1 Introduction to Quantum Information Processing QIC 710 / CS 667 / PH 767 / CO 681 / AM 871 Richard Cleve DC 2117 Lecture 16 (2011)

2 Distinguishing between two arbitrary quantum states

Holevo-Helstrom Theorem (1) 3 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

Holevo-Helstrom Theorem (2) 4 Claim: this succeeds with probability ½ + ¼ ||  −  || tr A key observation is Tr(  + −  − ) (  −  ) = ||  −  || tr Proof of Claim: Therefore, p s − p f = ½Tr(  + −  − ) (  −  ) = ½||  −  || tr From this, the result follows The success probability is p s = ½Tr(  +  ) + ½Tr(  −  ) & the failure probability is p f = ½Tr(  +  ) + ½Tr(  −  )

Purifications & Ulhmann’s Theorem 5 Any density matrix , can be obtained by tracing out part of some larger pure state: a purification of  Ulhmann’s Theorem*: The fidelity between  and  is the maximum of  φ  ψ  taken over all purifications  ψ  and  φ  Recall our previous definition of fidelity as F( ,  ) = Tr √  1/2   1/2  ||  1/2  1/2 || tr * See [Nielsen & Chuang, pp ] for a proof of this

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

7 Entropy and compression

Shannon Entropy 8 Let p = ( p 1,…, p d ) be a probability distribution on a set {1,…,d} Then the (Shannon) entropy of p is H( p 1,…, p d ) Intuitively, this turns out to be a good measure of “how random” the distribution p is: vs. H(p) = log dH(p) = 0 Operationally, H(p) is the number of bits needed to store the outcome (in a sense that will be made formal shortly)

Von Neumann Entropy 9 For a density matrix , it turns out that S(  ) = − Tr  log  is a good quantum analog of entropy Note: S(  ) = H( p 1,…, p d ), where p 1,…, p d 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

Classical compression (1) 10 Let p = ( p 1,…, p d ) be a probability distribution on a set {1,…,d} where n independent instances are sampled: ( j 1,…, j n )  {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  A nice way to prove the theorem, is based on two cleverly defined random variables … Intuitively, there is a subset of {1,…,d} n, called the “typical sequences”, that has size 2 n (H(p) +  ) and probability 1 −  * “Plain vanilla” version that ignores, for example, the tradeoffs between n and 

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

Classical compression (3) 12 By standard results in statistics, as n  , the observed value of g ( j 1,…, j n ) approaches its expected value, H(p) More formally, call ( j 1,…, j n )  {1,…,d} n  - typical if Then, the result is that, for all  > 0, for sufficiently large n, Pr [ ( j 1,…, j n ) 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 2 n (H(p) +  ) such sequences

Classical compression (4) 13 In summary, the compression procedure is as follows: The input data is ( j 1,…, j n )  {1,…,d} n, each independently sampled according the probability distribution p = ( p 1,…, p d ) The compression procedure is to leave ( j 1,…, j n ) 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 2 n (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

Quantum compression (1) 14 The scenario: n independent instances of a d -dimensional state are randomly generated according some distribution:  φ 1  prob. p 1     φ r  prob. p r 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 (a)compressing and then uncompressing the data (b)the original data left as is with probability more than ½ + ¼   0  prob. ½  +  prob. ½ Example:

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

Quantum compression (3) 16 The  - typical subspace is that spanned by where ( j 1,…, j n ) is  - typical with respect to ( q 1,…, q d ) By the same argument as in the classical case, the subspace has dimension  2 n (S(  ) +  ) and Tr(  typ   n )  1 −  Define  typ as the projector into the  - typical subspace  1  prob. q 1     d  prob. q d This is because  is the density matrix of

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