Quantum Information with Continuous Variable systems
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Quantum Information with Continuous Variable systems Carles Rodó Sarró
description
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.
Transcript of Quantum Information with Continuous Variable systems
Quantum Information with Continuous Variable systems
Carles Rodó Sarró
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Quantum Information with Continuous Variable systems
30 Abril 2010UAB
Carles Rodó Sarró
Anna Sanpera Trigueros
Supervisor:
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“Information is physical”
Rolf Landauer 1960.
Para ver esta película, debedisponer de QuickTime™ y deun descompresor GIF.
quantum bit (qubit)
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Outline
•Introduction and Motivation
•Correlations in CV systems
•Measurement induced Entanglement
•Conclusions
What and why Continuous Variable systems?
Classical and/or quantum correlations for communication.
The enhancement of quantum measurements.
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Introduction and Motivation
d-level system
one-mode system
Gaussian statesnon-Gaussian states
CV systems are those described by two canonical conjugated degrees of freedom
spin 1/2
Examples
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Introduction and Motivation
Hilbert space Phase space
Fourier-Weyl transform
vs • Infinite-dimensional and
• Complex space
• Operator character
• Infinite-dimensional but
• Real space but symplectic
• C-numbers but symmetrization
Wigner quasi-probability distribution
Density operator
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Gaussian states iff Gaussian Wigner distribution
displacement vector, DVcovariance matrix, CM
as a Gaussian distribution,1st and 2nd moments contain all
the information
Multi-mode
Single-mode
Introduction and Motivation
Gaussian states have a finite description
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Introduction and Motivation
Gaussian states Hilbert space Phase space
dimension
structure
states
positivity (hermiticity)
spectra
Gaussian operations
purity
Gaussian states are easy and cheap!
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Introduction and Motivation
non-Gaussian states Hilbert space Phase space
dimension
structure
states
positivity (hermiticity)
spectra
Gaussian operations
purity
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Outline
•Introduction and Motivation
•Correlations in CV systems
•Measurement induced Entanglement
•Conclusions
What and why Continuous Variable systems?
Classical and/or quantum correlations for communication.
The enhancement of quantum measurements.
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Correlations in CV systems
PPT-criterium
entanglement
entanglementNPPT
Pure states
Discrete Continuous
A. Peres PRL 77, 1413, 1993.
(time reversal)
M. Horodecki PLA 223, 1, 1996. R. Simon PRL 84, 2726, 2000.R. F. Werner. PRL 87, 3658, 2001.
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EPR entanglement
Input
Output
Example
Correlations in CV systems
Bipartite Gaussian states
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Gaussian states Hilbert space Phase space
dimension
structure
states
positivity (hermiticity)
spectra
Gaussian operations
purity
Correlations in CV systems
fidelity
separability
entanglement
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Correlations in CV systems
Tripartite qubit Tripartite Gaussian
convex and compact sets
A. Acín PRL 87, 040401, 2001.
G. Giedke PRA 64, 052303, 2001.
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Cryptography bipartite entanglement
Byzantine Agreement multipartite entanglement
Quantum protocols with CV
Entanglement between three or more players is used to achive a common decision detecting malicious contradictory actions.
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.
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Correlations in CV systems
#¿#¿###¿#¿##
#?#?###?#?##
Alice (A) Bob (B)
Eve (E)
Cryptography
Entanglement Based, Eckert91
Prepare and Measure, BB84•Security is guaranteed by the impossibility of measuring simultaneously non-commuting observables.
•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).
Two completely equivalent schemes
C. H. Bennett IEEE p175, 1984.
A. Ekert PRL 67, 661, 1991.
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Cryptography
Distributing bits from CV systems by digitalizing output measurements mapping entanglement to bits
correlations
Problem 2: Gaussian measurements on states fill a continuum.
Problem 1: In the Gaussian scenario it is not possible to distill maximally entangled states and proceed à la Eckert.
Nevertheless it was proven that a secret key scan be obtained
without distillation
Cryptography with Gaussian states à la Ekert
Solution
M. Navascués PRL 94, 010502, 2005.
measurements
bits
Solution
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Protocol: 1x1 mode
We have assumed Eve is entangled with Alice and Bob, thus Alice and Bob’s state is mixed.
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.
positiveNPPT
(entanglement)
4-mode pure state (purification)
Cryptography
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4. Eve’s state after Alice and Bob have projected onto is
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.
Eve’s distinguishability
error probability of non-coincident signs
Security of Classical Advantage Distillation
individual collective
Cryptography
A. Acín PRL 91, 167901, 2003.
3. Only unphysical perfect EPR give exact coincident outcomes. We assume a range of sufficient good correlations.
Protocol: steps
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Range of secure outcomes for Alice and Bob
Efficiency: average probability of obtaining a classical correlated bit (over the range of secure outcomes)
Open Sys. Inf. Dyn., 14 (69), 2007.
Cryptography
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Correlations in CV systems
Byzantine agreement
““AttaAttack”ck”
““AttaAttack”ck”
““AttaAttack”ck”
““AttaAttack”ck”
““AttaAttack”ck”
““AttaAttack”ck”
““AttaAttack”ck”
““AttaAttack”ck”
pairwise communication + secure classical channels
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Correlations in CV systems
Byzantine agreement““AttaAtta
ckck””““AttaAtta
ckck””
““RRetreetre
atat””““RRetreetre
atat””““AttaAtta
ckck””““AttaAtta
ckck””
?
““RetreRetre
atat””““RetreRetre
atat””
““AttaAtta
ckck””““AttaAtta
ckck””
““RetreRetre
atat””““RetreRetre
atat””
?
L. Lamport ACM 4, 382, 1982.
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 every loyal lieutenant obeys the order he sends.
Detectable broadcast
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Byzantine agreement
pure fully inseparable tripartite completely symmetric
Primitive
Solution with qutrits exists
Solution with Gaussian states?
Quantum solution
M. Fitzi PRL 87, 217901, 2001.
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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 primitiveConsidering any degree of entanglement
Byzantine agreement
Phys. Rev. A, 77 (062307), 2008.
measurements
trits
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Entanglement of non-Gaussian states
1x1 non-Gaussian bipartite states
for non-Gaussian states the separability problem is extremely hard
infinite moments!
•De-gaussifications of Gaussian states
•Mixtures of Gaussian states
lack of efficient entanglement measures
E. Shchukin PRL 95, 230502, 2005.
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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
correlationuncorrelationanticorrelation
Entanglement of non-Gaussian states
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bit quadrature correlations
Normalization
Zero on product states
Local symplectic invariance
Q measure (total correlations in CV bipartite systems)
average probability of obtaining a pair of classically correlated bit optimized over all possible choice of local quadratures
Entanglement of non-Gaussian states
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Gaussian statesPure case
monotonic in negativity i.e. measure
of entanglement
(origin) Product states•Separable mixed states•Pure entangled states•Maximally correlated states•18.000 random 2-mode Gaussian states
Q majorizes entanglement
measures classical
correlations only
Mixed casestandard form
invariant form
Entanglement of non-Gaussian states
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Pure non-Gaussian states
Photonic Bell states
Photon substracted states
Entanglement of non-Gaussian states
A. Kitagawa PRA 73, 042310, 2006.
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Experimental de-gaussified states
Mixtures of Gaussian states
The non-Gaussian operation allows to increase the
entanglement between Gaussian states
Mixed non-Gaussian states
Experiment Theory
Good resultsPhys. Rev. Lett., 100 (110505), 2008.
Entanglement of non-Gaussian states
Extremaility theorem
A. Ourjoumtsev PRL 98, 030502, 2007.
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Outline
•Introduction and Motivation
•Correlations in CV systems
•Measurement induced Entanglement
•Conclusions
What and why Continuous Variable systems?
Classical and/or quantum correlations for communication.
The enhancement of quantum measurements.
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Measurement induced entanglement
Multipartite entanglement•Scalable system•Magnetic adrdessing not possible
collective angular momentum
B. Julsgaard N 413, 400, 2001.
1 CV mode
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Measurement induced entanglement
Dipolar interaction
Matter-light interaction
Gaussian interactio
n
Light
x-polarized z-propagating
1 modeStokes
Atoms
x-polarized
1 modecollective angular momentum
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Measurement induced entanglement
Bipartite EPR entanglement
spin variance inequalities are violated for all a
a) Creation of entanglement (EPR)
b) Verification of entanglement
entanglement is induced as soon as light is measured
L.-M. Duan PRL 84, 2722, 2000.
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Measurement induced entanglement
Continuous Variable analysis
atom-light initial stateatom-light state after interaction
symplectic interaction
bipartite atomic state after interaction and measurement
TMS state with squeezing parameter
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Measurement induced entanglement
Eraser
Phys. Rev. A, 80 (062304), 2009.
Multipartite
GHZ-entanglement
Cluster-like entanglement
microtraps lenses
G. Birkl APB 86, 377, 2007.
Geometrical scheme
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Outline
•Introduction and Motivation
•Correlations in CV systems
•Measurement induced Entanglement
•Conclusions
What and why Continuous Variable systems?
Classical and/or quantum correlations for communication.
The enhancement of quantum measurements.
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ConclusionsCorrelations 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.
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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.
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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/0611277Journal-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:0707:2811Journal-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:0712.2404Journal-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:0907.4261Journal-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/0703177Journal-ref: Journal of Physics A: Mathematical and Theoretical 40, 8019 (2007)
