Experimental test of environment-assisted invariance

Envariance, or environment-assisted invariance, is a recently identified symmetry for maximally entangled states in quantum theory with important ramifications for quantum measurement, specifically for understanding Born's rule. We benchmark the degree to which nature respects this symmetry by using entangled photon pairs. Our results show quantum states can be $(99.66\pm 0.04)\%$ envariant as measured using the quantum fidelity, and $(99.963\pm 0.005)\%$ as measured using a modified Bhattacharya coefficient, as compared with a perfectly envariant system which would be 100% in either measure. The deviations can be understood by the less-than-maximal entanglement in our photon pairs.

Figure 1

Experimental setup. The entangled photon pairs are created using type-II spontaneous parametric down-conversion. The pump laser is focused on a periodically poled KTP crystal and pairs of entangled photons with anticorrelated polarizations are emitted. The pump is filtered using a band-pass filter, and polarization controls adjust for the alterations due to the coupling fibres. The entangled photon pairs are set so one photon is considered the system, and the other is considered the environment. After the source the unitary transformations are applied. A three-wave-plate combination is required to apply an arbitrary unitary transformation: quarter wave plate (QWP), half wave plate (HWP), QWP. A set of this combination of wave plates is mounted on each translation stage which can slide the wave plates in and out of the path of the incoming photons. The photons are then detected using polarizing beam splitters (PBS) and two wave plates to take projective measurements. The counts are then analyzed using coincidence logic.

Figure 2

Experimental measurement procedure. We investigated the impact of each unitary transformation by performing quantum state tomography at three different stages: directly on the initial state with no unitary transformations (I), on the state with a transformation applied to the system photon (II), and on a state with the same transformation applied to both the system and environment photon (III).

Figure 3

(a) The real and imaginary parts of the reconstructed density matrix of the initial state from the source (stage I of the procedure). It has $0.987$ fidelity [18] with the ideal. (b) The system photon is transformed using wave plates set to implement the rotation of ${90}^{\circ }$ about the $\widehat{x}$ axis, stage II. The resulting density matrix shown has $0.488$ fidelity with the ideal initial state, $0.501$ with the initial reconstructed state, and $0.995$ with the expected state, calculated by transforming the density matrix from (a). (c) The reconstructed density matrix after the same unitary from (b) is applied to both photons, stage III. This state has a $0.987$ fidelity with the ideal, $0.995$ with the reconstructed state from (a), and $0.997$ with the expected state calculated by transforming the state from part (a).

Figure 4

Analysis of the experimental results. (a)–(d) The fidelity analysis results for unitary rotations about $\widehat{x},\widehat{y},\widehat{z}$, and $\widehat{m}$ axes as functions of the rotation angle. The colored data points are the comparison between stage I and stage III (comparing the source state and the state after the unitary has been applied to both qubits). The open circles show a theoretical comparison. (e)–(h) The quantum Bhattacharyya results comparing stage I and stage III in the colored data points for each of the four axes, with the open circles being the theoretical comparison. For plots which include a comparison of stages I and II (applying the unitary to one qubit only) and theoretical comparisons, see the Appendix. The error bar for each graph is the standard deviation of comparisons of source state measurements during the experiment.

Figure 5

Generalized correlations for the singlet state as a function of $n$ using Son's theory [13]. The correlation as a function of $\theta$ is shown for $n=1$ (dashed blue line), $n=2$ (purple line), $n=5$ (dash-dotted brown line), and $n=10$ (dotted green line). The $n=2$ case corresponds to standard quantum mechanics.

Figure 6

Correlation functions versus the rotation angle $\varphi$. The experimental correlations are extracted from our data for the case where the rotation axis and the measurement bases are given by $[\widehat{z},(D,A)],[\widehat{z},(R,L)],[\widehat{y},(D,A)],[\widehat{y},(H,V)],[\widehat{x},(R,L)]$, and $[\widehat{x},(H,V)]$ shown as red squares, blue circles, green up-triangles, yellow down-triangles, black empty squares, and pink diamonds, respectively, as a function of the rotation angle $\varphi$. Error bars are of the size 0.003 which are too small to be seen in the figure. The best fit using Eq. (9) for each correlation is shown as a line whose color matches the corresponding data points. These fits yield estimates for the value of $n$ of {2.04, 2.01, 2.00, 2.01, 2.01, and 2.00}, respectively.

Figure 7

Summary of of the experimentally measured fidelity and the Bhattacharyya coefficient for a wide range of unitaries. (a)–(d) The fidelity analysis results for unitary rotations about $\widehat{x},\widehat{y},\widehat{z}$, and $\widehat{m}$ axes as functions of the rotation angle. The colored data points are the comparison between stage I and stage III, and stage I and stage II. The open circles show the theoretical comparison which takes the state from stage I and applies theoretical unitaries. (e)–(h) The Bhattacharyya results comparing stage I and stage III, and stage I and stage II, in the colored data points for each of the four axes, with the open circles being the theoretical comparison.