Quantum state tomography

2.3 Quantum entanglement

2.3.3 Quantum state tomography

Determining an unknown quantum stateρis not a trivial exercise. It is in principle impossible to determine the state of an unknown quantum system ρ if one only has a single copy of ρ. This is because there is no quantum measurement which can accurately distinguish non-orthogonal states like |0⟩ and (|0⟩+|1⟩)/√

2 [51].

Quantum state tomography is a procedure that allows one to experimentally esti- mate the state of an unknown quantum system through repeated measurements on copies of that system [109, 110]. Usually the state to be characterised is produced by an experiment, one can prepare many copies of that state by repeating the experiment. In order to uniquely identify the state, the set of measurements have to be tomographically complete, that is, the operators measured have to form an operator basis on the system’s Hilbert space so as to provide all the information about the system. Thus any operator – in particular the density operator – can be written as a linear combination of the basis operators with uniquely determined coefficients. For example, the operators σ₀/√

2, σ₁/√

2, σ₂/√

2, σ₃/√

2 form an operator basis for a qubit where σ₀ is the identity matrix and σ₁, σ₂ and σ₃ are

the Pauli matrices given by

σ₀ = [

1 0 0 1

]

, (2.48)

σ1 = [

0 1 1 0

]

, (2.49)

σ₂ = [

0 −i

i 0

]

, (2.50)

σ₃ = [

1 0

0 −1 ]

. (2.51)

The density matrix ρ of a qubit’s state can be written in terms of the matrices above as

ρ= tr(ρ)σ₀+ tr(σ₁ρ)σ₁+ tr(σ₂ρ)σ₂+ tr(σ₃ρ)σ₃

2 . (2.52)

Since tr(Oρ) is the expectation value of the observableO, one can estimate the value of tr(Oρ) by measuring the observable O a large number of times n and computing the average of the measured quantities. The expectation values of the three observablesσ₁,σ₂andσ₃can thus be obtained with a high level of confidence in the limit of large sample size. A good estimate of ρ can therefore be obtained provided that one has a large enough sample size.

In order to measure the observables corresponding to the Pauli matrices, one has to perform a projective measurement corresponding to the eigenstates of each matrix. The eigenvalues of all the Pauli matrices are either 1 or -1. Letu_n andv_n be the eigenvectors associated with the eigenvalues 1 and -1 respectively for the Pauli matrix σ_n where n = 1,2 and 3. One can write the Pauli matrix σ_n as the operator

σ_n =|u_n⟩⟨u_n| − |v_n⟩⟨v_n|. (2.53) Then

tr{σ_nρ} = tr{|u_n⟩⟨u_n|ρ} −tr{|v_n⟩⟨v_n|ρ} (2.54)

= ⟨u_n|ρ|u_n⟩ − ⟨v_n|ρ|v_n⟩.

The quantity ⟨u_n|ρ|u_n⟩ can be approximated by measuring the coincidence counts corresponding to the projection operator |u_n⟩⟨u_n| and normalising the re- sults by dividing by the total count rate (corresponding to tr{σ₀ρ}), that is,

⟨u_n|ρ|u_n⟩= count rate for projective measurement

total count rate . (2.55)

The identity operator can be written as

σ₀ =|u_n⟩⟨u_n|+|v_n⟩⟨v_n|. (2.56) Thus

tr{σ₀ρ}=⟨u_n|ρ|u_n⟩+⟨v_n|ρ|v_n⟩. (2.57) Therefore, one can estimate tr{σ_nρ} using the coincidence count rates by

tr{σ_nρ}= count rate foru_n−count rate forv_n

count rate foru_n+ count rate forv_n. (2.58) The expansion in Eq. 2.52 can be generalised to the case where one has 2 qubits and to the case of qudits. In the case of 2 qubits, it becomes [51]

ρ=∑

m,n

tr{σ_n⊗σ_mρ}σ_n⊗σ_m

4 , (2.59)

where n, m are chosen from the set 0,1,2,3. Each term in Eq. (2.59) can be estimated by measuring observables which are products of Pauli matrices. The tensor product of the Pauli matrices can also be written in terms of the eigenvectors as follows

σ_m⊗σ_n = (|u_m⟩⟨u_m| − |v_m⟩⟨v_m|)⊗(|u_n⟩⟨u_n| − |v_n⟩⟨v_n|)

= |u_mu_n⟩⟨u_mu_n| − |v_mu_n⟩⟨v_mu_n| − |u_mv_n⟩⟨u_mv_n|+|v_mv_n⟩⟨v_mv_n|. (2.60) Thus tr{σ_m⊗σ_nρ} can be written as

tr{σ_m⊗σ_nρ}=⟨u_mu_n|ρ|u_mu_n⟩−⟨v_mu_n|ρ|v_mu_n⟩−⟨u_mv_n|ρ|u_mv_n⟩+⟨v_mv_n|ρ|v_mv_n⟩. (2.61)

This quantity can be estimated from the coincidence count rate as was done in the case of one qubit above. In the case where n, m= 1,2,3, one gets

tr{σ_n ⊗σ_mρ}=

count rate foru_nu_m−count rate forv_nu_m−count rate foru_nv_m+ count rate forv_nv_m count rate foru_nu_m+ count rate forv_nu_m+ count rate foru_nv_m+ count rate forv_nv_m.

(2.62) In the case of a tensor product between a Pauli matrix and the identity matrix,

one can write

σ₀⊗σ_n = (|u_m⟩⟨u_m|+|v_m⟩⟨v_m|)⊗(|u_n⟩⟨u_n| − |v_n⟩⟨v_n|)

= |u_mu_n⟩⟨u_mu_n|+|v_mu_n⟩⟨v_mu_n| − |u_mv_n⟩⟨u_mv_n| − |v_mv_n⟩⟨v_mv_n| (2.63) and

tr{σ₀⊗σ_nρ}=⟨u_mu_n|ρ|u_mu_n⟩+⟨v_mu_n|ρ|v_mu_n⟩ − ⟨u_mv_n|ρ|u_mv_n⟩ − ⟨v_mv_n|ρ|v_mv_n⟩

= count rate foru_nu_m+ count rate forv_nu_m−count rate foru_nv_m−count rate forv_nv_m count rate foru_nu_m+ count rate forv_nu_m+ count rate foru_nv_m+ count rate forv_nv_m (2.64) Also,

σm⊗σ0 = (|um⟩⟨um| − |vm⟩⟨vm|)⊗(|un⟩⟨un|+|vn⟩⟨vn|)

= |u_mu_n⟩⟨u_mu_n| − |v_mu_n⟩⟨v_mu_n|+|u_mv_n⟩⟨u_mv_n| − |v_mv_n⟩⟨v_mv_n| (2.65) and

tr{σ_m⊗σ₀ρ}=⟨u_mu_n|ρ|u_mu_n⟩ − ⟨v_mu_n|ρ|v_mu_n⟩+⟨u_mv_n|ρ|u_mv_n⟩ − ⟨v_mv_n|ρ|v_mv_n⟩

= count rate foru_nu_m−count rate forv_nu_m+ count rate foru_nv_m−count rate forv_nv_m count rate forunum+ count rate forvnum+ count rate forunvm+ count rate forvnvm

(2.66)

In the case a d−dimensional state (qudits), one can expand the density matrix of the state in terms of the generalised Gell-Mann matricesτ_i, that is

ρ= 1 d

d∑²−1 i=0

tr(τ_iρ)τ_i, (2.67)

and in the case of n qudits, one can expand ρ in terms of the products of the generalised Gell-Mann matrices as follows

ρ=∑

m,n

tr(τm⊗τnρ)τm⊗τn

d² , (2.68)

where nowm, nare chosen from the set 0,1,2,· · ·d²−1. Ford= 3, the Gell-Mann matrices are given by

τ₀ =







1 0 0 0 1 0 0 0 1





^{, τ1 =}







0 1 0 1 0 0 0 0 0





^{, τ2 =}







0 −i 0

i 0 0

0 0 0





^,

τ3 =







1 0 0

0 −1 0

0 0 0





^{, τ4 =}







0 0 1 0 0 0 1 0 0





^{, τ5=}







0 0 −i

0 0 0

i 0 0





^{, (2.69)}

τ₆ = 1

√3







1 0 0

0 1 0

0 0 −2





^{, τ7=}







0 0 0 0 0 1 0 1 0





^{, τ8 =}







0 0 0

0 0 −i

0 i 0





^.

The most important drawback of the quantum state tomography is that the reconstructed density matrix often has negative eigenvalues due to experimental imperfections. In the present work, these negative eigenvalues are removed by adding the absolute value of the most negative eigenvalue to the diagonal elements of the reconstructed density matrix and renormalising the results. Furthermore, if the error bars of the resulting eigenvalues, computed from Poisson statistics, still pushed below zero, the mean and standard deviations of these eigenvalues are adjusted so that they remain above zero.

An alternative method of reconstructing the density matrix is the maximum likelihood estimate (MLE) [111]. It is based on the principle that the best estimate of the density matrix is the state that maximises the probability of the measured dataMin the presence of the constraints (for example a positive density matrix).

In other words, the best estimate ρ is the state that maximises the likelihood function [111, 112]

L(ρ)≡p(M|ρ). (2.70)

The MLE also has a major flaw: if due to experimental imperfection the state reconstructed is not a physically valid state (valid density matrix), the MLE will reconstruct that state as a pure state. For example if we consider qubits and that experimental imperfections lead to a state that lies outside of the Bloch sphere, the MLE will try to reconstruct the closest valid state instead and this is a state that lies on the surface of the Bloch sphere [112].

In fact, it was shown in Ref. [112] that the zero eigenvalues produced by the MLE are related to the negative eigenvalues produced by the the quantum state tomography. That is, for a given dataset, if the quantum state tomography pro- duces a density matrix with negative eigenvalues, then the MLE will produce a density matrix representing a pure state. This can be a problem if one is interested in a non-unitary evolution of the entanglement.