For the case where the experimental detection scheme does not distinguish which photon is which, and we are using non-photon number resolving detectors, the number of N-photon output cor- relations is equal to the number of unordered samples without repetition of N objects chosen from Mout objects,
Mout! N!(Mout −N)!
. (A.19)
Furthermore, when the photons are not distinguished by the de- tection scheme the full density matrix can never be determined, since no information about ‘which photon is which’ is measured. However, a subspace of the full density matrix can still be re- covered. Here we show the mathematical form of this reduced density matrix, and derive the number of free parameters in it. This allows us to compare the number of free parameters with the number of distinct output correlations that can be measured, thus determining if tomography is possible for a given number of output modes, Mout.
The independent elements in the density matrices of a single photon can be extracted mathematically by projection onto a set of matrices, |si hs0|, according to,
ρRe(s|s0) = Tr ˆ ρ |si hs 0|+|s0i hs| 2 = TrρˆOˆ(+s|s0) (A.20) ρIm(s|s0) = Tr ˆ ρ |si hs 0| − |s0i hs| 2 = TrρˆOˆ−(s|s0) . (A.21) where |si ∈ (0,K−1)are the labels for theK different orthogonal
§A.3 Scaling of device complexity with photon number 139
state in the system. Here ρRe(s|s0) is the real part of the density
matrix element (s|s0) and ρIm(s|s0) is the imaginary part. Since ˆρ is
Hermitian, by definitionρRe(s|s0) = ρRe(s0|s) and ρIm(s|s0) = ρIm(s0|s), thus the
number of independent parameters in the density matrix is(K+ 1)K/2 real parameters and K(K − 1)/2 imaginary parameters, for a total of K2 independent parameters.
Now for an N-photon density matrix, the matrix elements can be determined by projection onto operators of the form,
ˆ O±(s p1,sp2...spN|s0p1,s0p2...s0pN) = |sp1i hs0p1| ⊗ |sp2i hs0p2| ⊗...|spNi hs0pN| ±H.C. /2 (A.22) according to, ρ±(s 1,s2...sN|s01,s20...s0N) = Tr ˆ ρOˆ(±s p1,sp2...spN|s0p1,s0p2...s0pN) . (A.23) Here sp1 denotes the state of the photon labeled p1, and so on. Now there will be Min2N independent parameters in the density matrix. In the case where the photons are not distinguished by the detectors, then some elements of the N-photon density ma- trix will be indistinguishable from others. For example the two photon density matrix element, corresponding to projection onto
|0ip1h0|p1 ⊗ |0ip2h1|p2 + H.C will be impossible to distinguish from the element corresponding to projection onto |0ip1h1|p1 ⊗
|0ip2h0|p2 + H.C, since if the photon labels are removed, and the ordering of the tensor product ignored, both operators are the same. In general if permuting the labels of the photons leaves a density matrix element unchanged (aside from the ordering/la- beling of the tensor product elements) then those two elements cannot be distinguished by any measurement that doesn’t dis- tinguish the photons. Because of this, for the case where the measurement apparatus does not distinguish photons, the num- ber of free parameters of the density matrix is reduced according to the number of degenerate elements.
The number of free parameters in this reduced density matrix will be equal to the number of ways of choosing a tensor product
Figure A.2: Independent elements in a two-photon two-mode density matrix, where the photons are not distinguished by the detection scheme.a) list of the independent elements in the form of operators as in Eq. (A.24). This is for the specific case whereN=2 andMin =2. b) The structure of the typical
4-by-4 representation of the density matrix is shown in b) and c) (real and imaginary parts respectively), where the labels A, B, C... indicate which elements are degenerate, corresponding to a single element in the list from a). Density matrix elements with a red cross are always equal zero.
of N single photon density matrices, where the ordering of the single photon density matrices is not important. i.e each unique density matrix element can be written as
ρ±N 1,N2...NM2 in = 1 2tr ˆ ρ (|0i h0|)N1 ⊗ (|0i h1|)N2 ⊗...⊗(|0i hMin|)NMin ⊗... ⊗(|Mini hMin|) N Min2 ± H.C. (A.24) Here, ρ±N 1,N2...NM2 in
is a element of the density matrix, where Nn
specifies the number of photons with the single photon density matrix |sni hs0n|. Thus for a state with N photons N1 + N2 +...+
NM2
in = N. The density matrix elements are now specified only
in terms of how many photons are in each single photon state, the photon labeling / ordering is no longer a factor. An example of the structure of the density matrix described in Eq. (A.24) is shown in Fig. A.2, for the specific case where N = 2 and
Min = 2.
Note that in Eq. (A.24), since there are Min different modes the
photons could be in, then the number of distinct single photon density matrices is Min2 . Thus the number of unique N photon density matrix elements is equal to the number of ways of dis- tributing the N indistinguishable photons across these M2in dif- ferent single photon density matrices. In other words, N indis- tinguishable objects must be distributed into M2in distinguishable bins. Using standard combinatorics, the number of free parame-
§A.3 Scaling of device complexity with photon number 141
ters in the N-photon density matrix is therefore,
M2in+ N−1 Min2 −1 = (M 2 in+ N−1)! N!(M2in−1)! . (A.25) For tomography to be possible the number of distinct correla- tion measurements must be equal to or greater than the number of free parameters in the density matrix. Therefore, recalling the number of different correlations measurements given in Eq. (A.19), we require,
Mout!
N!(Mout− N)! ≥
(M2in+ N−1)!
N!(M2in−1)! . (A.26) Thus the minimum number of output modes required scales lin- early with the photon number as
Mout = M2in+ N−1, (A.27)
as this number of output ports corresponds to exact equality in Eq. (A.26).
For our experimentally fabricated device with two input ports (Min = 2) we get Mout = N + 3 according to Eq. (A.27). This
agrees perfectly with the numerical result calculated for N ≤ 25 that was shown in Ch. 6 Fig. 6.6(b) in the main text.