# Quantum information with Gaussian states.pdf

arXiv:0801.4604v1 [quant-ph] 30 Jan 2008Quantum information with Gaussian statesXiang-Bin Wang1Department of Physics, Tsingghua University, Beijing 100084, China;and Imai Quantum Computation and Information Project, ERATO-SORST, JST,Daini Hongo White Bldg. 201, 5-28-3, Hongo, Bunkyo-ku, Tokyo 113-0033, JapanTohya Hiroshima2, Akihisa Tomita, Masahito HayashiImai Quantum Computation and Information Project, ERATO-SORST, JST,Daini Hongo White Bldg. 201, 5-28-3, Hongo, Bunkyo-ku, Tokyo 113-0033, JapanAbstractQuantum optical Gaussian states are a type of important robust quantum stateswhich are manipulatable by the existing technologies. So far, most of the impor-tant quantum information experiments are done with such states, including brightGaussian light and weak Gaussian light. Extending the existing results of quantuminformation with discrete quantum states to the case of continuous variable quan-tum states is an interesting theoretical job. The quantum Gaussian states play acentral role in such a case. We review the properties and applications of Gaussianstates in quantum information with emphasis on the fundamental concepts, thecalculation techniques and the effects of imperfections of the real-life experimentalsetups. Topics here includethe elementary propertiesof Gaussian states and relevantquantum information device, entanglement-based quantum tasks such as quantumteleportation, quantum cryptography with weak and strong Gaussian states andthe quantum channel capacity, mathematical theory of quantum entanglement andstate estimation for Gaussian states.Contents1 Introduction 31.1 Elements of quantum information with 2-level states 61.2 Phase space representation and definition of Gaussian states 91.3 Coherent states 151 Email: xbwang@mail.tsinghua.edu.cn2 Email: tohya@qci.jst.go.jpPreprint submitted to Elsevier Science 3 February 20081.4 Squeezed states 181.5 Beam-splitter 231.6 Beam-splitter as an entangler 312 Entanglement-based quantum tasks 332.1 Teleportation with two-level states 332.2 CVQT with Gaussian states 362.3 Experiment 402.4 Dense coding with two-mode squeezed states 422.5 Quantum error correction codes 432.6 Gaussian cloning transformation 462.7 Entanglement-based quantum tasks with weak Gaussian states:post-selection vs non-post-selection 483 Quantum cryptography with weak coherent light 543.1 Introduction 543.2 QKD in practice 553.3 Eavesdropping and security 673.4 Ideas of unconditional security proof 714 Security proofs and protocols of QKD with weak and strongGaussian states 734.1 Entanglement purification and security proof of QKD 734.2 Secure key distillation with a known fraction of tagged bits 814.3 Decoy-state method 864.4 SARG04 protocol 994.5 QKD in position-momentum space 1014.6 Security proof of QKD with squeezed states 1045 Mathematical theory of quantum entanglement with Gaussian states 10925.1 General Properties of Quantum Entanglement 1105.2 Entanglement Properties of Gaussian States 1125.3 Conclusions 1276 Classical capacities of Gaussian channels 1286.1 Classical Capacities of Quantum Channels 1286.2 Gaussian Channels 1306.3 Gaussian Channels with Gaussian Inputs 1366.4 Dense Coding with Gaussian Entanglement 1406.5 Entanglement measure 1427 Estimation theory for Gaussian states 1427.1 Information quantities 1437.2 Measurement theory 1457.3 Formulation of estimation 1497.4 Independent and identical condition 1507.5 Bayesian method 1517.6 Group covariant method 1537.7 Unbiased method 1547.8 Simple hypothesis testing 1568 Acknowledgement 158References 1581 IntroductionQuantum information processing (QIP) is a subject on information processingwith quantum states[1]. In the recent years, the subject has attracted muchattentionof scientists fromvariousareas. Ithasbeen foundthatinsome impor-tant cases, quantum information processing can have great advantage to anyknown method in classical information processing. A quantum computer can3factorize a large number exponentially more efficiently than the existing clas-sical methods do[2]. This means, given a quantum computer, the widely usedRSA system in classical communication is insecure because one can factorize ahuge number very effectively by Shor’s alhorithm. Interestingly, quantum keydistribution (QKD) can help two remote parties share a random binary stringwhich is in principle unknown to any third party[3]. Private communicationbased on QKD is proven secure under whatever eavesdropping including quan-tum computing. Quantum teleportation can transfer unknown quantum stateto a remote party without moving the physical system itself[4].In classical information processing, all information are carried by classical bits,which are binary digits of either 0 or 1. The physical carrier of a classical bitcan be any physical quantity that has two different values, e.g., the electri-cal potential (positive or negative voltages). These are macroscopic quantitieswhich can be manipulated robustly by our existing technology. In quantuminformation processing, we use quantum states to carry either quantum infor-mation or classical information. Also, we use quantum entangled states as theresource to assist the effective processing of quantum information. In princi-ple, lots of different physical systems can be used to generate the requestedquantum states and quantum entanglement. In those tasks related to com-munication, light seems to be the best candidate for the physical system tocarry the quantum information and/or the quantum entanglement due to itsobvious advantage that it can be transmitted over a long distance efficiently.Therefore one may naturally consider to use a single-photon state as a quan-tum bit (qubit) and a two-photon entangled state as the entanglement resourceto assist the processing.Towards the final goal of real-life application of quantum information process-ing, a very important question in concern is how robustly we can manipulatethe quantum states involved. In practice, preparing the single-photon states ortwo-photon entangled states deterministically are technically difficult. Whatcan be prepared and manipulated easily is a Gaussian state, e.g., a coherentstate[5,6] (the state of light pulses from a traditional Laser) or a squeezed vac-uum state[6,7,8,9,10,11] of one mode or two modes. Mathematically, a stateis Gaussian if its distribution function in phase space or its density operatorin Fock space is in the Gaussian form. (We shall go into details of the phasespace and Fock space later in this chapter and also other chapters.) Gaussianstates have proven to be a type of important robust states which have beenextensively applied for various QIP tasks in labs. Actually, so far almost allthose important experiments of quantum information are done with Gaussianlight.There are two types of application of Gaussian states in practical QIP. Oneis to use weak Gaussian light as approximate qubits or two-photon entangledstates. The other is to use strong Gaussian light forcontinuous variable QIP. In4applying the weak Gaussian light, one can regard weak coherent light[5] as ap-proximate single-photon source and regard 2-mode entangled weak Gaussianlight as probabilistic 2-photon entanglement resource in polarization space.Weak Gaussian states have been used in many experiments such as the quan-tum teleportation with spontaneous parametric down conversion (SPDC)[12]and quantum cryptography with weak coherent states or SPDC[13]. How-ever, sometimes the higher order terms, i.e., the multi-photon states in thesingle-mode Gaussian light or the multi-pair states in the two-mode Gaussianlight take a very important role even though the Gaussian light in applica-tion is weak. For example, in QKD with weak coherent states, the final keycan actually be totally insecure if we ignore the role of multi-photon pulses.For another example, in quantum-entanglement based experiments such asquantum teleportation with SPDC, it is possible that actually there is no en-tanglement and the result is a post-selection result if we don’t consider thedetailed properties of the weak Gaussian states. We shall review these types ofeffects and possible ways to overcome the drawbacks. One can also use strongGaussian light to obtain the analog QIP results of discrete states for contin-uous variable states. One important advantage here is that strong Gaussianlight can be used as a resource of deterministic quantum entanglement there-fore the results are deterministic and non-post-selection. This is quite differentfrom QIP using weak Gaussian states where the results are often probabilisticand post-selection. We shall review the main theoretical results of continu-ous variable QIP with strong Gaussian light including quantum teleportation,cloning, error correction and QKD using a strong Gaussing light. We shall alsoreview the quantum entanglement properties of multi-mode Gaussian states,quantum channel capacity and quantum state estimation of Gaussian states.This review is arranged as follows: In the remaining part of this section, wepresent the elements of QIP and Gaussian states, including an overview ofquantum information with qubits and two-photon maximal entangled states,mathematic foundation and definition of Gaussian states and properties ofa beam-splitter as an elementary QIP device. In section 2, we review theentanglement-based quantum tasks with Gaussian states, which seem to be avery hot topic in the recent years. The section includes quantum teleportation,quantum error correction, quantum cloning and non-post-selection quantumtasks with weak Gaussian states. We then go into the most important appli-cation of QIP, practical quantum key distribution in section 3 and section 4.Section 3 introduces the elements and technology background of QKD withweak coherent light, section 4 reviews the protocols and security proofs ofQKD with weak and strong Gaussian light. We then change our directionto the mathematical theory of QIP with Gaussian states in section 5 and 6.Section 5 is on quantum entanglement of Gaussian states, which have beenextensively studied these years. Section 6 is on the properties of quantumGaussian channel, which is assumed to be a fundamental subject in quantumcommunication. The theory of quantum state estimation of Gaussian states is5reviewed in section 7. This review does not include quantum computation.1.1 Elements of quantum information with 2-level statesA qubit is simply a physical system that carries a two-level quantum state.For example, a photon can be regarded as a qubit in polarization space or anyother two dimensional space. In general, we have the following mathematicalform for the state of a qubit|ψ〉 = α|0〉+β|1〉 (1)and |α|2 +|β|2 = 1. Here |0〉 and |1〉 are orthonormal states of any two-levelsystem. Mathematically, one can use the following representation|0〉 =10;|1〉 =01. (2)Consequently, we can use matrices as representations of operations to a qubit.In particular, the unity matrix I represents for doing nothing, matrix σx =0 11 0 for a bit-flip operation, σz =1 00 −1 for a phase-flip operation andσy = 0 i−i 0 for both bit-flip and phase-flip.If we use the photon polarization, notations |0〉,|1〉 represent the horizontalpolarization and the vertical polarization (polarization of angle π/2) respec-tively. One can replace them with more vivid notations of |H〉,|V〉. For exam-ple, state cosθ|H〉+sinθ|V〉 is the state of polarization angle θ. In the aboveequation, |ψ〉 is linearly superposed by |0〉 and |1〉. The quantum linear su-perposition is different from the classical probabilistic mixture. For example,consider the polarizations of π/4 and 3π/4. They are the linear superpositionstates of |±〉 = 1√2(|H〉±|V〉). Given state |+〉, i.e., polarization of π/4, ifwe measure it in the |H〉,|V〉 basis, we have equal probability to obtain anoutcome of either |H〉 or |V〉. This is due to the fact|〈H|+〉|2 = |〈V|+〉|2 = 1/2. (3)However, the state |+〉, i.e., the polarization of π/4 is different from a mixedstate which is in a classical mixture of horizontal and vertical polarizations,with equal probability. Such a classical mixture can be realized in this way:Source A only emits photons of horizontal polarization and source B onlyemits photons of vertical polarization. In a remote place we receive photons6from both sources. But we don’t know which photon is from which source. Ouronly knowledge is that any photon has equal probability from A or B. In sucha case, any individual photon is in a classical mixture of state |H〉 and |V〉. Ifwe measure such a mixed state in {|H〉, |V〉} basis, the measurement outcomeis identical to that of apurestate ofπ/4 polarization. However, if we use aπ/4polarizer, the outcome will be different: the purestate of π/4 polarization willalways transmit the polarizer while the mixed state only has half a probabilityto transmit the polarizer. These can be interpreted mathematically by Eq.(3)and the following equation :|〈+|+〉|2 = 1. (4)Eq.(3) shows that both |H〉 and |V〉 have half a probability to transmit theπ/4−polarizer. This means, any classical mixture of |H〉 and |V〉 will alwaysonly have half a probability to pass through the π/4−polarizers. To have auniversal mathematical picture for both pure states and mixed states, we canuse density operator which reflects the classical probability distribution overdifferent quantum states. Suppose a certain source consists of n sub-sources.Any sub-source i will only produce the pure state |ψi〉. Whenever the sourceemits a photon, the probability that the photon being emitted from sub-sourcei is pi (p1 +p2 +···pn = 1). The state of any photon from such a source canbe described by the density operator summationtextipi|ψi〉〈ψi|. A density operator can berepresented by a matrix which is called as density matrix. The density matrixfor pure state |ψ〉 = α|0〉+β|1〉 is|ψ〉〈ψ| =αβ(α∗,β∗) =|α|2 αβ∗α∗β |β|2. (5)Given any pure state |ψ〉, there is always a unitary matrix U satisfying|ψ〉 = U|0〉. (6)This is to say, we can use the following criterion for a pure state |ψ〉〈ψ| =U|0〉〈0|U†: Given a density matrix M, if it is for a pure state, there exists aunitary matrix U so thatUMU† =1 00 0. (7)Besides qubits, quantum entanglement is often needed in non-trivial QIPtasks, such as quantum teleportation[4]. We shall consider the most well-known case of two-photon maximally entangled state in the polarization space.Consider the state for spatially separated two photons, A and B. We have 47orthogonal states to span the polarization space of the two-photon system:|φ±〉 = 1√2(|H〉A|H〉B ±|V〉A|V〉B);|ψ±〉 = 1√2(|H〉A|V〉B ±|V〉A|H〉B).All these 4 states are maximally entangled and we shall call any of them asan EPR (pair) state named after Eistein-Podolsky-Rosen[14], or a Bell state.We shall also call the measurement basis of these 4 EPR states Bell basis.We use |φ+〉 to demonstrate the non-trivial properties of an EPR pair. Forthe pair state |φ+〉, the polarization of photon A and photon B are