An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography

We show that $\Omega(rd/\epsilon)$ copies of an unknown rank-$r$, dimension-$d$ quantum mixed state are necessary in order to learn a classical description with $1 - \epsilon$ fidelity. This improves upon the tomography lower bounds obtained by Haah, et al. and Wright (when closeness is measured with respect to the fidelity function).


Background
We consider quantum state tomography: given n copies of a mixed state ρ, output a classical description of a state σ that is close to ρ. In this note we measure closeness between ρ and σ via their fidelity F (ρ, σ), defined as the supremum of | ϕ | ψ | 2 over all purifications |ψ , |ϕ of ρ, σ respectively 1 . In fidelity tomography, the goal is for the output state σ to satisfy F (ρ, σ) ≥ 1 − . Haah et al. [2] showed that n = O(rd log(d/ )/ ) is sufficient for fidelity tomography where r is the rank of the density matrix ρ. They also proved a n = Ω rd δ 2 log(d/rδ) lower bound for trace distance tomography, where the goal is to output a state σ whose trace distance ρ − σ 1 = 1 2 Tr(|ρ − σ|) with ρ is at most δ. The Fuchs-van de Graff inequalities then imply a n = Ω rd log(d/r ) lower bound for fidelity tomography; thus their upper and lower bounds are tight up to logarithmic factors. O'Donnell and Wright [4] proved that n = O(rd/δ 2 ) input samples are sufficient for trace distance tomography. In his PhD thesis, Wright [7] showed that n = Ω(rd) samples are necessary for both fidelity and trace distance tomography; this is optimal when the desired trace distance error δ or infidelity is fixed to a constant, but otherwise is loose when the error is treated as a quantity going to zero.
In this note we improve the lower bound of Haah et al. [2] and Wright [7] in the fidelity tomography case and show that n = Ω(rd/ ) input samples are needed. It is natural to conjecture that the optimal bound for fidelity tomography is n = Θ(rd/ ); however we leave obtaining a matching upper bound for future work.

The argument
We prove our lower bound via reduction to the pure state tomography scenario, in which the input samples ρ are guaranteed to be pure states (in other words, the rank of the density matrix is 1). It was proved by Bruß and Macchiavello [1] that n = Θ(d/ ) samples are necessary and sufficient to achieve fidelity 1 − ; this was based on a tight connection between optimal pure state tomography and optimal pure state cloning [3,5].
Suppose there is an algorithm A that, on input n copies of a rank-r, dimension-d mixed state ρ, outputs with high probability a classical description of a state σ that has fidelity 1 − with ρ. Then we use this algorithm to construct another algorithm B that performs tomography on pure, dimension-rd states using O n + r 2 input samples and achieves 1 − O( ) fidelity. The performance of algorithm B is subject to the bounds of Bruß and Macchiavello [1] -in other words, B must use at least Ω(rd/ ) input samples. Thus it must be that The algorithm B works as follows. Let |ψ XY denote the rd-dimensional pure input sample where X denotes an r-dimensional register and Y denotes a d-dimensional register.
1. The algorithm B takes n input samples |ψ XY and traces out the X register in each copy to obtain n copies of a mixed state ρ ∈ C d×d .
2. Run algorithm A on the n copies of ρ to obtain (with high probability) a classical description of a rank-r, dimension-d state σ that has fidelity 1 − with ρ.
3. Compute a classical description of the rank-r projector Π onto the support of σ.
4. Take O(r 2 / ) additional copies of the input state |ψ XY and measure the Y register of each copy using the projective measurement {Π, I − Π}, and keep the post-measurement states | ψ of the copies where the Π outcome was obtained.
5. Use the tomography procedure of [1] for dimension-r 2 pure states on the copies of | ψ where we treat the states as residing in the dimension-r 2 subspace Let |ϕ ∈ C r ⊗supp(Π) denote the result of this pure state tomography procedure. The algorithm B then outputs the classical description of |ϕ as its estimation of |ψ .
We analyze the algorithm B. Let denote the Schmidt decomposition of |ψ where {|u 1 , . . . , |u r } is an orthonormal basis for C r and {|v 1 , . . . , |v r } is an orthogonal set of vectors in C d . We can then write ρ as Note that ρ is a rank-r density matrix. Let σ = r i=1 µ i |w i w i | denote the estimate produced by Step 2. By the guarantees of algorithm A, we have that (with high probability) F (ρ, σ) ≥ 1 − . Let Π = r i=1 |w i w i | denote the projector onto the support of σ. By the definition of fidelity, there exists a purification |ϕ ∈ C r ⊗ C d of σ such that On the other hand, where the inequality uses Cauchy-Schwarz. Let | ψ = (I ⊗ Π) |ψ / I ⊗ Π |ψ , and observe that The number of copies of | ψ available in Step 5 is, with high probability, at least Ω(r 2 / ). Thus the estimate |ϕ computed by Step 5 will satisfy We now turn to a simple geometric proposition: where e iα is a complex phase satisfying | ψ | ψ | = e iα ψ | ψ . Similarly | ψ − e iβ |ϕ 2 ≤ 2η for some complex phase e iβ . Then by triangle inequality we have The proposition follows via rearrangement.
Therefore with high probability, the estimate |ϕ produced by algorithm B satisfies

Conclusion
We proved a Ω(rd/ ) sample complexity lower bound for fidelity tomography for rank-r, dimension-d mixed states where 1 − is the fidelity of the resulting estimate. This is proved via reduction to the Ω(d/ ) lower bound for pure state tomography established by [1]. In contrast, the lower bounds of [2] and [7] are based on communication complexity arguments. Natural questions include: (a) whether the upper bound for fidelity tomography can be improved to O(rd/ ), and (b) whether a Ω(rd/δ 2 ) lower bound can be established for trace distance tomography. One obstacle to extending our argument to the trace distance setting is that we do not know whether applying the projection Π to the state |ψ (if Π is the projector onto the support of a state σ that is δ-close to ρ in trace distance) yields a state that is O(δ)-close to |ψ in trace distance. The Gentle Measurement Lemma [6] implies that the post-measurement state is O( √ δ)-close to |ψ ; this ultimately yields a Ω(rd/δ) lower bound on trace distance tomography, which we believe is not optimal.