Introduction to Quantum Information Processing QIC 710 / CS 667 / PH 767 / CO 681 / AM 871

1. Introduction to Quantum Information ProcessingQIC 710 / CS 667 / PH 767 / CO 681 / AM 871 Lecture 16 (2011) Richard Cleve DC 2117 cleve@cs.uwaterloo.ca

2. Distinguishing between two arbitrary quantum states

3. 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

4. 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

5. Purifications & Ulhmann’s Theorem Any density matrix , can be obtained by tracing out part of some larger pure state: a purification of  Ulhmann’s Theorem*: Thefidelitybetween  andis the maximum of φψ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

6. Relationships between fidelity and trace distance 1 − F(,)  ||−||tr  √1 − F(,)2 See [Nielsen & Chuang, pp. 415-416] for more details

7. Entropy and compression

8. 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) Intuitively, this turns out to be a good measure of “how random” the distribution p is: vs. vs. vs. H(p) = log d 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)

9. 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 pn in the classical case, and • quantum state n in the quantum case

10. 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 (dnpossibilities, nlogd 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 nand

11. Classical compression (2) Define the random variable f :{1,…,d} R as f ( j ) =− log pj Note that Thus Define g:{1,…,d}n R as Also, g( j1,…, jn)

12. 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 Then, the result is that, for all > 0,for sufficiently large n, Pr[( j1,…, jn) is-typical] 1− • We can also bound the number ofthese-typical sequences: • By definition, each such sequence has probability  2−n(H(p) +) • Therefore, there can be at most 2n(H(p) +) such sequences

13. 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 inton(H(p) + )bits The error probability is at most , the probability of an atypical input arising

14. 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 0 prob.½ + prob.½ Example: 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 … • -good: • No procedure can distinguish between these two states • compressing and then uncompressing the data • the original data left as is • with probability more than ½ + ¼ 

15. Quantum compression (2) Theorem: for all > 0, for sufficiently large n, there is a scheme that compresses the data to n(S() + ) qubits, that is 2√ -good Express  in its eigenbasis as Define For the aforementioned example, 0.6n qubits suffices The compression method: With respect to this basis, we will define an -typical subspace of dimension 2n(S() +) =2n(H(q) +)

16. Quantum compression (3) 1 prob.q1    d prob.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() +) andTr(typ n)  1− The -typicalsubspace is that spanned by where ( j1,…, jn) is -typical with respect to (q1,…, qd) This is because  is the density matrix of

17. Quantum compression (4) Proof that the scheme is 2√ -good: (This is to be inserted)