Upper bounding the expected communication cost

which is increasing in the in-terval [0, 1). The last inequality follows from the fact that (1−β)|ψihψ| is a substate of ρ.

3.3.3 Upper bounding the expected communication cost

In Sections 3.3.1 and 3.3.2, we showed that the average communication of the Greedy Quantum Rejection Sampler for ρ, σ ∈ D(H) in the case where ρ is a pure state and in the case where dim(H) = 2 is at most S_max(ρ||σ) + 2 lg(S_max(ρ||σ) + 1) + O(1). This motivates the following question: Is S_max(ρ||σ) + 2 lg(S_max(ρ||σ) + 1) + O(1) an upper bound for the average communication cost of the Greedy Quantum Rejection Sampler in general?

This section contains some of the approaches we took to prove this conjecture. First we introduce a quantum rejection sampling protocol, Π_Smax, with average communication cost bounded by S_max(ρ||σ) + 2 lg(S_max(ρ||σ) + 1) + O(1) for arbitrary states ρ and σ such that supp(ρ) ⊆ supp(σ).

Let ρ, σ ∈ D(H) and K be a finite dimensional Hilbert space such that dim(K) ≥ dim(H). Recall that in any quantum rejection sampling protocol, Alice and Bob initially share a sequence, {|φi_i}_i∈N, of a fixed purification, |φi, of σ in C²⊗ K ⊗ H such that Bob’s marginal of each state is σ and Alice holds the rest of each state. Let α = 2^{−Smax(ρ||σ)}, then by definition αρ ≤ σ. The protocol Π_Smax is defined as follows. In the j-th iteration Alice uses the subroutine Π(αρ, σ) on |φi_j in order to prepare the substate αρ on Bob’s side.

If Alice is successful in preparing αρ she sends the index J = j to Bob using the prefix encoding function E₂, then Bob outputs his marginal of the j-th state and the protocol terminates. If Alice fails, she proceeds to the (j + 1)-th iteration.

Claim 3.3.11. The protocol Π_Smaxterminates with probability 1 and Bob’s output state is ρ.

Proof: The contribution of the j-th iteration in Bob’s output state, Xj, is equal to the probability of reaching the j-th iteration times αρ, i.e. X_j = (1 − s_j−1)αρ. Hence, s₀ = 1 and for every j ≥ 1,

s_j = s_j−1+ (1 − s_j−1)α = α + (1 − α)s_j−1 .

It is straightforward to see that for every j ≥ 0, s_j = 1 − (1 − α)^j. So we have

∞

X

j=1

X_j =

∞

X

j=1

(1 − α)^jαρ = ρ .

Claim 3.3.12. The expected communication cost of the protocol Π_Smax is bounded as E [ lE2(J ) ] ∈ S_max(ρ||σ) + 2 lg(S_max(ρ||σ) + 1) + O(1) .

Proof: We show that

E [lg(J)] ≤ Smax(ρ||σ) .

Then the claim follows by a similar argument as in claim3.3.4. We have E [lg(J)] ≤ lg (E[J]) (By Jensen’s inequality)

= lg

Now we return to upper bounding the expected communication cost of the Greedy Quantum Rejection Sampler. Let P : N −→ R be the probability distribution defined for every j ∈ N as P (j) = Tr(Xj), where X_j is the contribution of the j-th iteration to Bob’s output state in the Greedy Rejection Sampler. Note that in the Greedy Rejection Sampler, for every j ∈ N, Xj+1 is a feasible solution to the semidefinite program (P_j) defined in Section 3.2.2 . So Tr(X_j), the optimal value of (P_j), is more than or equal to Tr(X_j+1), for every j ∈ N, i.e. P is a non-increasing probability distribution over N.

Next we introduce a way of proving upper bounds on the expected communication cost of the Greedy Rejection Sampler. We first need to prove the following lemma.

Lemma 3.3.13. Let P₁, P₂ : N −→ R be two probability distributions on N, such that P1 

since P₁  P₂, for every i ∈N, by definition ai ≥ 0 . Hence, we have

Note that Lemma 3.3.13 in fact holds if we replace the logarithm function by any function g : [1, ∞) −→R which is non-decreasing over the interval [1, ∞).

Let Q : N −→ R be any non-increasing probability distribution on N such that P  Q, then Lemma 3.3.13states that X

i∈N

Q(i) lg(i) is an upper bound onE [lg(J)] for the Greedy Quantum Rejection Sampler.

Consider the probability distributions Q : N −→ R defined for every j ∈ N as Q(j) = (1 − α)^jα , for α = 2^{−Smax(ρ||σ)}, where (1 − α)^jα is the probability that in the protocol Π_Smax terminates with J = j. Note that the probability distribution ((1 − α)^jα)_j∈N is non-increasing. If we could show that P  Q, then by Lemma 3.3.13 , the upper bound of S_max(ρ||σ)+2 lg(S_max(ρ||σ)+1)+O(1) for the expected communication cost of the Greedy Quantum Rejection Sampler could be obtained. It turns out that there are states ρ and σ such that P  Q. We simulated the greedy rejection sampler for states ρ = ⁵₆|+ih+| +

1

6|−ih−| and σ = ²₃|0ih0| + ¹₃|1ih1|, where |+i = ^√1

2(|0i + |1i) and |−i = ^√1

2(|0i − |1i), respectively. Table3.1 contains the results.

A different approach we took for proving an upper bound of S_max(ρ||σ)+2 lg(S_max(ρ||σ)+

1) + O(1) is the following. The expected value of lg(J ) in any quantum rejection sampling protocol is given by

If we could show that for the Greedy protocol, for every j ≥ 1,

j

j Tr(X_j) α(1 − α)^j−1 Pj

Table 3.1: Greedy rejection sampler vs. Π_Smax in the first 10 iterations

then using a similar argument as in Claim 3.3.10, we could bound every j ≥ 1 as follows.

Since s_j is the probability that the protocol terminates within j iterations, we have

1 ≥ s_j =

Note that Tr(X₁) ≥ α since αρ is a feasible solution of the semidefinite program defin-ing X₁. Our first conjecture was that Tr(X_j) ≥ (1 − s_j−1)α, meaning that in the j-th iteration at least an α portion of the trace of the remaining state R_j−1is removed. This con-jecture turned out not to be true. In fact, we could find an instance of the problem for which even the weaker condition Rejec-tion Sampler. We simulated the greedy rejecRejec-tion sampler for states ρ = ⁵₆|+ih+| +¹₆|−ih−|

and σ = ²₃|0ih0| + ¹₃|1ih1|. Table3.2 contains the results.

Table 3.2: Simulation results for the first 10 iterations of the greedy quantum rejection sampler

What is interesting about this example is that ρ and σ are states in D(C²), while in Section3.3.2we have shown that the upper bound of Smax(ρ||σ)+2 lg(Smax(ρ||σ)+1)+O(1) does hold for the Greedy Rejection Sampler in this case. This shows that perhaps we need a stronger approach for proving the upper bound.

An alternative way of upper bounding the expected communication cost of the Greedy Rejection Sampler is to bound Tr(X_j) for every j ∈ N. Since in each iteration Tr(Xj) is the optimal value of a semidefinite program, a natural and more general approach would be using duality theory. Recall that in the Greedy Rejection Sampler, for every j ∈ N, in the j-th iteration, X_j is defined as an optimal solution of the semidefinite program (P )

defined as program (P ) in which X is free to be any Hermitian operator as follows.

(P⁰) : maximize : Tr(X) for (P⁰) and (D⁰), respectively. So by Theorem 2.1.6 strong duality holds and both (P⁰) and (D⁰) achieve their optimal values. Moreover, (D⁰) is equivalent to

(D⁰⁰) : minimize : Tr(B) + Tr((A − B) Y₁) of (D⁰). Let ˜X be an optimal solution of (P⁰). By complementary slackness condition, we have

Also, by definition of Π₊ and Π− we have

Π₊(A − B) Π₋ = Π₋(A − B) Π₊ = 0 . (3.3.7) Finally, by (3.3.6) and (3.3.7) we have

X˜ = (Π−+ Π₊) ( ˜X) (Π−+ Π₊)

= Π₋AΠ₋+ Π₊AΠ₋+ Π₋BΠ₊+ Π₊BΠ₊

= Π−AΠ−+ Π+BΠ−+ Π−BΠ++ Π+BΠ+

= Π−AΠ−− Π−BΠ−+ B

= A − B

2 − |A − B|

2 + B

= A + B

2 − |A − B|

2 .

Now we return to the original semidefinite program (P ). The dual program of (P ) is defined as

(D) : minimize : Tr(AY1) + Tr(BY2) subject to : Y₁+ Y₂ ≥ 1H

Y₁, Y₂ ≥ 0 .

The first difference between the pair (P, D) and (P⁰, D⁰) is that (P ) is not necessarily strictly feasible. For example if ρ is not full rank in supp(σ), then no feasible solution of (P ) is positive definite. So it is no longer guaranteed that the dual optimal value is achieved. The other difference is that Y₁+ Y₂ ≥ 1^H in (D). Since (D) is a minimization problem, one may conjecture that without changing the optimal value of (D) we may replace the constraint Y₁ + Y₂ ≥ 1H with Y₁ + Y₂ = 1H. But this is not generally true since for positive semidefinite operators A and B, the operator A + B

2 − |A − B|

2 is not necessarily positive semidefinite. However, since (Y₁, Y₂) = (1^H,1^H) is a strictly feasible solution for (D) and X = 2^{−(Smax(A,B))}A is a feasible solution for (P ), by Theorem 2.1.6 strong duality holds and the primal optimal value is achieved. The above observations suggest that the analysis of the dual program (D) is more complicated compared to (D⁰).

In order to give a taste of the difficulty in analyzing the dual program (D), next we consider a simple example in which ρ, σ ∈ D(C²) are 2 by 2 density operators and ρ is a pure state.

Let σ = (p) |0ih0| + (1 − p) |1ih1| ∈ D(C²) for p ∈ (0, 1), and ρ = |+ih+|. Let B = σ and A = ρ in the semidefinite programs (P ) and (D). Then 2^{−Smax(ρ,σ)} = 2p(1 − p) and

the optimal solution of (P ) is equal to ˜X = 2p(1 − p)|+ih+|. Let  > 0. As stated in the previous paragraph, the dual program (D) does not necessarily achieve its optimal value but since strong duality holds, there exists a dual feasible solution ( ˜Y₁(), ˜Y₂()) such that 0 ≤ Tr(ρ ˜Y₁()) + Tr(σ ˜Y₂()) − Tr( ˜X) ≤  . (3.3.8) following form in the |+i, |−i basis for real numbers b, c, d and e.

Y˜₁() = ζ b

or equivalently, (d + (2p − 1))² ≤ . We choose d = (1 − 2p). So when   1 and p 6= 1/2,

This example shows that the analysis of the semidefinite programs (P ) and (D), even in this simple case, is much more complicated compared to the pair (P⁰) and (D⁰).