In Geom. Funct. Anal. (GAFA), Special Volume—GAFA2000, 816–838 (2000). Quantum Information Theory: Results and Open Problems1 Peter Shor AT&T Labs—Research, Florham Park, NJ 07932 1 Introduction The discipline of information theory was founded by Claude Shannon in a truly re- markable paper [28] which laid down the foundations of the subject. We begin with a quote from this paper which is an excellent summary of the main concern of informa- tion theory: The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Quantum information theory is motivated largely by the same problem, the difference being that either the method of reproduction or the message itself involves fundamen- tally quantum effects. For many years, information theorists either ignored quantum effects or approximated them to make them susceptible to classical analysis; it was only in the last decade or so that the systematic study of quantum information theory began. We next give a quote from John R. Pierce which shows roughly the state of quantum information theory a quarter century ago. In a 1973 retrospective [25], celebrating the 25th anniversary of Shannon’s paper, Pierce says I think that I have never met a physicist who understood information the- ory. I wish that physicists would stop talking about reformulating infor- mation theory and would give us a general expression for the capacity of a channel with quantum effects taken into account rather than a number of special cases. In retrospect, this quote seems both optimistic and pessimistic. It was certainly pes- simistic in that there are now many physicists who understand information theory, and I believe that even when Pierce wrote this, there were several who did. Ironically, one of the first fundamental theorems of quantum information theory was proved in the same year [17]. On the other hand, Pierce was quite optimistic in that he seems to have believed that finding the capacity of a quantum channel would be fairly straight- forward for a physicist with the right background. This has not proven to be the case; even now, we do not have a general formula for the capacity of a quantum channel. However, there have been several recent fundamental advances made in this direction, and I describe these in this paper. 1A large part of this paper is included in the paper “Quantum Shannon Theory,” which will appear in the IEEE Information Theory Society Newsletter. 1 2 Shannon theory Shannon’s 1948 paper [28] contained two theorems for which we will be giving quan- tum analogs. The first of these is the source coding theorem, which gives a formula for how much a source emitting random signals can be compressed, while still permitting the original signals to be recovered with high probability. Shannon’s source coding the- ¡ ¡§¦©¨ ¦ £¢¥¤ ¤ orem states that outputs of a source can be compressed to length bits, and restored to the original with high probability, where ¢ is the entropy function. ¢ For a probability distribution with probabilities £ , , , , the entropy is ¦ ¢¥¤ #"%$'&£ )( (1) ! where information theorists generally take the logarithm base 2 (thus obtaining bits as the unit of information). The second of these theorems is the channel coding theorem, which states that with ¦ ¤, + high probability, uses of a noisy channel * can communicate bits reliably, where + is the channel capacity given by .-0/#1 ¡;: ¡¥¦)¦ ¤ ¤ * + (2) 243%57698 Here the maximum is taken over all probability distributions on inputs ¡ to the chan- ¡¥¦ ¡ ¤ nel, and * is the output of the channel given input . The mutual information 8 is defined as: ¡;:=<>¦? <@¦ <BA ¡¥¦ ¤ ¢¥¤ ¢§¤ (3) 8 ¡§¦¨ <@¦ ¡ <@¦ ¢¥¤ ¢¥¤ ¢¥¤ ( ( (4) ¡ < ¡ <@¦ <BA ¡¥¦ ¢¥¤ ¢¥¤ where ( is the entropy of the joint distribution of and , and is the < ¡ ¡ ¡ conditional entropy of , given . That is, if the possible values of are , then the conditional entropy is <CA ¡§¦D ¡IF¡ ¦ <CA ¡JK¡ ¦ ¢¥¤ ¤ ¢¥¤ (5) FEHG In this paper, I outline the progress that has been made in extending these formulae to quantum channels, while also taking a few side detours that address related prob- lems and results in quantum information theory. I will keep this paper at a fairly low technical level, so I only sketch the proofs for some of the results I mention. When the formula for mutual information is extended to the quantum case, two generalizations have been found that both give capacities of a quantum channel, al- though these capacities differ in both the resources that the sender and receiver have available and the operations they are permitted to carry out. One of these formulae generalizes the expression (3) and the other the expression (4); these expressions are equal in the classical case. 2 3 Quantum mechanics Before we can start talking about quantum information theory, I need to give a brief description of some of the fundamental principles of quantum mechanics. The first of these principles that we present is the superposition principle. In its most basic form, this principle says that if a quantum system can be in one of two distinguishable ALM ANOM AL©M!¨.QRANOM Q P states and , it can be in any state of the form P , where and A A ¨SA QHA UT AWVXM are complex numbers with P . Here is the notation that physicists use for a quantum state; we will occasionally be using it in the rest of this paper. ANOM Recall we assumed that AL©M and were distinguishable, so there must conceptually be some physical experiment which distinguishes them (this experiment need not be performable in practice). The principle says further that if we perform this experiment, ALM A A ANOM A QHA we will observe with probability P and with probability . Furthermore, ANOM after this experiment is performed, if state AL©M (or ) is observed the system will thereafter behave in the same way as it would have had it originally been in state ALM (or ANYM ). Mathematically, the superposition principle says that the states of a quantum system are the unit vectors of a complex vector space, and that two orthogonal vectors are distinguishable. In accordance with physics usage, we will denote quantum states by column vectors. The Dirac bra-ket notation denotes a column vector by AZ[M (a ket) Z]A and its Hermitian transpose (i.e., complex conjugate transpose) by \ (a bra). The Z A Z[M¥ Z \_^ ^a` ^]` inner product between two vectors, and ^ , is denoted , where is the conjugate transpose of ^ . Multiplying a quantum state vector by a complex phase factor (a unit complex number) does not change any properties of the system, so mathematically the state of a quantum system is a point in projective complex space. Unless otherwise stated, however, we will denote quantum states by unit vectors in a complex vector space bdc . We will be dealing solely with finite dimensional vector spaces. Quantum informa- tion theory is already complicated enough in finite dimensions without introducing the additional complexity of infinite-dimensional vector spaces. Many of the theorems we will be discussing do indeed generalize naturally to infinite-dimensional spaces. A qubit is a two-dimensional quantum system. Probably the most widely known qubit is the polarization of a photon, and we will thus be using this example in the remainder of the paper. For the polarization of a photon, there can only be two dis- tinguishable states. If one sends a photon through a birefringent crystal, it will take one of two paths, depending on its polarization. By re-orienting this crystal, these two distinguishable polarization states can be chosen to be horizontal and vertical, or they can be chosen to be right diagonal and left diagonal. In accordance with the superposi- tion principle, each of these states can be expressed as a complex combination of basis states in the other basis. For example, T T A[e Mg A=j M£¨ A[k!M f h i h i T T A=j M A[k!M A[l Mg m h i h i 3 T AYn opMg A=j M£¨ q A[k!M i i h h T A=j M q A[k!M A©r=stMg i i h h n s ouM A©r M Here, A and stand for right and left circularly polarized light, respectively; these are another pair of basis states for the polarization of photons. For example, when diagonally polarized photons are put through a birefringent crystal oriented in k j the ( direction, half of them will behave like vertically polarized photons, and half like horizontally polarized photons. If you have two quantum systems, their joint state space is the tensor product of their individual state spaces. For example, the state space of two qubits is bdv and of zy b three qubits is bxw . The high dimensionality of the space for qubits, , is one of the places where quantum computation attains its power. The polarization state space of two photons has as a basis the four states Akxk{M AkHj|M A=j}k!M A=j~jM ( ( ( This state space includes states such as an EPR (Einstein, Podolsky, Rosen) pair of photons T T Ak{j|M AjkHMW¦D AYe l M AYl e MW¦ m m f f ¤ ¤ h i h i ( (6) where neither qubit alone has a definite state, but which has a definite state when con- sidered as a joint system of two qubits. In this state, the two photons have orthogonal polarizations in whichever basis they are measured in. Bell [3] showed that the out- comes of measurements on the photons of this state cannot be reproduced by joint probability distributions which give probabilities for the outcomes of all possible mea- surements, and in which each of the single photons has a definite probability distribu- tion for the outcome of measurements on it, independent of the measurements which are made on the other photon.