Multipartite correlations and complementarity
Chiara Macchiavello University of Pavia
L. Maccone (University of Pavia) D. Bruss (University of Duesseldorf) B. Kraus, D. Sauerwein (University of Innsbruck)
OUTLINE
Provide interpretation of entanglement with classical correlations of measurement outcomes for complementary properties
Figures of merit: mutual information, Pearson coefficient
Role of quantum correlations and best separable states
Improvements: more than 2 complementary properties (qubits)
Generalisation to multipartite case
Entanglement
For two systems: pure states
mixed states
Previous approaches to entanglement based on non-locality, violation of Bell inequalities, positive partial transpose (PPT) criterion, etc
Here we focus on correlations of measurement outcomes for complementary observables
Reminder: complementary properties
If one knows the value of one property, all possible values of the other property are equally likely
E.g.: X, Y, Z for qubits
Simple examples
Maximally entangled state:
has perfect correlations both on 0/1 and +/-
Separable state
has perfect correlations for 0/1 and no correlation for +/-
Complementary correlations
Bipartite system
Compute classical correlations of measurement outcomes A-B and C-D
What figure of merit?
On system 1 measure either A or C
On system 2 measure either B or D
Complementary correlations for separable states
separable states entangled states
Do separable states achieve the bound? What kind of separable states, classically or quantum correlated?
Complementary correlations based on I are not convex and therefore they are not necessarily maximised on the border
Other separable states, with quantum correlations without entanglement, always lead to lower values
Optimal states are classically correlated (CC):
Complementary correlations for separable states
CQ and QC states:
QQ states:
Results
Two qubit states
Entanglement threshold
p=0,1 classically correlated otherwise q correlated
entangled for
entangled for
Pearson coefficient
Complementary correlations
perfect correlations
for product states
Conjecture 2)
1)
Results
Entanglement threshold
Notice: for separable states Person correlations are maximised by , classical and q correlations give the same value
Pearson vs mutual information
Even if the Pearson correlation measures only linear correlations, it is not weaker than information correlation:
The two-qubit state:
has negligible I for but Pearson correlations always >1!!
Improvement: 3 complementary properties
Add third complementary property: E and F (three Pauli operators for qubits)
Separable states fulfill
proved
conjectured
All entangled Werner states now exceed threshold of Pearson correlations
Multipartite correlation measure n: number of subsystems
N: number of complementary properties (MUBs)
Optimal states (maximal correlation measure)
1) Bipartite systems (n=2) with equal dimension d: maximally entangled states
2) Multipartite systems (n>2) with equal dimension: they can be mixed, e.g.
Info between one system and the rest
Pure multipartite states
2) Stabilised by local mutually unbiased operators
If a pure state has the following properties:
1) Completely mixed single-subsystems reduced states:
it has maximal correlation measure (sufficient condition)
Examples
1) n=3, d=2 QX
C2
Qz
Qy
3-tangle
2) n=3, d=3
3) n=d, d
Aharonov states
Correlation measure maximal in 2 bases, not in 3!
Correlation measure maximal in any N!
Correlation measure maximal in any N!
Entanglement detection
Starting from entropic uncertainty relations:
If the state is not fully separable
If the state is tripartite entangled
Tripartite entanglement detection
Eg: mixtures of GHZ and maximally mixed states:
Define:
Threshold value for p increases with increasing dimension
tripartite entanglement
SUMMARY AND OUTLOOK We introduced a classical info-theoretic approach to interpret entanglement by complementary observables
If correlations of measurement outcomes of complementary observables exceed a threshold value the state is entangled
Behaviour of classically and quantum correlated separable states depeds on the figure of merit (for I optimal separable states are CC) Improvement in the efficiency by adding a third complementary property (qubits)
Bipartite systems: maximal complementary correlations are necessary and sufficient condition for maximal entanglement
L. Maccone, D. Bruss & C.M., Phys. Rev. Lett. 114, 130401 (2015)
Multipartite systems: generalized correlation measures for any number of systems and complementary properties States that maximise complementary correlations are not necessarily pure D. Sauerwein, C.M., L. Maccone & B. Kraus, Phys. Rev. A 95, 042315 (2017)
Easy to implement! Z. Huang, L. Maccone, A. Karim, C.M.,R.J. Chapman & A. Peruzzo, Sci. Rep. 6, 27637 (2016)