Quantum Data Compression

Transcription

1 PHYS 476Q: An Introduction to Entanglement Theory (Spring 2018) Eric Chitambar Quantum Data Compression With the basic foundation of quantum mechanics in hand, we can now explore different applications. The first application we consider is the task of data compression. At the end of the first lecture we studied data compression when the source was some random variable X n that represented the outcome of n i.i.d. experiments. Our aim in this lecture is to generalize the problem and consider the task of data compression when the source consists of quantum systems. In order to better understand this problem and how it differs from the classical version, we first describe a general distinction between classical and quantum information. Contents 1 Classical and Quantum Information A distinction in terms of state distinguishability The von Neumann entropy The Compression of Quantum Information Problem Description The Fidelity between Two Quantum States The Schumacher Compression Protocol Data Compression and Entanglement 8 4 Exercises 8 1 Classical and Quantum Information 1.1 A distinction in terms of state distinguishability Since the dawn of mankind, humans have been using their environment to store and share concrete personal thoughts. For instance, in writing this lecture, I am taking my thoughts and translating them into a string of symbols appearing on my computer screen, called words. Someone who reads these words in the future (like you) will hopefully be able to revert them back into concrete ideas and understand what I am currently thinking. In general, communication between two individuals involves the exchange of codewords such phonetic words, strings of 0/1 s, hand signals, etc. The two individuals have a pre-established dictionary of what the different codewords mean, and this allows them to communicate. Importantly, all forms of communication that we experience share the same fundamental property that any two codewords describe states of a physical system can be perfectly distinguished. For example, when Alice speaks to Bob, she imprints her message into the sound waves traveling from her to Bob. Each word she utters corresponds to a different wave profile, and Bob is able to perfectly

2 distinguish between the different profiles to decipher what word she says. As another example, consider the faces of a two-sided coin. Any yes/no message can be encoded into the face of a coin, and this message can be perfectly decoded simply by looking at the coin since heads and tails are visually distinguishable from one another. In this course we will identify classical information as being one particular state of a physical system whose possible states are, in principle, perfectly distinguishable from one another. Distinguishing between the possible states may be very challenging and require precise experimental equipment, but if it is theoretically possible by the laws of physics for a discrimination to be made, then we say the states of the system represent classical information. From this perspective, information is united with physics since physical systems are the carriers of information. How information evolves in time can therefore be understood using the laws of physics. On small length scales, however, the laws of physics are described by quantum mechanics, and one prominent feature of quantum mechanics is the existence of non-distinguishable physical states. To better understand this, we need a precise meaning of distinguishability when dealing with states of a quantum system. Suppose that Bob has a quantum system with state space H B and Alice prepares his system in either state ψ 0 y or ψ 1 y. We say that Bob is able to perfectly distinguish these states if there exists a two-outcome POVM tπ 0, Π 1 u such that 0 = xψ 0 Π 1 ψ 0 y = xψ 1 Π 0 ψ 1 y. (1) Physically this says that with zero probability will Bob obtain measurement outcome 1 if Alice prepares state ψ 0 y, and likewise with zero probability will Bob obtain measurement outcome 0 if Alice prepares state ψ 1 y. Since Π x is a positive operator for x P t0, 1u, the previous equation implies that 0 = Π 1/2 0 ψ 1 y = Π 1/2 1 ψ 0 y. We then use the completion relation Π 0 + Π 1 = I to compute xψ 0 ψ 1 y = xψ 0 (Π 0 + Π 1 ) ψ 1 y = xψ 0 Π 0 ψ 1 y + xψ 0 Π 1 ψ 1 y = 0. (2) Hence, a necessary condition for ψ 0 y and ψ 1 y to be perfectly distinguishable is that they are orthogonal. Conversely, if xψ 0 ψ 1 y = 0, then ( ψ 0 yxψ 0 + ψ 1 yxψ 1 ) ψ x y = ψ x y for x P t0, 1u, which says that P = ψ 0 yxψ 0 + ψ 1 yxψ 1 is the projector onto the two-dimensional subspace spanned by t ψ 0 y, ψ 1 yu. Thus, the two states can be perfectly distinguished by the projective measurement given by projectors t ψ 0 yxψ 0, ψ 1 yxψ 1, I Pu. We therefore conclude that any two states ψ 0 y and ψ 1 y are perfectly distinguishable if and only if they are orthogonal. In general, most pairs of states ψ 0 y and ψ 1 y in Hilbert space will not be orthogonal. Yet, we would still like to think of quantum systems as containing some sort of information, even if it is information that is not distinguishable in the sense described above. We thus define quantum information as simply being one particular state of a quantum system. Under these definitions, the state of a quantum system can also represent classical information provided the state of the system is restricted to some set of orthogonal possibilities. That is, if a system is prepared in state ψ x y with probability p x and the t ψ x yu x are pairwise orthogonal, then the state of the system can be regarded as classical information. 1.2 The von Neumann entropy The sample spaces we considered in the first lecture involved completely distinguishable states of matter, such as faces of coin, numbers on a card, etc. The random variable X associated with the 2

3 different outcomes in such experiments thus refers to classical information. One of the most important quantities in the theory of classical information is the Shannon entropy, which for random variable X, describes the optimal compression rate of X n into m-bit sequences. The notion of entropy can be generalized to the study of quantum information. In the quantum case, the random variable X is replaced by an ensemble of states E = t ψ x y, p x u xpx, which physically describes a quantum system S represented by Hilbert space H S that is prepared in state ψ x y P H S with probability p x. The quantum entropy of this ensemble is a function of the ensemble average ρ = x p x ψ x yxψ x, and it is defined as S(ρ) = Tr[ρ log ρ]. (3) This quantity is called the von Neumann entropy of the density matrix ρ as it was originally proposed by John von Neumann. By recalling the spectral decomposition of ρ = λ i =0 p λ i P λi, we can express its von Neumann entropy as S(ρ) = Tr p λi P λi log p λj P λj = λ i =0 λ i =0 λ j =0 p λi log p λi. (4) In other words, the von Neumann entropy of ρ is equal to the Shannon entropy of the distribution defined by the eigenvalues of ρ. It is then easy to verify that S(ρ bn ) = ns(ρ) (5) for any integer n. Our goal now is to show that the von Neumnann entropy plays a role analogous to the Shannon entropy in the task of quantum data compression. 2 The Compression of Quantum Information 2.1 Problem Description Let us now turn to the task of data compression. Recall that in the classical setting, the problem involved an i.i.d. source of some classical variable X that ranges over set X with probability p X. In the quantum setting, an i.i.d. quantum source E bn consists of n quantum systems each independently prepared according to ensemble E. In other words, the state across all n systems is described by the ensemble E bn := t ψ x ny, p x nu x n PX n (6) where ψ x ny = Â n i=1 ψ x i y and p x n = ± n i=1 p x i for any sequence x n = (x 1, x 2,, x n ) P X n. Classical data compression of an i.i.d. random variable X n involves the encoder/decoder sequence x n f g px n δ x n. In the first step, the encoder f maps the sequence x n to an m-bit sequence f (x n ) P t0, 1u m. This sequence is then sent to the decoder g that generates the output sequence px n. On average px n differs from x n with probability no greater than δ. Our goal will be to describe a similar procedure for the quantum source E bn in which the ensemble states are mapped to the space of m-qubit states and then reversed back to E bn. 3

4 Let d be the dimension of H S so that each ψ x ny lies in the nd-dimensional space (H S ) bn. The quantum analogs of the encoding/decoding functions f and g are encoding/decoding unitary transformations U and V. One immediate difference between the classical and quantum problems is that range of f : X n Ñ t0, 1u m has a different size than its domain, whereas when U acts on (H S ) bn, its domain and range are both nd-dimensional vector spaces. Consequently, it is not immediately clear how the encoding U can generate any sort of compression of the quantum source. This is accomplished by splitting the output of U into two subsystems: (i) an m-qubit system Q whose collective state represents the compressed quantum data, and (ii) an (nd 2m)- dimensional system E that represents any remaining data that is lost in the compression process. Thus, we consider unitaries of the form U : (H S ) bn Ñ H Q b H E with H Q (C b2 ) bm and H E C b(nd 2m). Similarly, the decoder is a unitary V : H Q b H E1 Ñ (H S ) bn with H 1E C b(nd 2m), and the entire compression/decompression sequence is given by E ψ x ny U E 1 Q V ρ x n δ ψ x nyxψ x n. Let us closely analyze each step in this procedure. For a given state ψ x ny drawn from E bn, the unitary U is applied and generates the state ψx 1 nyqe = U ψ x ny. Notice that ψ 1 y is a state on the bipartite system QE. The Q subsystem is forwarded to the decoder while the E subsystem is discarded. If this entire process works correctly, then the quantum information of the original state ψ x ny will be compressed into the subsystem Q and so discarding E will not affect the recovery of ψ x ny. Thus, the state sent to the decoder is described by the reduced density matrix Tr E ψx 1 nyxψ1 x n QE. Since the decoder V must act on an nd-dimensional state space, an additional system E 1 must be introduced and combined with system Q to form the full input to the decoder. The system E 1 is in some initial state 0yx0 E1 so that the output of the decoder is given by ( ρ x n = V Tr E ψx 1 nyxψ1 x n QE b 0yx0 E1) V :. (7) The goal is for ρ x n to be a close approximation to the input state ψ x ny. But what does it mean to say that one quantum state is a close approximation to another? While there are different answers to this question, in this course we will deal primarily with the fidelity measure of two quantum states. Thus, in order to properly a success figure of merit to the quantum data compress scheme just described, we must briefly digress and discuss the notion of quantum state fidelity. 2.2 The Fidelity between Two Quantum States Suppose a d-dimensional quantum system S is prepared in some unknown pure state. The experimenter (call her Esther) wants to test if the system is in some particular state ψy. She performs the ψy-test, which involves making the projective measurement defined by the projectors tp 0 = ψyxψ, P 1 = I ψyxψ u. The system is said to pass the ψy-test iff she obtains measurement outcome 0. What is the probability that some state φy passes the ψy-test? It is given by Probt φy passes the ψy-testu = xφ P 0 φy = xψ φy 2. 4

5 as Given this observation, we are motivated to define the fidelity 1 of two pure states ψy and φy F 2 ( ψy, φy) := xψ φy 2. (8) Operationally it measures the probability that φy would pass the ψy-test; or equivalently the probability that ψy would pass the φy-test. Similarly, suppose we have an ensemble E = t φ i y, p i u i of pure states. If a system is in state φ i y with probability p i, then the total probability that the system passes the ψy-test is ProbtE passes the ψy-testu = p i xψ φ i y 2 = p i xψ φ i yxφ i ψy = xφ ρ φy. Thus for a pure state ψy and mixed state ρ, their fidelity is defined as i i F 2 ( ψy, ρ) := xψ ρ ψy, (9) and it measures the probability a system passes the ψy-test when it is described by the mixed state ρ. Note that F( ψy, ρ) = 1 iff ψyxψ = ρ and F( ψy, ρ) = 0 iff ψy is orthogonal to the support of ρ. Also notice that F 2 ( ψy, ρ) is linear in its second argument so that F 2 ( ψy, ρ) = p i F 2 ( ψy, φ i y). (10) 2.3 The Schumacher Compression Protocol i Let us now return to the problem of quantum data compression. We will use the notion of fidelity to establish a quantitative measure for how well a given encoding and decoding process works. In any compression scheme of E bn = t ψ x ny, p x nu x n, each input state ψ x ny is transformed into the output state ρ x n described by Eq. (7). Ideally we would want ρ x n = ψ x nyxψ x n, but at the least we would hope that ρ x n should pass the ψ x ny-test with high probability, i.e. F( ψ x ny, ρ x n) 1. However, ψ x ny itself is drawn from the ensemble E bn with probability p x n, and this probability may be very low. So we might tolerate a low fidelity F( ψ x ny, ρ x n) if the state ψ x ny is transmitted with low probability. What we will prioritize in this problem is the average preserved fidelity of the input/output states. We thus say that a quantum compression scheme has compression fidelity 1 δ if x n PX n p x n F( ψ x ny, ρ x n) 1 δ. (11) If there exists an encoder U : (H S ) bn Ñ H Q b H E with H Q (C b2 ) bm and decoder V : H Q b H E1 Ñ (H S ) bn such that Eq. (11) holds, then we say that (1 δ)-compression is achievable at rate R = m n. Theorem 1 (Quantum Data Compression). Let E = t ψ x y, p x u xpx be an ensemble of states with ensemble average ρ = xpx p x ψ x yxψ x. For any δ 0 and R S(ρ), δ-good compression can always be achieved at rate R. Proof. The compression protocol that we describe here was first proposed by Benjamin Schumacher [Sch95], and it is hence called Schumacher compression. Let us begin by taking a spectral decomposition of the ensemble average ρ = X x=1 p x ψ x yxψ x = 1 In the literature, typically the fidelity is defined as the square root of F 2 ( ψy, φy). Y y=1 p y yyxy. (12) 5

6 Here the p y are the eigenvalues of ρ and the t yyu Y y=1 form a complete orthonormal basis of HS, which we will henceforth take as the computational basis of H S. It is important to emphasize that while both p x and p y form probability distributions, the ψ x y are not necessarily pairwise orthogonal whereas the yy are since they are eigenstates of a positive operator. Taking n copies of ρ, we have the spectral decomposition ρ bn = y n p y n y n yxy n. (13) Note that S(ρ bn ) = ns(ρ) = H(Y), (14) where H(Y) is the Shannon entropy of a random variable Y having distribution p y. Let ɛ, δ 0 be arbitrary. Our strategy will now be to apply the properties of typicality to the i.i.d. variable Y n. Recall that a sequence y n P Y n is called ɛ-typical if 2 n(h(y)+ɛ) p y n 2 n(h(y) ɛ). (15) The ɛ-typical set A (n) ɛ consists of all ɛ-typical sequences. Relating this to the problem of quantum data compression, we form the direct sum decomposition where (H S ) bn = H typ ` H atyp, (16) H typ = t y n y : y n P A (n) ɛ u (17) is called the typical subspace, and H atyp = H K typ. By the typicality theorem, we are ensured that (1 δ)2 n(s(ρ) ɛ) A (m) ɛ = dim[h typ ] = Tr[Π typ ] 2 n(s(ρ)+ɛ) (18) for sufficiently large n. As as in the classical data compression protocol, there exists an encoder function f that maps each element of A (n) ɛ to a unique m-bit sequence, where m = rn(s(ρ) + ɛ)s. However, unlike in the classical problem, we will want this function to be fully invertible on all of Y n so that it can be used to define a unitary transformation. Thus, for y n R A (n) ɛ we simply let f (y n ) be some unique number in the set t1, 2,, su where s is the number of atypical sequences. We can then define a unitary transformation U : (H S ) bn Ñ H Q b H E whose action is given by y n y Ñ # f (y n )y Q 0y E if y n P A (n) ɛ 0y Q f (y n )y E if y n R A (n) ɛ where f (y n )y Q is the state of some m-qubit system, and f (y n )y E is some state of the extra system E. Since the mapping f is invertible, this indeed defines a valid unitary transformation. We make two very important observations: (19) (a) If τy, ˆτy P H typ, then τyx ˆτ = U : [( Tr E U τyx ˆτ U : ) b 0yx0 E] U; (20) 6

7 (b) If y n R A (n) ɛ then since f (y n ) ranges over the set t1, 2,, su whenever y n R A (n) ɛ. x0 f (y n )y E = 0, (21) Now let us return to the ensemble E bn = t ψ x ny, p x nu x n whose states in (H S ) bn we want to compress. By the direct sum decomposition of Eq. (17), we can write each input state ψ x ny as ψ x ny = α x n τ x ny + β x n τx K ny, (22) where τ x ny P H typ and τ K x ny P H atyp. The α x n, β x n satisfy α x n 2 + β x n 2 = 1, and intuitively α x n 2 quantifies the fraction of ψ x ny that lies in the typical subspace. We next apply the encoding map U to the state and trace out system E. From Eq. (21) observed above, we find that the encoded state ψ 1 x nyqe = U ψ x ny has a reduced density matrix ρ Q x n = Tr E ψ 1 x nyxψ1 x n QE = α x n 2 Tr E U τ x nyxτ x n U : + β x n 2 0yx0. (23) We next introduce the system E 1 in state 0yx0 E1 so that the state arriving at the decoder is ρ QE1 x n = α x n 2 ( Tr E U τ x nyxτ x n U : ) b 0yx0 + β x n 2 0yx0 b 0yx0. (24) For the decoder, we apply the inverse of the encoding, U :. Hence by Eq. (20) our final state is with a fidelity given by ρ 1 x n = α x n 2 τ x nyxτ x n + β x n 2 U : 00yx00 U, (25) F 2 ( ψ x ny, ρ 1 x n) = xψ x n ( α x n 2 τ x nyxτ x n + β x n 2 U : 00yx00 U α x n 4 = (1 β x n 2 ) 2 ) ψ x ny 1 2 β x n 2. (26) From this we deduce the average fidelity to be x n PX n p x n F 2 ( ψ x ny, ρ 1 x n) 1 2 x n PX n p x n β x n 2. (27) What remains then is to bound the last term in this inequality. To do so, let Π typ denote the projector onto H typ, commonly called the typical projector. By the typicality theorem, for sufficiently large n we have 1 δ PrtY n P A (n) ɛ u = Tr[Π typ ρ] = x n PX n p x nxψ x n Π typ ψ x ny = x n PX n p x n α x n 2 = 1 p x n β x n 2. (28) Combining the last two equations, we have the desired result that p x n F 2 ( ψ x ny, ρ 1 xn) 1 2δ. (29) x n PX n 7

8 3 Data Compression and Entanglement One of the nicest properties of Schumacher is that it can be used in the compression of entangled subsystems. Suppose Alice wants to share a particular bipartite entangled state Ψy with Bob. Connecting Alice and Bob is a quantum channel that allows her to send a quantum system to Bob, such as a fiber optic cable that supports the transmission of single polarized photons. Thus Alice can achieve her goal by first preparing the state Ψy AA1 in her local laboratory AA 1 and then sending system A 1 to Bob. When Bob receives the transmitted system, they end up sharing the entangled state Ψy AB. Here we have relabeled the system A 1 Ñ B through this transmission in order to indicate that Bob holds the system in his laboratory. Alice wants to go one step further and share multiple copies of Ψy with Bob. That is, the goal if for Alice and Bob to hold Ψy bn, or a state with high fidelity to it. Using Schumacher compression, this can be accomplished with efficient use of the quantum channel connecting Alice and Bob. Let us write start with a Schmidt decomposition Ψy AA1 = Y y=1 a py e y y A yy A1, (30) where t e y yu y is a Schmidt basis for system A and t yyu y is a Schmidt basis for system A 1. Then for n copies we have Ψy bn = y n a py n e y ny y n y. (31) Written as a density matrix, we have ΨyxΨ bn = y n,ŷ n PA (n) ɛ a py n pŷn e y nyxeŷn A b y n yxŷ n A1 + X, (32) where X is some hermitian operator for which Tr[ e y nyxeŷn b IX] = 0 for any y n, ŷ n P A (n) ɛ. We now apply Schumacher compression on the second system with the latter belonging to Bob after the decoding step. Doing so generates the bipartite mixed state ρ AB = a Ψbn py n pŷn e y nyxeŷn A b y n yxŷ n A1 + ˆX. (33) y n,ŷ n PA (n) ɛ It is straightforward to compute the fidelity F 2 ( Ψy by, ρ Ψ bn) PrtY n P A (n) ɛ u 2 (1 δ) 2 1 2δ. (34) Thus, by sending quantum information to Bob at any rate R S(ρ) of qubits per channel use, Alice and Bob can share an entangled state ρ AB with high fidelity to multiple copies of Ψy. 4 Exercises Exercise 1 Prove that a collection of n quantum states are perfectly distinguishable if and only if they are pairwise orthogonal. 8

9 Exercise 2 Give a proof for the fact that S(ρ bn ) = ns(ρ). This property is sometimes called additvity of the Shannon entropy. Exercise 3 Consider the qubit state ψ(θ)y = cos θ 0y + sin θ 1y for θ P [0, 2π). Form the density matrix ρ(θ) = 1 ( 0yx0 + ψ(θ)yxψ(θ) ). 2 (a) Express the entropy S(ρ) as a function of θ. (b) For what values of θ does S(ρ) attain its minimum and maximum values? Exercise 4 Consider again the trine ensemble consisting of states e 0 y = 0y, e 1 y = 1? 3 2 0y + 2 1y, e 2y = 1? 3 2 0y 1y. (35) 2 What is the fidelty between the state r+y = 1? 2 ( 0y + i 1y) and the mixed state ρ = i=0 e iyxe i? Exercise 5 Provide a detailed proof of Eqns. (20) and (21). Exercise 6 An alternative interpretation of Schumacher compression can be given in terms of purity extraction []. Starting from n copies of a mixed state ρ acting on space H, a purity extraction protocol generates m copies of a pure state 0y by applying a unitary transformation and discarding subsystems. A rate R is said to be an achievable purity extraction rate if for every δ 0, there exists a unitary transformation U : H bn Ñ H ba b H bb such that F 2 ( Tr A [Uρ bn U : ], 0yx0 bm) 1 δ (36) and R = m n. Prove that pruity extraction is always possible for any R d = dim[h]. log d S(ρ), where References [Sch95] Benjamin Schumacher. Quantum coding. Phys. Rev. A, 51: ,