VANISHING ROLE OF ENTANGLEMENT IN
QUANTUM METROLOGY
Antonio Acin1
Remigiusz Augusiak1
Manabendra Nath Bera2
Janek Kolodynski1
Maciek Lewenstein1
Alexander Streltsov1
1ICFO
– The Institute of Photonic Sciences, Barcelona, Spain
2HRI – Harish-Chandra Research Institute, Allahabad, India
ICE-2 Conference, Bilbao, June 2015
ASYMPTOTICALLY VANISHING SIGNIFICANCE OF
ENTANGLEMENT IN QUANTUM METROLOGY
Antonio Acin1
Remigiusz Augusiak1
Manabendra Nath Bera2
Janek Kolodynski1
Maciek Lewenstein1
Alexander Streltsov1
1ICFO
– The Institute of Photonic Sciences, Barcelona, Spain
2HRI – Harish-Chandra Research Institute, Allahabad, India
ICE-2 Conference, Bilbao, June 2015
QUANTUM METROLOGY PROTOCOL
Unitary encoding of the parameter:
[e.g. (squeezed) photons in Mach-Zehnder interferometry, (spin-squeezed) atoms in Ramsey spectroscopy]
Ultimate bound on estimation precision in the limit of sufficiently large statistics (ν → ∞):
Quantum Cramer-Rao Bound
Quantum Fisher Information (QFI)
[optimised over all measurement/inference strategies, parameter-independence of QFI due to unitary encoding]
[”sufficiently large statistics” ???]
IMPORTANCE OF ENTANGLEMENT
QFI is additive and convex:
For separable states precision is bounded by the
Standard Quantum Limit:
SQL
Entanglement yields improved precisions:
GHZ state:
yields the ultimate Heisenberg Limit.
But...
W state:
[notice that both are Genuinely Multipartite Entangled (GME) !!!]
For large systems, N >> 1, it is the precision scaling that we should care about, i.e.:
HL
MAIN CLAIM OF OUR WORK (AND THIS TALK)
• In order to attain “exactly the HL” (1/N2) in the above metrological protocol as N→∞,
both the relative size and the amount of entanglement cannot be vanishing
asymptotically with N.
• In order to attain “almost the HL” (1/N2-ε for any ε>0) in the above metrological
protocol as N→∞, both the relative size and the amount of entanglement may be
vanishing asymptotically with N.
In the asymptotic N limit, one may achieve any super-classical
scaling of precision arbitrarily close to the Heisenberg Limit despite
arbitrarily little entanglement being contained in the system.
“Relative size” of entanglement
“Amount” of entanglement
k-producibility,
Largest Entangled Block (LEB) of particles
Geometric measure of entanglement, EG
k-PRODUCIBILITY AND THE RELATIVE SIZE OF LEB
An N-particle pure state is termed k-producible if it can be written as a tensor product:
An N-particle mixed state is termed k-producible if it can be expressed as a convex sum of kproducible pure states:
We define a state to have the
Largest Entangled Block, LEB = l, if:
(marked in green)
think of a Werner state:
[O. Guhne, G. Toth, and H. J. Briegel, NJP 7, 229 (2005)]
[O. Guhne and G. Toth, PRA 73, 052319 (2006)]
as entanglemement depth [A. S. Sørensen and K. Mølmer, PRL 86, 4431 (2001)]
p or (1-p) ~ O(exp[N]) for separability or GME
[O. Guhne, M. Seevinck, NJP 12, 053002 (2010)]
METROLOGICAL IMPLICATIONS OF k-PRODUCIBILITY
[G. Toth, PRA 85, 022322 (2012); P. Hyllus et al, PRA 85, 022321 (2012)]
For super-classical precision-scaling
k-producibility must grow with N:
But we should really care about relative size of the LEB – the ratio to the total number of particles:
Can be asymptotically vanishing for any ε>0 !!!
To attain “exact HL” RLEB must be asymptotically approaching a constant, but to attain a precision-scaling
“arbitrarily close to HL” RLEB potentially may be taken to be arbitrary small for sufficiently large N.
GEOMETRIC MEASURE OF ENTANGLEMENT, EG
The geometric measure of entanglement is defined as:
Geometric interpretation:
notice such notion can be generalised to any convex set of states,
e.g. k-producible ones (Geometric measure of k-producibility)
CONTINUITY OF QFI ON QUANTUM STATES
We prove that the difference of the QFI for any two states is upper-bounded by their
relative distance defined via fidelity (and hence Bures distance, trace distance …):
Aside for specialists:
In general, we prove ξ=8 using purification-based definition of QFI
[A. Fujiwara, PRA 63, 042304 (2001); B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nature Phys. 7, 406 (2011)],
but if one of the states is pure we may tighten the bound to ξ=6 via
the convex-roof-based definition of QFI
[G. Toth and D. Petz, PRA 87, 032324 (2013); S. Yu arXiv:1302.5311 (2013)]
Crucially, this allows us to bound the QFI of a state via its geometric measure of entanglement EG:
Thus, from the point of view of the asymptotic precision scaling:
Can be asymptotically
vanishing for any ε>0 !!!
To attain “exact HL” EG must be asymptotically approaching a constant, but to attain a precision-scaling
“arbitrary close to HL” EG potentially may be taken to be arbitrary small for sufficiently large N.
METROLOGICAL IMPLICATIONS OF THE QFI-CONTINUITY
Consider two sequences of states, out of which one contains only states of fixed LED, RLED=l/N :
Hence, for sequence
to attain an “arbitrarily close to HL”
precision-scaling:
Thus, continuity predicts that
can approach asymptotically any state of fixed producibility
(even separable l=1)…????? How is this possible??
attain the boundary of
from inside, whereas
attain the boundary from outside at
slow enough rate, so that the producibility of
may arbitrarily increase.
[Our results suggest exponential collapse of the producibility hierarchy with N. This is known for the set of separable states]
(L. Gurvits and H. Barnum, PRA 72, 032322 (2005); S. J. Szarek, PRA 72, 032304 (2005))
STATES THAT DO THE JOB
Pure states – moving along the boundary of
hierarchy:
Mixed states – moving “right in the middle” through the sets of
[L. E. Buchholz, T. Moroder, and O. Gühne, arXiv:1412.7471]
hierarchy:
CONCLUSIONS
• In order to attain “exactly the HL” (1/N2) in a standard metrological protocol as
N→∞, both the relative size (RLEB) and the amount (EG) of entanglement
cannot be vanishing asymptotically with N.
• In order to attain “almost the HL” (1/N2-ε for any ε>0) in a standard
metrological protocol as N→∞, both the relative size (RLEB) and the amount
(EG) of entanglement may be vanishing asymptotically with N.
• We hope our work suggests what sort of entanglement properties should be
preserved when decoherence effects are included in large systems, e.g.
uncorrelated noise forces LEB to be a noise-dependent constant as N→∞.
• Our results explain why bound entangled states can fully reach “the exact HL”
[Ł. Czekaj et al, (2014), arXiv:1403.5867].
• But can one propose experimentally motivated state sequences that allow to
benefit from such an effect... (Dicke, spin-squeezed, Twin-Fock, squeezed...).
• What about the Bayesians?
THANK YOU FOR YOUR ATTENTION