Experimental test of uncertainty relations for general unitary operators

Lei Xiao; Kunkun Wang; Xiang Zhan; Zhihao Bian; Jian Li; Yongsheng Zhang; Peng Xue; Arun Kumar Pati

doi:10.1364/OE.25.017904

1. Introduction

In quantum theory there are two kinds of uncertainty relations: one is the preparation uncertainty relation and the other is the measurement uncertainty relation. The uncertainty relation that is typically taught in text book is the preparation uncertainty which suggests that it is impossible to prepare quantum system for which uncertainties in two non-commuting observables can be arbitrarily small [1]. This is well documented via the Heisenberg-Robertson-Schrödinger uncertainty relation that exists between any two non-commuting observables of a quantum-mechanical system [2–4]. The uncertainty relations are useful for a wide range of applications in quantum technologies including quantum cryptography [5, 6], quantum entanglement [7–9], quantum computation [10, 11], and general physics. Uncertainty relations were tested experimentally with neutronic [12–14] and photonic qubits [15–20]. These experiments test the measurement uncertainty relations. Recently, stronger preparation uncertainty relations for all incompatible observables are proved which go beyond the Heisenberg-Robertson uncertainty relations [21] and have been verified experimentally with linear optics [22]. Thus, the stronger preparation uncertainty displays ‘more’ limitations on the statistical spread in observables in an ensemble of similarly prepared systems.

The uncertainty relations are the hallmarks of quantum physics and have been well investigated. On the other hand, unitary is a fundamental tenet of quantum theory. Every evolution of a closed quantum state can always be represented by the unitary operators and evolution of open system can be represented by acting a unitary on an enlarged system consisting of the quantum system as a subsystem. If one prepares a quantum system and evolves the state under two non-commuting unitary operators, can one reveal the preparation uncertainty without measuring physical observables? That is indeed possible with the uncertainties associated with the unitary operators. Therefore, the uncertainty relations of the unitary operators are naturally of broad interest to investigate. Recently, the uncertainty relation for two arbitrary unitary operators acting on physical states of any Hilbert space are derived by Bagchi and Pati in [23]. The uncertainty relation for two arbitrary unitary operators can reveal the preparation uncertainty and thus the new uncertainty will unify two fundamental features of quantum world, namely, the interference and the uncertainty. In this paper, we experimentally demonstrate that this uncertainty relation, relating on that sum of variances, is valid in a state-dependent manner and the lower bound is guaranteed to be nontrivial for two observables being incompatible on the state of the system being measured. The behaviour we find agrees with the predictions of quantum theory and obeys the new uncertainty relation.

2. Theoretical framework

Consider a quantum system prepared in the state |ψ〉 ∈ ℋ with ℋ being a finite or infinite dimensional Hilbert space. Let us consider two unitary operators U and V that act on the quantum state. The uncertainties associated with U and V in the state |ψ〉 are defined as

\begin{array}{l} Δ U^{2} = 1 - {| 〈 ψ | U | ψ 〉 |}^{2} = 1 - Tr (| ψ_{U} 〉 〈 ψ_{U} | ψ 〉 〈 ψ |) \\ Δ V^{2} = 1 - {| 〈 ψ | V | ψ 〉 |}^{2} = 1 - Tr (| ψ_{V} 〉 〈 ψ_{V} | ψ 〉 〈 ψ |) \end{array}

where |ψ_U〉 = U |ψ〉, |ψ_V〉 = V |ψ〉 with 0 ≤ ΔU² ≤ 1 and 0 ≤ ΔV² ≤ 1. The uncertainty in any unitary operator has simple physical meaning: it is the distance between the original and the unitarily evolved quantum state. The uncertainties in two unitary operators U and V are lower bounded as [23]

Δ U^{2} + Δ V^{2} \geq 1 + {| 〈 ψ_{U} | ψ_{V} 〉 |}^{2} - 2 cos ϕ | Δ^{(3)} |

where

{| 〈 ψ_{U} | ψ_{V} 〉 |}^{2} = 〈 ψ | V^{†} U | ψ 〉 〈 ψ | U^{†} V | ψ 〉 = Tr (V^{†} U | ψ 〉 〈 ψ | U^{†} V | ψ 〉 〈 ψ |)

Δ^{(3)} = 〈 ψ | ψ_{U} 〉 〈 ψ_{U} | ψ_{V} 〉 〈 ψ_{V} | ψ 〉 = Tr (U | ψ 〉 〈 ψ | U^{†} V | ψ 〉 〈 ψ | V^{†} | ψ 〉 〈 ψ |)

is the 3-point Bargmann invariant and ϕ = ArgΔ⁽³⁾.

The physical meaning of the uncertainty relation for two general unitary operators is that one cannot prepare a quantum state with two distinguishable metrics and the sum of them is arbitrarily small. Alternately, one can interpret this as sum of visibility in the interference setup due to two non-commuting unitary operators is non-trivially upper bounded. For certain class of states, uncertainty relation for two unitary operators can reduce to the uncertainty relations for two Hermitian operators. The advantage of the preparation uncertainty based on the unitary operators is that we do not need to measure two incompatible observables on identically prepared quantum systems. We simply prepare and let the system to evolve under two different unitary operators to test the intrinsic preparation uncertainty relation in Eq. (2).

Without loss of generality, we show an example by choosing two unitary operators [23]

U = (\begin{matrix} 1 & 0 & 0 \\ 0 & e^{- i \frac{2}{3} π} & 0 \\ 0 & 0 & e^{- i \frac{4}{3} π} \end{matrix}), V = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix})

and a family of qutrit states being measured |ψ_θ〉 = cos θ |0〉 − sin θ |2〉 with θ ∈ [0, π].

The unitary operators U and V belong to a particularly important class of unitary operators so called the clock and shift matrices [23], which are the non-Hermitian generalizations of the Pauli matrices to higher dimensions. They are the cornerstones in the quantum mechanics of the finite dimensional Hilbert space. The bases of the two unitary operators are the mutually unbiased bases with respect to each other, and they are related to each other via the discrete Fourier transform. It is important to study the uncertainty relation with respect to the mutually unbiased bases since the mutually unbiased bases have found wide applications in quantum information [24–26]. This explains why we choose the unitary operators U and V shown in Eq. (5) as an example to study the uncertainty relation in Eq. (2).

Now we focus on the feasibility of demonstration of the new uncertainty relation. The terms of left-hand side (LHS) of the inequality, i.e., the variances ΔU² and ΔV² can be calculated by the measured expectation values of U and V. The second term of the right-hand side (RHS) of the inequality can be calculated by the expectation values of the operator V^†U. The third term of the RHS of the inequality can be obtained by tomography of the initial state |ψ_θ〉, the evolved states |ψ_U〉 and |ψ_V〉, respectively. Thus we can demonstrate the uncertainty relation for two unitary operators.

3. Experimental implementation

We report the experimental test of the uncertainty relation for two three-level unitary operators [23] with photonic qutrits. A photonic qutrit is represented by three modes of the single photons. The basis states |0〉, |1〉, and |2〉 are encoded by the horizontal polarization of the photon which is in the lower spatial mode, the horizontal polarization of the photon in the upper spatial mode, and the vertical polarization of the photon in the upper spatial mode, respectively. Figure 1 shows the experimental step which involves three stages of demonstration: the specific state preparation, state evolution, measurement on the system of interest (projection measurement or state tomography).

Fig. 1 Experimental setup. The herald single photons are produced via type-I spontaneous parametric down-conversion in a β-barium-borate nonlinear crystal and are injected into the optical network. The first polarizing beam splitter, half-wave plate (H₁) and beam displacer (BD₁) are used to prepare a qutrit state |ψ_θ〉. Two sandwich-type sets of wave plates (Q₁-H₂-Q₂ and Q₁-H₃-Q₂) are used to realize the evolution operator U. The halfwave plates (H₄–H₇) and two beam displacers (BD₂ and BD₃) are used to realize the evolution operator V (or V^†). The projection measurement {P₁, P₂} can be realized by four half-wave plates (H₈–H₁₁) and two beam displacers (BD₄–BD₅). Tomography of the qutrit state can be realized by half-wave plates (H₁₂–H₁₄), quarter-wave plate (Q₃), BD₆ and a polarizing beam splitter.

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In the state preparation stage, polarization-degenerated photon pairs at a wavelength of 801.6nm are produced via a type-I spontaneous parametric down-conversion process [27–32]. The trigger-signal photon pair is registered by a coincidence count at two single-photon avalanche photodiodes. Total coincidence counts are about 10⁴ over a collection time of 5s. The signal photon as a single photon source is heralded in the measurement setup with the detection of trigger photon. The specific state is prepared by letting heralded single photons to pass through a polarizing beam splitter and a half-wave plate (H₁) with the certain setting angle and then to be split by a birefringent calcite beam displacer into two parallel spatial modes—upper and lower modes due to their polarizations [33, 34]. After passing through a beam displacer, the vertically polarized photons are directly transmitted and horizontally polarized photons undergo a 3mm lateral displacement into a neighboring mode. Therefore the photons are prepared in the state |ψ_θ〉. We choose eleven values of θ ∈ [0, π], i.e., total eleven states for testing the uncertainty relation proposed in [23]. The matrix form of the operation of the half-wave plate is $(\begin{matrix} cos 2 ϑ & sin 2 ϑ \\ sin 2 ϑ & - cos 2 ϑ \end{matrix})$ and the setting angle of H₁ used for generating the state |ψ_θ〉 is −θ/2. The matrix form of the operation of the quarter-wave plate is $(\begin{matrix} {cos}^{2} φ + i {sin}^{2} φ & (1 - i) sin φ cos φ \\ (1 - i) sin φ cos φ & {sin}^{2} φ + i {cos}^{2} φ \end{matrix})$ .

For the stage of the state evolution, firstly we consider the LHS of the inequality, i.e., the sum of the uncertainties ΔU² + ΔV². It can be calculated by the expectation values 〈U〉 and 〈V〉. The expectation values of U and V can be measured by applying the projection measurement {P₁, P₂} on the evolved states |ψ_U〉 and |ψ_V〉, respectively, where P₁ = |ψ_θ〉〈ψ_θ | and P₂ = 𝕀 − P₁.

The evolution U applied on the initial state |ψ_θ〉 is a phase operation which adds a phase factor e^−i4π/3 on the state |2〉, e^−i2π/3 on |1〉 and keeps |0〉 unchanged. In the stage of state evolution, the operator U can be realized by two sandwich-type set of quarter-wave plates and half-wave plate with certain setting angles, i.e., Q₁-H₂-Q₂ and Q₁-H₃-Q₂ inserting in different spatial modes, respectively. A sandwich-type set of wave plates involving two quarter-wave plates and a half-wave plate can realize any arbitrary two-polarization-mode transformation on the photons propagating in the same spatial mode.

The other evolution operation V is a flip operation and can be realized by two beam displacers (BD₂–BD₃) and four half-wave plates (H₄–H₇). The beam displacers are used to split the photons with different polarizations into different spatial modes and to combine the photons with certain two of the polarization modes in the same spatial mode. Then two-polarization-mode transformations can then be implemented using half-wave plates (H₅ and H₆) acting on the two polarization modes propagating in the same spatial mode. The other half-wave plates (H₄ and H₇) are used to compensate the optical delay.

In the stage of measurement, the projection measurement can be realized by two beam displacers (BD₄–BD₅) and four half-wave plates (H₈–H₁₁), where H₈–H₁₀ are used to realize bit-flip operation on the polarization modes of photons, and H₁₁ combining with BD₅ maps the state |ψ_θ〉 to the certain spatial mode accomplishing the projection measurements with single-photon avalanche photodiodes. The setting angles of the wave plates for evolution and projection measurements are shown in Table 1.

Table 1. The setting angles of the wave plates for the unitary evolutions and projection measurements. Here “−” denotes the corresponding wave plate is removed from the optical circuit.

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The outcomes represented by the click of the single-photon avalanche photodiode D₂ give the measured probabilities $p_{1}^{U}$ and $p_{1}^{V}$ , which equal to the probabilities Tr(U|ψ_θ〉〈ψ_θ | U^† |ψ_θ〉〈ψ_θ|) and Tr(V |ψ_θ〉〈ψ_θ | V^† |ψ_θ〉〈ψ_θ|). To measure the expectation values 〈U〉 or 〈V〉, we remove the setup for realizing V or U, respectively.

Now we consider the RHS of the inequality. There are three terms each of which can be obtained from the experimental data. The first term is a constance. The second term, i.e., the expectation value of the operator V^†U can be obtained with a similar way. The photons pass through the setup for realizing V^†U and are projected to the initial state |ψ_θ〉. The measured | 〈ψ_U | ψ_V〉|² is equivalent to the probability of the photons being measured in |ψ_θ〉. The probability is obtained by normalizing photon counts in the single-photon avalanche photodiode D₂ to the total photon counts.

The last term of the RHS of the inequality depends on the Bargmann invariant Δ⁽³⁾ which cannot be measured directly. However, one can obtain it via the state tomography and reconstruction of the initial state |ψ_θ〉 and the evolved states |ψ_U〉 and |ψ_V〉. To realize the qutrit state tomography, we project the states into nine basis states {|0〉, |1〉, |2〉, $(i | 0 〉 + | 1 〉) / \sqrt{2}$ , $(i | 0 〉 + | 2 〉) / \sqrt{2}$ , ( $(| 1 〉 + i | 2 〉) / \sqrt{2}$ , $(| 0 〉 + | 1 〉) / \sqrt{2}$ , $(| 0 〉 + | 2 〉) / \sqrt{2}$ , $(| 1 〉 + | 2 〉) / \sqrt{2}$ }. To realize the state tomography, we replace the block of “projection measurement” by the block of “state tomography” in Fig. 1. Firstly, one projects the states into the bases {|1〉, |2〉, $(| 1 〉 + i | 2 〉) / \sqrt{2}$ , $(| 1 〉 + | 2 〉) / \sqrt{2}$ }. The photons in the upper mode directly pass through the wave plates (Q₃ and H₁₄) to apply the rotation on the qutrit state and a polarizing beam splitter to map the state into the certain basis states. The photon count in the single-photon avalanche photodiode D₃ is proportional to the probabilities of the projection measurements. To project the photonic states into the rest six bases, the photons pass through the wave plates (H₁₂–H₁₄, Q₃) and a beam displacer (BD₆) to apply the rotation on the qutirt state and a polarizing beam splitter to map the state into the certain basis states. The setting angles of the wave plates for state tomography are shown in Table 2. The photon count in the single-photon avalanche photodiode D₄ is proportional to the probabilities of the projection measurements.

Table 2. The setting angles of the wave plates for state tomography.

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In Fig. 2, we show the experimental results of the verification of the uncertainty relation for general unitary operators in (2). The solid black line corresponds to the theoretical prediction of the LHS of the inequality, i.e., ΔU² + ΔV². The black triangles represent the sum of the measured uncertainties of ΔU² and ΔV² with the eleven states |ψ_θ〉. The red dashed line corresponds to the theoretical prediction of the RHS of the inequality. The red squares represent the experimental results of the RHS of inequality with the eleven states |ψ_θ〉. For θ = 0, π/2, π, the inequality becomes an equality, which shows the new uncertainty inequality (2) is tight. The experimental results agree remarkably well with the theoretical prediction. Thus we provide the first experimental demonstration of the new uncertainty relation for general unitary operators.

Fig. 2 Experimental results. The solid black line corresponds to the theoretical prediction of the LHS of the inequality, i.e., ΔU² + ΔV². The black triangles represent the sum of the measured uncertainties of ΔU² and ΔV² with the eleven states |ψ_θ〉. The red dashed line corresponds to the theoretical prediction of the RHS of the inequality. The red squares represent the experimental results of the RHS of inequality with the eleven states |ψ_θ〉. Error bar indicates the statistical uncertainty which is obtained based on assuming Poissonian statistics.

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4. Conclusion

Uncertainty relation imposes quantitative limitation on the mutual exclusiveness of sharp preparations for two incompatible observables. Since its inception, the preparation uncertainty relation has been usually tested using Hermitian operators in quantum theory. However, it is possible to test the limitation on the joint sharp preparation of quantum system using the uncertainty relation for unitary operators. We have demonstrated a method for experimentally testing the uncertainty relation for general unitary operators. This has allowed us to test the uncertainty inequality for general unitary operators. Our demonstration provides the first evidence for the validity of the uncertainty relation for two arbitrary unitary operators. We hope that the experimental confirmation of a fundamental limitation of preparation uncertainty using unitary operators will provide deep insights into the nature of intrinsic uncertainty that prevails at fundamental state preparation level.

Funding

Natural Science Foundation of China (NSFC) (11474049 and 11674056); Natural Science Foundation of Jiangsu Province (BK20160024); Open Fund from State Key Laboratory of Precision Spectroscopy, East China Normal University.

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	U	V	V ^† U
Q₁	45.00°	−	45.00°
Q₂	45.00°	−	45.00°
H₂	−165.00°	−	−165.00°
H₃	45.00°	−	45.00°
H₄	−	0.00°	45.00°
H₅	−	−45.00°	45.00°
H₆	−	−45.00°	45.00°
H₇	−	0.00°	0.00°
H₈	45.00°	45.00°	0.00°
H₉	45.00°	45.00°	45.00°
H₁₀	45.00°	45.00°	45.00°
H₁₁	45.00° + θ/2	45.00° + θ/2	45.00° + θ/2

	Q₃	H₁₂	H₁₃	H₁₄
\|0〉	0.00°	0.00°	45.00°	45.00°
\|1〉	0.00°	−	−	0.00°
\|2〉	0.00°	−	−	45.00°
$(i \| 0 〉 + \| 1 〉) / \sqrt{2}$	0.00°	0.00°	45.00°	22.50°
$(i \| 0 〉 + \| 2 〉) / \sqrt{2}$	0.00°	45.00°	45.00°	22.50°
$(\| 1 〉 + i \| 2 〉) / \sqrt{2}$	45.00°	0.00°	45.00°	22.50°
$(\| 0 〉 + \| 1 〉) / \sqrt{2}$	0.00°	−	−	22.50°
$(\| 0 〉 + \| 2 〉) / \sqrt{2}$	45.00°	45.00°	45.00°	22.50°
$(\| 1 〉 + \| 2 〉) / \sqrt{2}$	45.00°	−	−	22.50°

	U	V	V ^† U
Q₁	45.00°	−	45.00°
Q₂	45.00°	−	45.00°
H₂	−165.00°	−	−165.00°
H₃	45.00°	−	45.00°
H₄	−	0.00°	45.00°
H₅	−	−45.00°	45.00°
H₆	−	−45.00°	45.00°
H₇	−	0.00°	0.00°
H₈	45.00°	45.00°	0.00°
H₉	45.00°	45.00°	45.00°
H₁₀	45.00°	45.00°	45.00°
H₁₁	45.00° + θ/2	45.00° + θ/2	45.00° + θ/2

	Q₃	H₁₂	H₁₃	H₁₄
\|0〉	0.00°	0.00°	45.00°	45.00°
\|1〉	0.00°	−	−	0.00°
\|2〉	0.00°	−	−	45.00°
$(i \| 0 〉 + \| 1 〉) / \sqrt{2}$	0.00°	0.00°	45.00°	22.50°
$(i \| 0 〉 + \| 2 〉) / \sqrt{2}$	0.00°	45.00°	45.00°	22.50°
$(\| 1 〉 + i \| 2 〉) / \sqrt{2}$	45.00°	0.00°	45.00°	22.50°
$(\| 0 〉 + \| 1 〉) / \sqrt{2}$	0.00°	−	−	22.50°
$(\| 0 〉 + \| 2 〉) / \sqrt{2}$	45.00°	45.00°	45.00°	22.50°
$(\| 1 〉 + \| 2 〉) / \sqrt{2}$	45.00°	−	−	22.50°

Experimental test of uncertainty relations for general unitary operators

Abstract

1. Introduction

2. Theoretical framework

3. Experimental implementation

4. Conclusion

Funding

References and links

Cited By

Figures (2)

Tables (2)

Equations (5)

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