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Complementarity-based complementarity: the choice of mutually unbiased observables shapes quantum uncertainty relations

Laura Serino, Giovanni Chesi, Benjamin Brecht, Lorenzo Maccone, Chiara Macchiavello, Christine Silberhorn

TL;DR

The paper investigates whether entropic and variance-based uncertainty relations for three mutually unbiased bases depend on the specific choice of observables in a five-dimensional system, a possibility suggested by inequivalent MUB triplets. It combines theory of inequivalent MUBs, numerical minimization, and experimental verification using a time-frequency photonic platform with a multi-output quantum pulse gate to project onto MUB eigenstates. The authors demonstrate two distinct lower bounds for sums of entropies and variances corresponding to two inequivalent triplets, revealing a phenomenon termed complementarity-based complementarity and challenging the assumption of uniform URs across triplets. These findings have implications for quantum cryptography, metrology, and the foundational understanding of quantum complementarity, and point to extensions to higher dimensions.

Abstract

Quantum uncertainty relations impose fundamental limits on the joint knowledge that can be acquired from complementary observables: perfect knowledge of a quantum state in one basis implies maximal indetermination in all other mutually unbiased bases (MUBs). Uncertainty relations derived from joint properties of the MUBs are generally assumed to be uniform, irrespective of the specific observables chosen within a set. In this work, we demonstrate instead that the uncertainty relations can depend on the choice of observables. Through both experimental observation and numerical methods, we show that selecting different sets of three MUBs in a 5-dimensional quantum system results in distinct uncertainty bounds, i.e. in varying degrees of complementarity, in terms of both entropy and variance.

Complementarity-based complementarity: the choice of mutually unbiased observables shapes quantum uncertainty relations

TL;DR

The paper investigates whether entropic and variance-based uncertainty relations for three mutually unbiased bases depend on the specific choice of observables in a five-dimensional system, a possibility suggested by inequivalent MUB triplets. It combines theory of inequivalent MUBs, numerical minimization, and experimental verification using a time-frequency photonic platform with a multi-output quantum pulse gate to project onto MUB eigenstates. The authors demonstrate two distinct lower bounds for sums of entropies and variances corresponding to two inequivalent triplets, revealing a phenomenon termed complementarity-based complementarity and challenging the assumption of uniform URs across triplets. These findings have implications for quantum cryptography, metrology, and the foundational understanding of quantum complementarity, and point to extensions to higher dimensions.

Abstract

Quantum uncertainty relations impose fundamental limits on the joint knowledge that can be acquired from complementary observables: perfect knowledge of a quantum state in one basis implies maximal indetermination in all other mutually unbiased bases (MUBs). Uncertainty relations derived from joint properties of the MUBs are generally assumed to be uniform, irrespective of the specific observables chosen within a set. In this work, we demonstrate instead that the uncertainty relations can depend on the choice of observables. Through both experimental observation and numerical methods, we show that selecting different sets of three MUBs in a 5-dimensional quantum system results in distinct uncertainty bounds, i.e. in varying degrees of complementarity, in terms of both entropy and variance.
Paper Structure (7 sections, 1 theorem, 22 equations, 5 figures, 1 table)

This paper contains 7 sections, 1 theorem, 22 equations, 5 figures, 1 table.

Table of Contents

  1. Uncertainty relations for inequivalent triplets of MUBs
  2. Experimental verification
  3. Results and discussion
  4. Conclusions
  5. MUBs, explicit form
  6. Numerical minimization
  7. Effects of the inequivalence of MUBs on the EURs

Key Result

Lemma 1

Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be the two inequivalent subsets of triplets of MUBs in $d=5$. Be and the two disjoint sets of probability vectors with $X_1 Y_1 Z_1 \in \mathcal{S}_1$ and $X_2 Y_2 Z_2 \in \mathcal{S}_2$, where $p_{T,\psi_0}^{(j)}$ is the Born probability of the superposition of the $j$-th eigenstate of a basis $T \in\{A,B,C,D,E,F\}$ with a five-dimensional pure state $|\p

Figures (5)

  • Figure 1: Alice and Bob are both interested in joint values of observables with MUBs as eigenstates. In dimension 5 there are 6 of them: Bob measures observables $\hat{A},\hat{B},\hat{C}$, Alice measures $\hat{D},\hat{E},\hat{F}$. Alice gets more information: she can divide the $DEF$ phase space in smaller uncertainty blocks than Bob's, even though they are both looking at MUBs.
  • Figure 2: Monte-Carlo evaluation of the sums of entropies in \ref{['euralberto']} and \ref{['eursurprise']} on pure states chosen randomly using the Haar measure. The histograms (250 bins) represent the number of states whose sum of entropies of three maximally complementary observables are equal to the value in the abscissa. (a) Simulation over $10^9$ random states. (b) Detail of the tails of the distributions for a simulation over $10^{10}$ random states. The orange (blue) curves refer to the entropies of $A,B,C$ ($D,E,F$). The black dashed line is at $2\log_25$ and is approached by the left tail of the orange distribution, the blue dashed line approaches the lower bound of \ref{['eursurprise']}. The dotted lines are the (matching) average values of the two distributions at $\sim 5.55$.
  • Figure 3: (a) Histograms of the Monte-Carlo evaluation of the sum of the entropies of two MUBs in dimension $d=5$ (250 bins over $10^9$ Haar distributed random pure states). As expected, all histograms match (here the histograms refer to $A,B$ in orange and $C,D$ in blue). The lower bounds of the left tails (orange-blue vertical dashed lines) approach the Maassen-Uffink bound $\log_25$ (black dashed line). (b) Histograms (250 bins over $10^9$ states) of the sum of variances of three MUBs, evaluated over $A,B,C$ (orange) with lower bound 1.67, and $D,E,F$ (blue) with bound 1.37.
  • Figure 4: Schematic of the experimental setup. The signal (1545) and pump (860) pulses are generated by a Ti:Sapphire ultrafast laser with an optical parametric oscillator (OPO) at a repetition rate of 80. Two waveshapers generate the frequency-bin states in input from the signal pulse and the frequency-bin basis for the measurement from the pump pulse, respectively. In the mQPG waveguide, the signal modes are up-converted into a distinct output frequency based on their overlap with each pump mode. The output beam (552nm) is separated from the unconverted signal and pump beams by a shortpass filter (SP) and then detected by a commercial CCD spectrograph (Andor Shamrock 500i).
  • Figure 5: Sum of the entropies calculated in $d=5$ for the two MUB triplets $CDF$ (left) and $ABE$ (right) for different types of input states: eigenstates of the MUBs in the selected triplet (green squares), eigenstates of the other MUBs (blue diamonds), random states (yellow circles), and low-entropy states that violate the previous assumption of bound \ref{['euralberto']} (red triangles). The filled markers show the experimental data, whereas the hollow markers describe the predicted results based on the characterized imperfect POVMs. The dashed green line and dash-dotted red line indicate the two lower bounds \ref{['euralberto']} and \ref{['eursurprise']}, respectively.

Theorems & Definitions (2)

  • Lemma 1
  • proof