[Paper Review] Entanglement in mutually unbiased bases

Marcin Wieśniak, Tomasz Paterek+1 more|arXiv (Cornell University)|Feb 10, 2011

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper demonstrates that a complete set of mutually unbiased bases (MUBs) in a composite quantum system always contains a fixed, nonzero amount of entanglement, independent of the specific basis choice. Using the properties of complex projective 2-designs and linear entropy, it proves that the average entanglement across all MUB states is constant and solely determined by subsystem dimensions, with explicit constructions provided for prime-squared dimensions using a single entangling operation.

ABSTRACT

One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon is most strongly manifested when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independently of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe basic properties of mutually unbiased bases composed only of product states. The constructions are illustrated with explicit examples in low dimensions. We believe that properties of entanglement in mutually unbiased bases might be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.

Motivation & Objective

• To investigate the role of entanglement in complete sets of mutually unbiased bases (MUBs) in composite quantum systems.
• To determine whether the amount of entanglement in MUBs is invariant across different sets, particularly in non-prime-power dimensions.
• To develop an experimentally friendly method for constructing complete MUB sets in dimensions d = p² using a single entangling operation.
• To characterize MUBs composed solely of product states and understand their structural limitations.
• To explore whether entanglement properties in MUBs could help resolve the long-standing question of the existence of complete MUB sets in non-prime-power dimensions.

Proposed method

• The authors use the mathematical framework of complex projective 2-designs to show that the average entanglement over a complete MUB set equals the average entanglement over all pure states in the Hilbert space.
• They employ the linear entropy of the reduced density matrix as a measure of entanglement for individual states in the MUBs.
• For d = p², they construct complete MUB sets by applying a single control-phase gate (P₃) repeatedly to product-state bases, generating either product states or maximally entangled states.
• The construction is based on a recursive application of the phase gate to transform product bases into entangled MUBs, with explicit examples in d=9 (qutrit × qutrit).
• They analyze the entanglement distribution by computing the linear entropy of the reduced density matrix for each state in the MUB set.
• Theoretical proofs rely on symmetry and invariance under unitary transformations, leveraging known results from statistical mechanics on the uniform distribution of entanglement in pure states.

Experimental results

Research questions

• RQ1 Is the total amount of entanglement in a complete set of MUBs independent of the choice of basis, or does it vary across different constructions?
• RQ2 Can a complete set of MUBs in composite dimensions d = d_A × d_B be constructed using only a single entangling operation, and what is the resulting entanglement structure?
• RQ3 What are the structural and entanglement properties of MUBs composed exclusively of product states?
• RQ4 Does the fixed entanglement content in MUBs provide a criterion for the existence or non-existence of complete MUB sets in non-prime-power dimensions?
• RQ5 How does the entanglement distribution among MUB states compare to the average entanglement of a random pure state in the Hilbert space?

Key findings

• The average entanglement over all states in any complete set of MUBs is constant and equal to the average entanglement of a random pure state, due to the 2-design property of MUBs.
• In composite dimensions d = d_A × d_B, every complete set of MUBs contains a fixed, nonzero amount of entanglement that is independent of the specific basis choice.
• For dimensions d = p², the authors construct complete MUB sets using only one control-phase gate (P₃) applied to product-state bases, resulting in sets composed of either product states or maximally entangled states.
• In d=9 (qutrit × qutrit), the construction yields three bases of maximally entangled states and six bases of product states, with explicit matrix representations provided.
• The entanglement content is quantitatively determined by the linear entropy of the reduced density matrix, and for d=9, the average linear entropy across all MUB states is approximately 0.555, consistent with the theoretical average for random pure states.
• MUBs composed solely of product states are shown to be highly constrained, with only a limited number of such bases possible in a complete set.