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Introduction to Quantum Information Processing CS 667 / PH 767 / CO 681 / AM 871 Lecture 16 (2009) Richard Cleve DC 2117 [email protected] 1 Distinguishing between two arbitrary quantum states 2 Holevo-Helstrom Theorem (1) Theorem: for any two quantum states and , the optimal measurement procedure for distinguishing between them succeeds with probability ½ + ¼|| − ||tr (equal prior probs.) Proof* (the attainability part): Since − is Hermitian, its eigenvalues are real Let + be the projector onto the positive eigenspaces Let − be the projector onto the non-positive eigenspaces Take the POVM measurement specified by + and − with the associations + and − * The other direction of the theorem (optimality) is omitted here 3 Holevo-Helstrom Theorem (2) Claim: this succeeds with probability ½ + ¼|| − ||tr Proof of Claim: A key observation is Tr(+ − −)( − ) = || − ||tr The success probability is ps = ½Tr(+ ) + ½Tr(− ) & the failure probability is pf = ½Tr(+ ) + ½Tr(− ) Therefore, ps − pf = ½Tr(+ − −)( − ) = ½|| − ||tr From this, the result follows 4 Purifications & Ulhmann’s Theorem Any density matrix , can be obtained by tracing out part of some larger pure state: d j j j 1 m m j Tr2 j j j j j j j 1 j 1 a purification of Ulhmann’s Theorem*: The fidelity between and is the maximum of φψ taken over all purifications ψ and φ * See [Nielsen & Chuang, pp. 410-411] for a proof of this Recall our previous definition of fidelity as F(, ) = Tr√ 1/2 1/2 ||1/2 1/2||tr 5 Relationships between fidelity and trace distance 1 − F(, ) || − ||tr √1 − F(, )2 See [Nielsen & Chuang, pp. 415-416] for more details 6 Entropy and compression 7 Shannon Entropy Let p = (p1,…, pd) be a probability distribution on a set {1,…,d} Then the (Shannon) entropy of p is H(p1,…, pd) d p j 1 j log p j Intuitively, this turns out to be a good measure of “how random” the distribution p is: vs. H(p) = log d vs. vs. H(p) = 0 Operationally, H(p) is the number of bits needed to store the outcome (in a sense that will be made formal shortly) 8 Von Neumann Entropy For a density matrix , it turns out that S() = − Tr log is a good quantum analog of entropy Note: S() = H(p1,…, pd), where p1,…, pd are the eigenvalues of (with multiplicity) Operationally, S() is the number of qubits needed to store (in a sense that will be made formal later on) Both the classical and quantum compression results pertain to the case of large blocks of n independent instances of data: • probability distribution pn in the classical case, and • quantum state n in the quantum case 9 Classical compression (1) Let p = (p1,…, pd) be a probability distribution on a set {1,…,d} where n independent instances are sampled: ( j1,…, jn) {1,…,d}n (d n possibilities, n log d bits to specify one) Theorem*: for all > 0, for sufficiently large n, there is a scheme that compresses the specification to n(H(p) + ) bits while introducing an error with probability at most Intuitively, there is a subset of {1,…,d}n, called the “typical sequences”, that has size 2n(H(p) + ) and probability 1 − A nice way to prove the theorem, is based on two cleverly defined random variables … * “Plain vanilla” version that ignores, for example, the tradeoffs between n and 10 Classical compression (2) Define the random variable f :{1,…,d} R as f ( j ) = − log pj Note that E[ f ] d p j 1 Define g:{1,…,d}n d j f ( j ) p j log p j H p1 ,, pd j 1 R as g j1 ,, jn Thus E[ g ] H p1 ,, pd Also, g( j1,…, jn) log p j p j 1 n 1 n f ( j1 ) f ( jn ) n 11 Classical compression (3) By standard results in statistics, as n , the observed value of g( j1,…, jn) approaches its expected value, H(p) More formally, call ( j1,…, jn) {1,…,d}n -typical if g j1 ,, jn H p ε Then, the result is that, for all > 0, for sufficiently large n, Pr[( j1,…, jn) is -typical] 1− We can also bound the number of these -typical sequences: • By definition, each such sequence has probability 2−n(H(p) + ) • Therefore, there can be at most 2n(H(p) + ) such sequences 12 Classical compression (4) In summary, the compression procedure is as follows: The input data is ( j1,…, jn) {1,…,d}n, each independently sampled according the probability distribution p = (p1,…, pd) The compression procedure is to leave ( j1,…, jn) intact if it is -typical and otherwise change it to some fixed -typical sequence, say, ( j ,…, j) (which will result in an error) Since there are at most 2n(H(p) + ) -typical sequences, the data can then be converted into n(H(p) + ) bits The error probability is at most , the probability of an atypical input arising 13 Quantum compression (1) The scenario: n independent instances of a d-dimensional state are randomly generated according some distribution: φ1 prob. p1 φr prob. pr Example: 0 prob. ½ + prob. ½ Goal: to “compress” this into as few qubits as possible so that the original state can be reconstructed with small error in the following sense … The expected* trace distance between the reconstructed state and the state that was actually generated should be small * Defined as the expected value of the trace distance, taken with respect to the randomness of the generation procedure 14 Quantum compression (2) r Define pi φ i φ i i 1 Theorem: for all > 0, for sufficiently large n, there is a scheme that compresses the data to n(S() + ) qubits, with expected trace distance √2 For the aforementioned example, 0.6n qubits suffices The compression method: d Express in its eigenbasis as q j ψ j ψ j j 1 With respect to this basis, we will define an -typical subspace of dimension 2n(S() + ) = 2n(H(q) + ) 15 Quantum compression (3) The -typical subspace is that spanned by ψ j1 , , ψ jn where ( j1,…, jn) is -typical with respect to (q1,…, qd) Define typ as the projector into the -typical subspace By the same argument as in the classical case, the subspace has dimension 2n(S() + ) and Tr(typ n) 1− This is because is the density matrix of 1 prob. q1 d prob. qd 16 Quantum compression (4) Calculation of the expected fidelity: p i1in i i typ i i 1 n 1 i1in n p i1in Tr typ i1in i1in i1in Abbreviations: pi1i = pi1 pi n n φi1in = φi1φin Tr typ pi1in i1in i1in i1in Tr typ n 1 Using || − ||tr √1 − F(, )2, we can upper bound the expected trace distance by √2 17