This book deals with the study of quantum communication protocols with Continuous Variable (CV) systems. Continuous Variable systems are those described by canonical conjugated coordinates x and p endowed with infinite dimensional Hilbert spaces, thus involving a complex mathematical structure. A special class of CV states, are the so-called Gaussian states. With them, it has been possible to implement certain quantum tasks as quantum teleportation, quantum cryptography and quantum computation with fantastic experimental success. The importance of Gaussian states is two- fold; firstly, its structural mathematical description makes them much more amenable than any other CV system. Secondly, its production, manipulation and detection with current optical technology can be done with a very high degree of accuracy and control. Nevertheless, it is known that in spite of their exceptional role within the space of all Continuous Variable states, in fact, Gaussian states are not always the best candidates to perform quantum information tasks. Thus non-Gaussian states emerge as potentially good candidates for communication and computation purposes.
2. Quantum Information with Continuous Variable systems
Carles Rodó Sarró
UAB
30 Abril 2010
Supervisor: Anna Sanpera Trigueros
2
3. “Information is physical”
Rolf Landauer 1960.
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quantum bit (qubit)
3
4. Outline
•Introduction and Motivation
What and why Continuous Variable systems?
•Correlationsquantum correlations for communication.
Classical and/or in CV systems
•Measurement induced Entanglement
The enhancement of quantum measurements.
•Conclusions
4
5. Introduction and Motivation
d-level system
spin 1/2
one-mode system
CV systems are those described by two
canonical conjugated degrees of freedom
Gaussian states
non-Gaussian states
Examples
5
6. Introduction and Motivation
Hilbert space Phase space
Density operator Wigner quasi-probability distribution
Fourier-Weyl transform
vs • Infinite-dimensional and
• Complex space
• Operator character
• Infinite-dimensional but
• Real space but symplectic
• C-numbers but symmetrization
6
7. Introduction and Motivation
Gaussian states iff Gaussian Wigner distribution
Single-mode
as a Gaussian distribution,1st and 2nd
moments contain all the information
Multi-mode
displacement vector, DV
covariance matrix, CM
Gaussian states have a finite description
7
8. Introduction and Motivation
Gaussian states Hilbert space Phase space
dimension
structure
states
positivity (hermiticity)
spectra
Gaussian operations
purity
Gaussian states are easy and cheap!
8
9. Introduction and Motivation
non-Gaussian states Hilbert space Phase space
dimension
structure
states
positivity (hermiticity)
spectra
Gaussian operations
purity
9
10. Outline
•Introduction and Motivation
What and why Continuous Variable systems?
•Correlationsquantum correlations for communication.
Classical and/or in CV systems
•Measurement induced Entanglement
The enhancement of quantum measurements.
•Conclusions
10
11. Correlations in CV systems
Pure states
PPT-criterium (time reversal)
Discrete Continuous
entanglement
A. Peres PRL 77, 1413, 1993.
NPPT entanglement
M. Horodecki PLA 223, 1, 1996. R. Simon PRL 84, 2726, 2000.
R. F. Werner. PRL 87, 3658, 2001.
11
12. Correlations in CV systems
Bipartite Gaussian states
Example
Input
EPR entanglement Output
12
13. Correlations in CV systems
Gaussian states Hilbert space Phase space
dimension
structure
states
positivity (hermiticity)
spectra
Gaussian operations
purity
fidelity
separability
entanglement
13
14. Correlations in CV systems
Tripartite qubit Tripartite Gaussian
convex and compact sets
A. Acín PRL 87, 040401, 2001.
G. Giedke PRA 64, 052303, 2001.
14
15. Quantum protocols with CV
Cryptography bipartite entanglement
Entanglement is used in the protocol to distribute a private random key between
two parties in a secure way i.e. malicious manipulations are detected.
Byzantine Agreement multipartite entanglement
Entanglement between three or more players is used to achive a common
decision detecting malicious contradictory actions.
15
16. Correlations in CV systems
Cryptography
Two completely equivalent schemes
#¿#
#¿# #?#
#?#
Prepare and Measure, BB84
• Security is guaranteed by the impossibility
of measuring simultaneously non-
commuting observables.
Alice (A) Bob (B) C. H. Bennett IEEE p175, 1984.
Entanglement Based, Eckert91
• Security is guaranteed by the nature of quantum
correlations and proved by violation of Bell
inequalities.
• Unconditional security is achieved with maximally
entangled states (distillation).
Eve (E)
A. Ekert PRL 67, 661, 1991.
16
17. Cryptography
Cryptography with Gaussian states à la Ekert
Problem 1: In the Gaussian scenario it is not possible to distill
maximally entangled states and proceed à la Eckert.
Solution
Nevertheless it was proven that a secret
key scan be obtained without distillation M. Navascués PRL 94, 010502, 2005.
Problem 2: Gaussian measurements on states fill a continuum.
Solution
Distributing bits from CV systems by digitalizing output measurements
mapping entanglement to bits correlations
measurements bits
17
18. Cryptography
Protocol: 1x1 mode
Any NPPT of NxM modes can be map with GLOCC to a
1xN mode preserving entanglement.
Thus it suffices to consider the case 1x1 mixed state.
We have assumed Eve is entangled with Alice and Bob, thus
Alice and Bob’s state is mixed.
4-mode pure state (purification)
positive
NPPT
(entanglement)
18
19. Cryptography
Protocol: steps
1. Alice and Bob perform homodyne measurement of their x
quadratures. They associate to a positive/negative value the bit 0/1.
A string of sign-bit correlations is induced.
2. Bob publicly announces only the modulus of his outcomes.
3. Only unphysical perfect EPR give exact coincident outcomes.
We assume a range of sufficient good correlations.
4. Eve’s state after Alice and Bob have projected onto is
Security of Classical Advantage Distillation
error probability of non-coincident signs
Eve’s distinguishability
individual collective
A. Acín PRL 91, 167901, 2003.
19
20. Cryptography
Efficiency: average probability of obtaining a classical correlated bit
(over the range of secure outcomes)
Range of secure outcomes for Alice and Bob
Open Sys. Inf. Dyn., 14 (69), 2007.
20
21. Correlations in CV systems
Byzantine agreement
““Attac
Attac ““Attac
Attac
k”
k” k”
k”
““Attac
Attac ““Attac
Attac
k” k”
k”
k”
pairwise communication + secure classical channels
23
22. Correlations in CV systems
Byzantine agreement
““Attac
Attac
”
kk” ?
L. Lamport ACM 4, 382, 1982.
““Attack
““Retrea
Retrea ““Attac
Attac
”
kk”
?
Attack ”
tt”
””
““Retreat
Retreat
““Retrea
Retrea
”” ”
tt”
The commanding general sends an order to
his n-1 lieutenants such that:
(i) All loyal lieutenants obey the same order.
(ii) If the commanding general is loyal, then Detectable broadcast
every loyal lieutenant obeys the order he
sends.
24
23. Byzantine agreement
Quantum solution
Primitive
Solution with qutrits exists M. Fitzi PRL 87, 217901, 2001.
pure fully inseparable tripartite completely symmetric
Solution with Gaussian states?
25
24. Byzantine agreement
measurements trits
It’s not possible to achieve this trit-primitive with Gaussian states
We proposed the first protocol that uses tri-partite genuine
Gaussian entanglement by invoking twice a bit primitive and
mapping it into the desired primitive
Considering any degree of entanglement
Phys. Rev. A, 77 (062307), 2008.
26
25. Entanglement of non-Gaussian states
for non-Gaussian states the separability problem is
extremely hard
lack of efficient entanglement measures
infinite moments!
E. Shchukin PRL 95, 230502, 2005.
•De-gaussifications of Gaussian states
1x1 non-Gaussian bipartite states
•Mixtures of Gaussian states
28
26. Entanglement of non-Gaussian states
We study the relation between the performance on
extracting classical correlated bits from entangled CV
states with the correlations embedded in the states
We compute the conditional joined
probabilities that measuring arbitrary rotated
quadratures (with uncertainty ), Alice and
Bob can associate the bit 0/1 to a
positive/negative result.
We define the (normalized) degree of bit correlations
correlation
uncorrelation
anticorrelation
29
27. Entanglement of non-Gaussian states
Q measure (total correlations in CV bipartite systems)
bit quadrature correlations
average probability of obtaining a pair of classically correlated bit optimized
over all possible choice of local quadratures
Normalization
Zero on product states
Local symplectic invariance
30
28. Entanglement of non-Gaussian states
Gaussian states
Pure case
monotonic in negativity i.e.
measure of entanglement
Mixed case
standard form
invariant form
Q majorizes entanglement
(origin) Product states
•Separable mixed states measures classical
•Pure entangled states correlations only
•Maximally correlated states
•18.000 random 2-mode
Gaussian states
31
29. Entanglement of non-Gaussian states
Pure non-Gaussian states
Photonic Bell states
Photon substracted states
A. Kitagawa PRA 73, 042310, 2006.
32
30. Entanglement of non-Gaussian states
Mixed non-Gaussian states
Experimental de-gaussified states
Experiment Theory
A. Ourjoumtsev PRL 98, 030502, 2007.
The non-Gaussian operation allows
to increase the entanglement
between Gaussian states
Mixtures of Gaussian states Extremaility theorem
Good results
Phys. Rev. Lett., 100 (110505), 2008.
33
31. Outline
•Introduction and Motivation
What and why Continuous Variable systems?
•Correlationsquantum correlations for communication.
Classical and/or in CV systems
•Measurement induced Entanglement
The enhancement of quantum measurements.
•Conclusions
34
32. Measurement induced entanglement
B. Julsgaard N 413, 400, 2001.
collective angular momentum
Multipartite entanglement
1 CV mode
•Scalable system
•Magnetic adrdessing
not possible
35
34. Measurement induced entanglement
Bipartite EPR entanglement
a) Creation of entanglement (EPR)
entanglement is induced as
soon as light is measured
b) Verification of entanglement
spin variance inequalities are violated for all a
L.-M. Duan PRL 84, 2722, 2000.
37
35. Measurement induced entanglement
Continuous Variable analysis
atom-light initial state atom-light state
after interaction
symplectic interaction
bipartite atomic state after interaction and measurement
TMS state with squeezing parameter
38
37. Outline
•Introduction and Motivation
What and why Continuous Variable systems?
•Correlationsquantum correlations for communication.
Classical and/or in CV systems
•Measurement induced Entanglement
The enhancement of quantum measurements.
•Conclusions
41
38. Conclusions
Correlations in CV systems
• I have first shown that the sharing of entangled Gaussian variables and the use of
only Gaussian operations permits efficient Cryptography against individual and finite
coherent attacks.
• I have proposed the first tripartite protocol to solve detectable broadcast with
entangled Continuous Variable using Gaussian states and Gaussian operations only.
There exists a broad region in the space of the relevant parameters (noise,
entanglement, range of the measurement shift, measurement uncertainty) in which
the protocol admits an efficient solution.
• I have proposed an operational quantification of the correlations encoded in several
relevant non-Gaussian states being this a monotone for pure Gaussian states and
majorizing negativity for mixed ones.
• The measure considered, based on (and accessible in terms of) second moments
and homodyne detections only, provides an exact quantification of entanglement in
a broad class of pure and mixed non-Gaussian states, whose quantum correlations
are encoded non-trivially in higher moments too.
42
39. Conclusions
Measurement induced entanglement
• I have studied multipartite mesoscopic entanglement using a quantum atom-light
interface. Exploiting a geometric approach in which light beams propagate through
the atomic samples at different angles makes it possible to establish and verify EPR
bipartite entanglement explicitily through the complete covariance matrix, GHZ
and cluster-like multipartite entanglement.
• Finally I have shown that the multipartite entanglement created can be
appropriately tailored and even completely erased by the action of a second pulse
with an appropriate different intensity.
43
40. References
1. Efficiency in Quantum Key Distribution Protocols with Entangled Gaussian States.
C. Rodó, O. Romero-Isart, K. Eckert, and A. Sanpera.
Pre-print version: arXiv:quant-ph/0611277
Journal-ref: Open Systems & Information Dynamics 14, 69 (2007)
2. Operational Quantification of Continuous-Variable Correlations.
C. Rodó, G. Adesso, and A. Sanpera.
Pre-print version: arXiv:0707:2811
Journal-ref: Physical Review Letters 100, 110505, (2008)
3. Multipartite continuous-variable solution for the Byzantine agreement problem.
R. Neigovzen, C. Rodó, G. Adesso, and A. Sanpera.
Pre-print version: arXiv:0712.2404
Journal-ref: Physical Review A 77, 062307, (2008)
4. Manipulating mesoscopic multipartite entanglement with atom-light interfaces.
J. Stasińska, C. Rodó, S. Paganelli, G. Birkl, and A. Sanpera.
Pre-print version: arXiv:0907.4261
Journal-ref: Physical Review A 80, 062304, (2009)
5. A covariance matrix formalism for atom-light interfaces.
J. Stasińska, S. Paganelli, C. Rodó, and A. Sanpera.
Journal-ref: Submitted to New Journal of Physics
6. Transport and entanglement generation in the Bose-Hubbard model.
O. Romero-Isart, K. Eckert, C. Rodó, and A. Sanpera.
Pre-print version: quant-ph/0703177
Journal-ref: Journal of Physics A: Mathematical and Theoretical 40, 8019 (2007)
44
Notas del editor
-This thesis deals exclusively with CV systems.
-In this presentation I’m going to show the work performed in the last years here in the QIG under supervision of Anna Sanpera.
-I would like to begin motivating this thesis with an old sentence of one of the precursors of the Quantum Information theory. -It was Rolf Landauer in the 60s who coined the idea that information is rather an abstract concept a physical process, thus governed by physical laws. -This idea changes the way information is encoded, processed and extracted by using quantum systems giving rise to new phenomena and with as the new elementary unit the Qubit or quantum bit.
-The outline of the thesis is as follows. I’ve separeted the thesis in the following parts. -1. I will first review CV systems focusing in the analogies between discrete and continuous systems. Answering the question... -2. Our aim is to propose novel algorithms with CV entanglement and address the important question of how to separate classical from quantum correlations in CV. -3. I present measurement induced entanglement. Our aim is to show that multipartite entanglement can be induced on separated mesoscopic atomic samples of gases. -4. Finally I will give a summary of conclusions.
-A qubit can be described in a complex space of dimension 2 encoding it e.g. in the spin orientation. -There are quantum states which do not have discrete structure but CV. -CCR implies operators of infinite dimension and continuous spectrum for measurements. -Examples: the position-momentum of a massive particle or the so-callled quadratures of the electromagnetic field.
-Let me describe in some detail the mathematical support of CV. -When dealing with CV, 2 possible formulations arise. -Let me review the advantatges of working in the phase-space formalism.
-Lets define properly a Gaussian state. -I just focus in a single mode. -One con straightforwardly extend the Gaussian formalism to a multi-mode.
-Lets summarize what we know so far. -For Gaussian states Phase-Space is not only an alternative description but a better one. -From density operator to covariance matrix. -Only coment that Gaussian unitaries contain quadratic generators corresponding to linear optics devices. Thus Gaussian states are easy mathematically and easily implemented. Gaussian states are easy and cheap.
-Lack of advantages for non-Gaussian states. -With that we finish the description of CV uncorrelated states.
-As it’s well known, there exists entangled states which display quantum correlations. The same structure appears in the CV scenario. -What is time reversal in CV? simply changes the sign of the momentum.
-Let us illustrate with an example, say, with light. A squeezed state is minimal uncertainty like a coherent but with position/momentum squeezed/antisqueezed.
-Our table completes the perfect analogy for Gaussian states. Showing that phase space is a good framework to describe also multipartite states and entanglement.
-One can extend to more partitions. Entanglement classes arises and classification, quantification and separability becomes a very difficult task.
-Once the formalism has been presented my interest during the PhD has been to use CV to investigate protocols that cannot be solved classically. -1. Def. Secure communications are perse very important. -2. Def. The importance concerns syncronitzed comunications on networks.
-Prepare and Measure is experimentaly done but checking security is difficult. We are interested in entanglement based scheme with CV. Reasons: fundamental & security is easy to check.
-An algorithm: set of rules for solving a problem in a finite number of steps. Any quantum protocol ends up dealing with bits, so one important point in CV protocols is the way one extract bits from CV states. -ED=number of singlets that can be extracted (entanglement distillation). KD=number of secrey bits that can be extracted.
-Any mixed bipartite state of 2 modes is the reduction of a pure state of 4 modes.
-Unphyisical perfect EPR in CV will give perfect bit correlations. But eve is there, so no maximally entanglement is possible. The outcomes are not coincidents but lie within a range. - Individual: where Eve performs individual measurements, possibly non-Gaussian, over her set of states. -Collective: where Eve waits until the distribution has been performed and, decides which collective measurement gives her more information on the final key.
-Without entering into much details. Our results are summarized in this two graphs. -We define two quantities to analyze the efficient solution.
-We swich gears and move to a multiartite CV entanglement protocol concerning syncronitzed comunications.
-Indistinguishable situations.
-The security of this protocol is extremly tedious. The part I’m going to show concerns the escencials for the solution. -Aharonov state not feasible. -Loock & Braunstein introduce a scheme for creation of N mode entangled GS.
-Surprisingly there is a very independent bound on the entanglement content in contrast with Cryptography.
-The best approach to the separability problem of arbitrary bipartite CV states was introduced by Shchukin and Vogel. They provide necessary and sufficient condition for the negativity of the partial transposition through an infinite series of inequalities based on determinants of successively increasing size matrices containing high order moments of the state. But NPPT necessary but not sufficient for entanglement. -Our aim is to study correlations in this small region of entangled non-Gaussian states.
-Hierarchy of moments. One expect decreasing importance but not begining the importance with high moments. -In agreement with extremality of Gaussian states. \\psi_h has entanglement in high order moments.
-De-gaussification in a delocalized fashion, thus mixed. -Explicar la diferencia de grafica. -An optical parametric amplifier (OPA) produces a two-mode squeezed state. A small fraction (R<<1) of the beams interfere in a balanced beam splitter (BS). An avalanche photodiode (APD) detects a photon in one of the BS outputs. Subtracting a photon in a de-localized fashion.
-We are used to see that the quantum measure breaks the entanglement. -In 2001 it was experimentally demonstrated in the group of Polzik how one can entangle two atomic gas samples spacially separated letting light interact with the samples + measurement on light. -Verification (B is needed).
-Let us see why one can treat the problem from a CV point of view.
-Explicar Colors. -Heisenberg+Maxwell-Bloch equations. -The quantum character is reflected at the level of fluctuations. Outgoing light carries information of the samples state. -Outcome of homodyne measurement only affects the value of the displaçement. The CV is independent and so the entanglement. -For entanglement verification one need simultanious measurements of two variances. Then the samples must be oposite polarized for this operators commute and a magnetic field is necessary to access to the appropriate variances.
-Lets detail the Gaussian character of the process. -After interaction its easy to check that the state is separable.
-Reverible. Gamma si, displacement no.
-We want to mimic the action of the magnetic field letting the light interact at a certain angle.