Quantum Information with Continuous Variable systems
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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.
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Quantum Information with Continuous Variable systems
- 1. Quantum Information with Continuous Variable systems Carles RodĂł SarrĂł
- 2. Quantum Information with Continuous Variable systems Carles RodĂł SarrĂł UAB 30 Abril 2010 Supervisor: Anna Sanpera Trigueros 2
- 3. “Information is physical” Rolf Landauer 1960. disponer esta pelĂcula, debe de Paradescompresor GIF. y un ver de QuickTime™ 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. AcĂn 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. AcĂn 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
- 33. Measurement induced entanglement Atoms x-polarized collective angular momentum 1 mode Light x-polarized z-propagating Stokes 1 mode Matter-light interaction Dipolar interaction Gaussian interaction 36
- 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
- 36. Measurement induced entanglement Geometrical scheme Eraser Multipartite GHZ-entanglement microtraps lenses Cluster-like entanglement Phys. Rev. A, 80 (062304), 2009. G. Birkl APB 86, 377, 2007. 39
- 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
Editor's Notes
- -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.