Lecture 4: Quantum typicality and quantum data compression Mark M. Wilde∗ One of the fundamental tasks in classical information theory is the compression of information. Given access to many uses of a noiseless classical channel, what is the best that a sender and receiver can make of this resource for compressed data transmission? Shannon's compression theorem states that the Shannon entropy is the fundamental limit for the compression rate in the IID setting. That is, if one compresses at a rate above the Shannon entropy, then it is possible to recover the compressed data perfectly in the asymptotic limit, and otherwise, it is not possible to do so. This theorem establishes the prominent role of the entropy in Shannon's theory of information. In the quantum world, it very well could be that one day a sender and a receiver would have many uses of a noiseless quantum channel available,1 and the sender could use this resource to transmit compressed quantum information. But what exactly does this mean in the quantum setting? A simple model of a quantum information source is an ensemble of quantum states fpX (x); j xig, i.e., the source outputs the state j xi with probability pX (x), and the states fj xig do not necessarily have to form an orthonormal basis. Let us suppose for the moment that the classical data x is available as well, even though this might not necessarily be the case in practice. A naive strategy for compressing this quantum information source would be to ignore the quantum states coming out, handle the classical data instead, and exploit Shannon's compression protocol. That is, the sender compresses the sequence xn emitted from the quantum information source at a rate equal to the Shannon entropy H(X), sends the compressed classical bits over the noiseless quantum channels, the receiver reproduces the classical sequence xn at his end, and finally reconstructs the sequence n j xn i of quantum states corresponding to the classical sequence x . The above strategy will certainly work, but it makes no use of the fact that the noiseless quantum channels are quantum! It is clear that noiseless quantum channels will be expensive in practice, and the above strategy is wasteful in this sense because it could have merely exploited classical channels (channels that cannot preserve superpositions) to achieve the same goals. Schumacher compression is a strategy that makes effective use of noiseless quantum channels to compress a quantum information source down to a rate equal to the von Neumann entropy. This has a great benefit from a practical standpoint|the von Neumann entropy of a quantum information source is strictly lower than the source's Shannon entropy if the states in the ensemble are non-orthogonal. In order to execute the protocol, the sender and receiver simply need to know the density operator P ρ ≡ x pX (x)j xih xj of the source. Furthermore, Schumacher compression is provably optimal in ∗Mark M. Wilde is with the Department of Physics and Astronomy and the Center for Computation and Technol- ogy, Louisiana State University, Baton Rouge, Louisiana 70803. These lecture notes are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Much of the material is from the book preprint \From Classical to Quantum Shannon Theory" available as arXiv:1106.1445. 1How we hope so! If working, coherent fault-tolerant quantum computers come along one day, they stand to benefit from quantum compression protocols. 1 Figure 1: The most general protocol for quantum compression. Alice begins with the output of some quantum information source whose density operator is ρ⊗n on some system An. The inaccessible reference system holds the purification of this density operator. She performs some CPTP encoding map E, sends the compressed qubits through 2nR uses of a noiseless quantum channel, and Bob performs some CPTP decoding map D to decompress the qubits. The scheme is successful if the difference between the initial state and the final state is negligible in the asymptotic limit n ! 1. the sense that any protocol that compresses a quantum information source of the above form at a rate below the von Neumann entropy cannot have a vanishing error in the asymptotic limit. Schumacher compression thus gives an operational interpretation of the von Neumann entropy as the fundamental limit on the rate of quantum data compression. Also, it sets the term \qubit" on a firm foundation in an information-theoretic sense as a measure of the amount of quantum information \contained" in a quantum information source. We begin this lecture by giving the details of the general information processing task corre- sponding to quantum data compression. We then prove that the von Neumann entropy is an achievable rate of compression. 1 The Information Processing Task We first overview the general task that any quantum compression protocol attempts to accomplish. Three parameters n, R, and corresponding to the length of the original quantum data sequence, the rate, and the error, respectively, characterize any such protocol. An (n; R + δ; ) quantum compression code consists of four steps: state preparation, encoding, transmission, and decoding. Figure 1 depicts a general protocol for quantum compression. An State Preparation. The quantum information source outputs a sequence j xn i of quantum states according to the ensemble fpX (x); j xig where n A A1 An n j x i ≡ j x1 i ⊗ · · · ⊗ j xn i : (1) The density operator, from the perspective of someone ignorant of the classical sequence xn, is 2 equal to the tensor power state ρ⊗n where X ρ ≡ pX (x)j xih xj: (2) x Also, we can think about the purification of the above density operator. That is, an equivalent mathematical picture is to imagine that the quantum information source produces states of the form RA X p R A j'ρi ≡ pX (x)jxi j xi ; (3) x where R is the label for an inaccessible reference system (not to be confused with the rate R!). The RA ⊗n resulting IID state produced is (j'ρi ) . Encoding. Alice encodes the systems An according to some CPTP compression map EAn!W where W is a quantum system of size 2nR. Recall that R is the rate of compression: 1 R = log d − δ; (4) n W where dW is the dimension of system W and δ is an arbitrarily small positive number. Transmission. Alice transmits the system W to Bob using n(R + δ) noiseless qubit channels. Decoding. Bob sends the system W through a decompression map DW !A^n . The protocol has error if the compressed and decompressed state is -close in trace distance RA ⊗n to the original state (j'ρi ) : RA⊗n W !A^n An!W RA⊗n 'ρ − (D ◦ E )( 'ρ ) ≤ . (5) 1 2 The Quantum Data Compression Theorem We say that a quantum compression rate R is achievable if there exists an (n; R + δ; ) quantum compression code for all δ; > 0 and all sufficiently large n. Schumacher's compression theorem establishes the von Neumann entropy as the fundamental limit on quantum data compression. Theorem 1 (Quantum Data Compression) Suppose that ρA is the density operator of the quantum information source. Then the von Neumann entropy H(A)ρ is the smallest achievable rate R for quantum data compression: inffR : R is achievableg = H(A)ρ: (6) In this lecture, we will only prove the direct part of this theorem (that is, we will exhibit a coding scheme that achieves the von Neumann entropy rate of compression). The main technical tool required in proving the direct part is the notion of quantum typicality, which we describe in the next section. 2.1 Quantum Typicality 2.1.1 The Typical Subspace Our first task is to establish the notion of a quantum information source. It is analogous to the notion of a classical information source, in the sense that the source randomly outputs a quantum state according to some probability distribution, but the states that it outputs do not necessarily have to be distinguishable as in the classical case. 3 Definition 2 (Quantum Information Source) A quantum information source is some device that randomly emits pure qudit states in a Hilbert space HX of size jX j. We use the symbol X to denote the quantum system for the quantum information source in addition to denoting the Hilbert space in which the state lives. Suppose that the source outputs states j yi randomly according to some probability distribution pY (y). Note that the states j yi do not necessarily have to form an orthonormal set. Then the density operator ρX of the source is the expected state emitted: X X ρ ≡ EY fj Y ih Y jg = pY (y)j yih yj: (7) y There are many decompositions of a density operator as a convex sum of rank-one projectors (and the above decomposition is one such example), but perhaps the most important decomposition is a spectral decomposition of the density operator ρ: X X X ρ = pX (x)jxihxj : (8) x2X The above states jxiX are eigenvectors of ρX and form a complete orthonormal basis for Hilbert X space HX , and the non-negative, convex real numbers pX (x) are the eigenvalues of ρ . X We have written the states jxi and the eigenvalues pX (x) in a suggestive notation because it is actually possible to think of our quantum source as a classical information source|the emitted X states fjxi gx2X are orthonormal and each corresponding eigenvalue pX (x) acts as a probability for choosing jxiX . We can say that our source is classical because it is emitting the orthogonal, X and thus distinguishable, states jxi with probability pX (x).