: Dariusz Chru ściński (Nicolaus Copernicus University in Toruń, Poland ) 49 th Symposium on Mathematical Physics Toruń, Poland

Introduction  Achieving robust manipulation of quantum dynamics is still a far reaching challenge which offers a remarkable amount of applications. Multipartite Quantum Systems Superposition of states Entanglement Suitable criteria to check Entanglement  Entanglement as a Resource 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? G. Vidal & R.F. Werner, Phys. Rev. A. 65, 032314, 2002. W.K. Wootters, Phys. Rev. Lett. 80, 2245, 1998.

Sommario Outline  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

The Quantum Marginal Problem  Given a multipartite quantum system and a set of marginals: Are they compatible with a e.g. compound state? ?  The answer is always positive:  The restrictions on the class of possible joint states can be described efficiently in terms of spectral inequalities.

The Quantum Marginal Problem: Simple cases  Example: Pure states of a bipartite system. Reduced density Matrices Schmidt decomp.: They always have the same spectrum:  Mixed Two-Qubit QMP ( S. Bravyi, 2004 )  Marginal constraints implies restrictions on the purity of the joint state.

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

Parametrization of States  Bloch Vector Parametrization – Simple geometrical interpretation, Easy control of composite systems. Positive definite Convex subset Bloch/ Coherence vector  Determining the boundary of the allowed values for the set { β i } is a complex problem in general : Characteristic Polynomial Decartes ’ rule of signs 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: Correlation Matrix  Positive semidefinit. conditions can be derived using Cholesky factorization algorithm . Lower triangular with non positive semidefinite iff negative diagonal entries n-1 inequalities : Conditions on the entries of Δ corr  The result is valid for arbitrary dimension.  This allows us to construct varieties of states according to the constrained marginals.

Parametrization of States with constrained marginals  E.g. two qubits case: • Valid if all L ii 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 Werner States MEMS Tangle ( C 2 ) Linear Entropy W.J. Munro, D.F.V. James, A.G. White & P.G. Kwiat, Phys. Rev. A, 56, 4452, 1991.

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): Eigenvalue dec.  MEMS are within the class of X-states : we can restrict ourselves to this class for two- qubits. F. Verstraete, K. Audenart, T.D. Bie & B.D. Moor, Phys. Rev. A, 56, 030302, 2001. W.J. Munro, D.F.V. James, A.G. White & P.G. Kwiat, Phys. Rev. A, 56, 4452, 2001. S. Ishizaka & T. Hiroshima, Phys, Rev. A, 62, 22310, 2000.

Two-Qubit Case Given Marginals Class of X-States Positivity Conditions Parametrized in terms of ε :  We compare such inequalities with the study of concurrence.

Two Qubit Case  Concurrence of an X-state: ρ AB is entangled iff  The maximum of concurrence is obtained when s=1

Two Qubit Case  Concurrence of an X-state: ρ AB is entangled iff  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?

Generalizing X-States  Consider the transformation by Verstraete et. Al : Basis with 2 Maximally entangled States (Bell States)  Let us generalize naively this transformation to 3x3 case : States with Cyclic Structure (Circulant) 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.

Circulant States  Ciclic property of two qubit X-states:  Let us construct 3x3 states with such a property: D. Chru ściński & A. Kossakowski, Phys. Rev. A 76.3, 032308, 2007.

Circulant States  3x3 Circulant states : Positive iff  Partial Transpose is again Circulant: 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) our 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.

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