developed the quantum version of Shannon's theorem. In this paper we ..... to the quantum noiseless coding theorem, in the limit of large block sizes Alice needs.
Quantum Data Compression John A. Vaccaro1, Yasuyoshi Mitsumori2,3, Stephen M. Barnett4 , Erika Andersson4, Atsushi Hasegawa2,3, Masahiro Takeoka2,3, and Masahide Sasaki2,3 1
Quantum Physics Group, STRC, University of Hertfordshire, College Lane Hatfield AL10 9AB, UK 2 Communications Research Laboratory, Koganei, 4-2-1 Nukuikita, Koganei, Tokyo 184-8795, Japan 3 CREST, Japan Science and Technology Corporation, 3-13-3 Shibuya, Tokyo 150-0002, Japan 4 Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 0NG, Scotland
Abstract. The last two decades has witnessed the emergence a new paradigm in information theory based on quantum theory. We review a fundamental element of quantum information theory, source coding, which entails the compression of quantum data. We also briefly outline a recent experiment that demonstrates this fundamental principle. Keywords: Quantum information theory, quantum data compression, quantum source coding, linear optics.
1 Introduction A new paradigm in information theory based on the principles of quantum theory has emerged in the last two decades. The most widely known goal of this new research area is to produce a quantum computer. This is a device which has been shown to have the ability, in principle, to solve some difficult problems efficiently [1]. A great deal of effort is underway to build the fundamental gates (quantum gates) needed for its operation and develop the theory. Already there have been significant milestones such as the factoring of the number 15 by a quantum computer based on nuclear magnetic resonance [2] and a quantum gate between two ions in an ion trap [3] to name but two. This new theory has also had an impact on data security where techniques for the secure distribution of a random key (quantum key distribution) [4]. The security is guaranteed, essentially, by the physical impossibility to clone quantum information and so eavesdroppers can be detected, in principle. The efficient storage and transmission of information lies right at the heart of information theory. Indeed, the information content of a message and the minimum requirements to represent that information is central to Shannon’s seminal paper [5] and forms his noiseless coding theorem. A few years ago Schumacher [6] and Jozsa [7] developed the quantum version of Shannon’s theorem. In this paper we briefly review quantum noiseless coding and describe some possible experimental implementations as well as some recent experimental results. We begin in Section 2 with a review of some of the basic principles of quantum information theory applicable here, and we discuss A. Albrecht and K. Steinh¨ofel (Eds.): SAGA 2003, LNCS 2827, pp. 98–107, 2003. c Springer-Verlag Berlin Heidelberg 2003
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the Schumacher quantum noiseless coding theorem and coding schemes in Section 3. In Section 4 we discuss experimental implementations and conclude in Section 5.
2 Review of Essentials of Quantum Information Theory Qubits. In classical information theory, each letter in a binary message can be a 0 or a 1. The defining property of classical messages is that a letter can only be one of these numbers. Quantum messages, in contrast, are not so restricted. We now review how a quantum memory element, the qubit, can be in a state representing a superposition of 0 and 1. One way to appreciate the meaning of a superposition is to consider a physical system in this state. There are many examples available, but for our purposes the state of a photon is the most useful. A photon is a particle of light. The granularity of light is not familiar to our daily experience due to the high rate of photons arriving in our eyes each second under normal circumstances. Nevertheless, devices called photodetectors can detect individual photons. Photodetectors can be operated in a way to give a voltage spike (or an audible “click” by a speaker) whenever a photon is detected. The devices we consider for manipulating photons consists of mirrors and beam splitters and combiners. A typical mirror consists of a reflective coating (usually a metal layer) on a glass slide. A perfectly reflecting mirror has a sufficiently thick coating to reflect all the light incident on it. In the absence of a coating, i.e. just the glass, none (or a negligible amount) of the light is reflected; the light is simply transmitted through through the glass. In the intermediate regime, a coating can be made sufficiently thin that, for example, 50% of an intense beam of light is reflected and 50% is transmitted. Such a semi-silvered mirror is called a 50-50 beam splitter, since it splits the beam equally into two. Imagine a single photon directed at a 50-50 beam splitter. To determine which path the photon takes, we can place a photodetector in each of the reflected and the transmitted paths as shown in the Fig. 1a. The photon will be found to be either transmitted or reflected with equal probability. One, and only one, of the photodetectors will “click”. We cannot predict a priori which path the photon will take, and each photon encountering the beam splitter does so independently of all other photons. We label the reflected path as a logical 0 and the transmitted path as a logical 1. This simple demonstration illustrates how a photon can represent a bit of information. (a)
(b) photodetector
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Fig. 1. A single photon representing 1 qubit.
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On the other hand, we can recombine the reflected and transmitted beams using mirrors and a further semi-silvered mirror (a beam combiner) as shown in the Fig. 1b. Light has a wave light property in addition to its granularity. In fact, light is electromagnetic radiation and the waves are due to the oscillations in the electric and magnetic fields. The crucial feature deciding how the two beams recombine is given by the relative position of the crests and troughs of the waves. If the lengths of the two possible paths 0 and 1 are such that the crests of both waves arrive simultaneously at the beam combiner, the photons will emerge in the downwards direction as shown in the figure. Alternatively, if the crests of one wave coincides with troughs of the other at the beam combiner, the photons will emerge in the upwards direction indicated by the dotted line in the figure. By slightly changing the relative path lengths, the photons can be switched from the downwards to the upwards path. This implies that when the photons arrive at the beam combiner, they have information of both path lengths. This phenomenon occurs for single photons indicating that a single photon exists simultaneously in both paths. We are driven to conclude that, when the photon is between the beam splitter and beam combiner, the photon represents both logical values 0 and 1 simultaneously. This state of the photon is called a superposition state. A bit with this added quantum feature is called a quantum bit, or qubit. The crucial point to be made here is that if we measure (i.e. examine) the path a photon is travelling along, we get a definite (but stochastic) answer pertaining to a single path. In other words, if we ask which logical value is stored in a qubit, we find a 0 or 1. If we don’t ask which path, but allow the photon to proceed and undergo some manipulation (e.g. be recombined), we find evidence that the photon has represented both logical values 0 and 1 simultaneously. We mention in passing that it is this basic superposition property of qubits that gives rise to the enormous potential of quantum computers. In contrast to a n-bit memory that can store one of 2n numbers, a n-qubit device can represent all 2n numbers simultaneously. Moreover it is possible to perform operations on the 2n numbers in parallel. However a full discussion of these ideas is beyond the scope of this work. Formal Framework. The logical elements 0 and 1 are represented in quantum information theory using the symbols |0 and |1 for two orthonormal vectors. A superposition state is represented as 1 |ψ = u |0 + v |1
(1)
The coefficients u and v can be complex. Fig. 2a illustrates the state |ψ for real coefficients. In the photon example, the complex arguments represent phase changes (or time delays) of the associated optical waves. The set of all superposition states forms a complex vector space. Indeed, the representation of |ψ as a column vector is simply u → − ψ = . An inner product ψ|φ on this space is defined for two arbitrary vectors v |ψ and |φ as follows. First we note that since the states |0 and |1 are orthonormal and their inner products are 0|0 = 1|1 = 1 0|1 = 0|1 = 0 .
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If |φ = a |0 + b |1 and |ψ is given by Eq. (1) then the inner product between |ψ and |φ is defined to be ψ|φ = u∗ a + v∗b . where * indicates complex ∗ conjugation. Inthe column vector representation, ψ|φ ≡ → − →∗ − → − u a − → ∗ ψ · φ where φ = and φ = . v∗ b
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Fig. 2. A vectorial representation of the state vectors |ψ = u |0 + v |1 and |L± = α |0 ± β |1.
It is usual to insist that the sum of the squares of the moduli of the coefficients is equal to unity |u|2 + |v|2 = 1, that is, all state vectors have a unit norm. This restricts the set of vectors to a subset of the vector space. A measurement to determine which logical value is stored in the qubit yields the results 0 and 1 with probabilities P(0) = 0|ψ 2 = |u|2 and P(1) = 1|ψ 2 = |v|2 , respectively. The operations allowed by quantum theory are those which take a state vector of unit norm into another state vector of unit norm. This brief review of the formal framework is sufficient for our purposes of describing quantum source coding. The reader is referred to standard texts on the subject for more details [8].
3 Quantum Source Coding Classical and Quantum Messages Messages comprise a sequence of letters Li , L2 , L3 , · · · . Each classical letter Ln belongs to an alphabet (or set) of N letters Ln ∈ {a, b, c, · · · }. The task of source coding is to represent the message with the shortest sequence of symbols such as the binary digits 0 and 1. Essentially, common letters are represented as short sequences of symbols and uncommon ones as longer sequences. Shannon’s noiseless coding theorem shows that the length of the shortest coded message is bounded below by MH bits, where M is the number of letters in the message and H is the Shannon entropy which is given by H = − ∑ P(n) log2 P(n) n
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for a source that produces messages with the letter probabilities P(a), P(b), P(c), · · · . H is the average information content per letter. It takes its maximum value of log2 N when all letters are equally likely, in which case no compression is possible. For example, an alphabet of N = 256 equally-likely letters would require 8 bits, or 1 byte, per letter to encode; this is equivalent to the standard ASCII coding. Quantum coding [6] applies to quantum messages comprising a sequence of quantum letter states |Ln belonging to an alphabet {|L1 , |L2 , |L3 , · · · } with corresponding probabilities P1 , P2 , P3 , · · · . If the letter states form an orthogonal set, the analysis of the quantum message is precisely that as for a classical message. In particular, if the letter states are equally likely, P1 = P2 = P3 = · · · , no compression is possible. New quantum features arise, however, when the letter states are not orthogonal. Significantly, compression is possible even in the case of equally-likely letter states. We restrict our discussion to qubit letters, |Ln = αn |0 + βn |1, for the brevity. The state of each letter can be represented as a matrix, called a density matrix, in the following way: |Ln Ln | = |αn |2 |0 0| + αn β∗n |0 1| + α∗n βn |1 0| + |βn |2 |1 1| or using the 2 × 2 matrix representation, the matrix Rn = (α∗n , β∗n )T (αn , βn ). The average letter state is given by the weighted sum of the density matrices of the letters: ρˆ = ∑ Pn |Ln Ln | n
=
∑ Pn|αn |2
|0 0| +
n
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In terms of 2 × 2 matrices, the average letter state is given by the positive semi-definite ˆ of the density operator ρˆ repmatrix Rav = ∑n Pn Rn . The von Neumann entropy S(ρ) resents a measure of the amount of statistical uncertainty in a quantum state. It is given by ˆ = − ∑ λi log2 λi s(ρ) i
where λi are the (positive) eigenvalues of the matrix Rav . The quantum noiseless coding theorem [6, 7] implies that by coding the quantum ˆ qubits are enough to encode each block in the message in blocks of K letters, KS(ρ) limit K → ∞. Jozsa and Schumacher considered letter states of the form [7]: 1 |L± = α |0 + β± |1
(2)
where β± = ±β, α2 + β2 = 1. For clarity we assume α and β are real numbers. These states are illustrated in Fig. 2b. Let the letter states occur with equal likelihood so that the density operator representing the average letter state is ρˆ = α2 |0 0| + β2 |11|. ˆ = −α2 log2 α2 − β2 log2 β2 . If the letThe corresponding von Neumann entropy is S(ρ) 2 2 ter states are orthogonal L |L+ = α − β = 0 which gives α2 = β2 = 12 . In this case
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ˆ = 1 and so 1 qubit is needed to encode each letter faithfully. Clearly a message of S(ρ) equally-likely, orthogonal letter states cannot be compressed to a smaller code. However, the von Neumann entropy of ρˆ is 0.4690 bits for the case α2 = 0.9 [7]. According to the quantum noiseless coding theorem, in the limit of large block sizes Alice needs approximately 1/2 qubit per letter state to faithfully transmit the message to Bob. If the coding is restricted to finite length blocks, the encoding and subsequent decoding will introduce errors. The exercise then is to consider the fidelity of the coding-decoding operation. 2 Qubit Blocks. Let us examine the compression of a block of 2 qubits. In an analogous manner to the possible numbers, 00, 01, 10, 11, able to be stored in 2 bits, the basis states of a 2-qubit system are |00, |01, |10 and |11. Here we have written the tensor (outer) product of the states of two qubits |n ⊗ |m as |nm. Thus the 2 letter states can be written as |Ls1 ⊗ |Ls2 = (α |0 + βs1 |1) ⊗ (α |0 + βs2 |1) = α2 |00 + αβs2 |01 + αβs1 |10 + βs1 βs2 |11
(3)
where sn ∈ {+, −}. For β < α the most likely 3-dimensional subspace Λ2 spanned by the largest 3 eigenvalues can be shown to be the subspace spanned by |00, |01 and |10. The 2-qubit block can be encoded approximately onto Λ2 as follows. The procedure which gives the highest fidelity is to perform a measurement on the 2-qubit system to see if the state lies in Λ2 or in the subspace Λ2 spanned by the remaining basis states, which, in this case is just |11. The probabilities for these two results are PΛ2 = 1 − β2 = α2 and PΛ2 = β2 , respectively. If the state lies in Λ2 , the state |00 is used as the coded block state, otherwise the (normalized) letter state after projection onto Λ2 , |C2 =
1 2 (α |00 + αβs2 |01 + αβs2 |10) , α
is used. The factor 1/α in the last expression is required for a unit norm. The coded block state lies in a 3 dimensional space, which is effectively a log2 3 = 1.58 qubit system. The decoding for this scheme is simple: no action is taken. The extension of the 2 qubit letter state |Ls1 ⊗ |Ls2 onto Λ cannot be recovered and this incurs errors. The fidelity of the coding-decoding operation is given by the average of the square of the inner products of the original 2-qubit letter state Eq. (3) with all possible decoded states. Formally the expression is 2 2 1 F2 = ∑ PΛ2 (Ls1 | ⊗ Ls2 |) |C2 + PΛ2 Ls1 | ⊗ Ls2 |00 . 4 s1 ,s2 We find the value of F2 = 1 − β4 = 0.99 for α2 = 0.90. This scheme uses 0.79 qubits per letter state. 3 Qubit Blocks. We now consider blocks of 3 letters. This is the case considered by Jozsa and Schumacher [7] and we review it briefly.
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The 3 letter states can be written as |Ls1 ⊗ |Ls2 ⊗ |Ls3 = α3 |000 + α2 βs3 |001 + α2 βs2 |010 + αβs2 βs3 |011 +α2 βs1 |100 + αβs1 βs3 |101 + αβs1 βs2 |110 + βs1 βs2 βs3 |111
(4)
In this case the 3 qubit block state can be encoded, approximately, onto the state of 2 qubits in the following way. First we identify the most likely 4-dimensional subspace Λ3 ; this is spanned by |000, |001, |010 and |100. The encoding is carried out by performing a measurement to determine if the block state Eq. (4) lies in Λ3 or the subspace Λ3 spanned by the remaining basis states |011, |101, |110 and |111. If it is found in the former subspace, the (normalized) state after the measurement is 1 (α |000 + βs3 |001 + βs2 |010 + βs1 |100) . |C3 = 1 + 2β2 This state is transformed into a 2-qubit state using the unitary transform Uˆ which maps |100 → |011, |011 → |100 and leaves all other basis states unchanged. Ignoring the first qubit (which is in the state |0) then leaves the remaining 2-qubit system in the state C3 = 1 (α |00 + βs3 |01 + βs2 |10 + βs1 |11) 1 + 2β2 If the measurement yields the result that the block state Eq. (4) lies in the subspace Λ3 , the state |00 is used as the 2-qubit coded block state. The probabilities for the outcomes of the measurement are PΛ3 = α4 (1 + 2β2) and PΛ3 = 1 − PΛ3 for the subspaces Λ3 and Λ3 , respectively. The decoding is performed by preparing a new qubit in the state |0 and performing the inverse unitary operation Uˆ −1 on the system of 3 qubits (i.e. on the new qubit and the 2-qubit coded block). The result is the state |C3 with probability PΛ3 and the state |000 with probability PΛ3 . The fidelity of the decoded block is found to be F3 = α8 (1 + 2β2)2 + α6 β4 (1 + 2α2 ) = 0.97 for α2 = 0.90. This scheme uses 0.67 qubits per letter state, and so it is slightly more efficient that the 2-qubits scheme, although it also has lower fidelity. This method can be taken further encoding blocks of K letters for larger values of K.
4 Physical Implementations We now illustrate how the previous schemes could be implemented using a linear optical scheme, that is using single photons, beam splitters (and combiners) and mirrors. As shown in section II, a qubit will be represented by a single photon travelling in a superposition of two possible paths. To produce a qubit with the coefficients α and β, we use beam splitters for which the transmission and reflection probabilities are |α|2 and β2 (or vise versa, depending on their position in the optical circuit). The sign of the coefficient β− = −β is produced using a piece of glass to delay the wave in the 1 path by half a wavelength, as shown in Fig. 3a.
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extra photon
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Fig. 3. Implementing (a) a single letter state and (b) the 2 qubit scheme.
2 Qubit Scheme. A 2 qubit system requires the photon to be in a superposition of 4 paths, as suggested by Eq. (3). This is produced by splitting the 0 and 1 paths into a further two paths each and using glass delays as appropriate as shown in Fig. 3b. The compression of the 4 state system into a 3 state system is performed using a photodetector to determine the presence or absence of a photon in path 11. This corresponds to the measurement to see if the block state is in the subspaces Λ2 or Λ2 . The detection of a photon indicates the latter case and an extra photon is switched into path 00 to represent the coding of the state |00, as indicated by the dotted line in Fig. 3b. The decoding is produced simply by making available an empty path representing the state |11. 3 Qubit Scheme. The 3 qubit block can be produced by further splitting each of the 4 paths into two. This allows a single photon to be in a superposition of 8 paths. The unitary transformation Uˆ is produced simply by interchanging the two paths 100 and 011 as shown in Fig. 4. The projection onto Λ3 or Λ3 is produced using the 4 photodetectors. If one of the photodetectors records a photon (indicating the block state lies in Λ3 ), an extra photon is switched into the optical path 00. The encoded state lies in a 4 dimensional space, which is the state-space of a 2-qubit system. The decoding is carried out in a reverse manner. The inverse unitary operation Uˆ −1 is again produced by interchanging two paths and 4 (dark) paths are added to reconstruct an 8 path system. We have recently performed an experimental implementation of the 3 qubit scheme that, while it is different to the one described here, it is based on essentially the same principles. The experiment makes use of an extra degree of freedom available to photons that we have ignored up to now. The electromagnetic waves comprising the photons oscillate transversely to the direction of travel. The orientation of the oscillations is called the polarization of the photon and gives rise to an intrinsic polarization qubit. Thus a 3 qubit system can be constructed from 2 path qubits (i.e. 4 paths) and 1 polarization qubit (2 orthogonal polarization directions). Details can be found in [9]. The extension of our 2 and 3-qubit optical schemes to blocks of larger K is straight forward requiring 2K optical paths. However the scheme quickly becomes experimentally difficult to perform owing to the degree of control required over the many optical paths.
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Fig. 4. Implementing the 3 qubit scheme.
Rather than use a single photon to represent a multiple qubit block, a more scalable system comprises one photon for each qubit. However the various operations needed for coding and decoding require nonlinear interactions between the photons. Such nonlinear interactions between individual photons have not yet been produced experimentally, although there are proposals for engineering them using available technology.
5 Conclusion The emergence of quantum information theory presents many exciting challenges, both experimentally and theoretically. It opens up a new paradigm in information theory forcing us to completely review what we understand to be information. Source coding is an essential element of information theory, for which it is known that a classical message of equally-likely letters is not compressible. In contrast, the Schumacher quantum noiseless source coding theorem shows that a quantum message of equally-likely, but non-orthogonal letter states is indeed compressible. We have reviewed this issue beginning with an intuitive description of qubits and we have given examples of experimental implementations using single photons. This work was supported by the British Council, the Royal Society of Edinburgh, the Scottish Executive Education and Lifelong Learning Department and the EU Marie Curie Fellowship program.
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References 1. P.W. Shor, SIAM Journal of Computation, 26, 1484-1509, (1997). 2. L.M.K. Vandersypen, M. Steffen, G. Breyta, C.S. Yannoni, M.H. Sherwood, and I.L. Chuang, Nature 414, 883 (2001). 3. F. Schmidt-Kaler, H. H¨affner, M. Riebe, S. Gulde, G.P.T. Lancaster, T. Deuschle, C. Becher, C.F. Roos, J. Eschner, R. Blatt, Nature 422, 408 (2003). 4. N. Gisin, G.G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74, 145 (2002). 5. C.E. Shannon, Bell System Technical Journal, 27, 379 (1948). 6. B. Schumacher, Phys. Rev. A51, 2738 (1995). 7. R. Jozsa and B. Schumacher, J. Mod. Opts. 41, 2343 (1994). 8. See e.g. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University press, Cambridge, 2000). 9. Y. Mitsumori, J.A. Vaccaro, S.M. Barnett, E. Andersson, A. Hasegawa, M. Takeoka and M. Sasaki, “Experimental demonstration of quantum source coding”, quant-ph/0304036.
