Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0505213 · Physical Review A

Entanglement and the power of one qubit

Animesh Datta, Steven T. Flammia, Carlton M. Caves

Published Oct 18, 2005• 372 citations
arXivHTMLPDF

Abstract

The ``power of one qubit'' refers to a computational model that has access to only one pure bit of quantum information, along with $n$ qubits in the totally mixed state. This model, though not as powerful as a pure-state quantum computer, is capable of performing some computational tasks exponentially faster than any known classical algorithm. One such task is to estimate with fixed accuracy the normalized trace of a unitary operator that can be implemented efficiently in a quantum circuit. We show that circuits of this type generally lead to entangled states, and we investigate the amount of entanglement possible in such circuits, as measured by the multiplicative negativity. We show that the multiplicative negativity is bounded by a constant, independent of $n$, for all bipartite divisions of the $n+1$ qubits, and so becomes, when $n$ is large, a vanishingly small fraction of the maximum possible multiplicative negativity for roughly equal divisions. This suggests that the global nature of entanglement is a more important resource for quantum computation than the magnitude of the entanglement.

DOI
10.1103/physreva.72.042316

Related papers

Studies on the Role of Entanglement in Mixed-state Quantum Computation

Animesh Datta

In this thesis, I look at the role of quantum entanglement in mixed-state quantum computation. The model we consider is the DQC1 or `power of one qubit' model. I show that there is minimal bipartite entanglement in a typical instance of the DQC1 circuit and even put an upper bound on the possible amount of entanglement. This limited amount of entanglement, however, does not imply that the system is classically simulatable. A matrix product state (MPS) algorithm for a typical instance of the DQC1 system requires exponential classical resources. This exposes a gap between the amount of entanglement and the amount of purely nonclassical correlations in a quantum system. This gap, I suggest, can be filled by quantum discord. I calculate it in a typical instance of the DQC1 circuit and find that the amount of discord is a constant fraction of the maximum possible discord for a system of that size. This allows an interpretation of quantum discord as the resource that drives mixed-state quantum computation. I also study quantum discord as a quantity of independent interest. Its role in the phenomenon of entanglement distribution is studied through an easily comprehensible example. This thesis also contains discussions on the relation between the complexity classes P, BQP and DQC1. Additional material is presented on the connections between the DQC1 model, Jones polynomials and statistical mechanics. The thesis concludes with a discussion of a few open problems related to the DQC1 model, the quantum discord and their scope in quantum information science.

arXiv:0807.4490•Jul 28, 2008•20 citations

Entangling power in deterministic quantum computation with one qubit

Chang‐shui Yu, X. X. Yi, He‐Shan Song +1 more

Deterministic quantum computing with one qubit (DQC1) is a mixed-state quantum computation algorithm that evaluates the normalized trace of a unitary matrix and is more powerful than the classical counterpart. We find that the normalized trace of the unitary matrix can be directly described by the entangling power of the quantum circuit of DQC1, so the nontrivial DQC1 is always accompanied by nonvanishing entangling power. In addition, it is shown that the entangling power also determines the intrinsic complexity of this quantum computation algorithm, i.e., larger entangling power corresponds to higher complexity. In addition, it is also shown that nonvanishing entangling power always exists in other similar tasks of DQC1.

arXiv:1307.1196•Physical Review A•Feb 19, 2013•8 citations

Power of one qumode for quantum computation

Nana Liu, Jayne E. Thompson, Christian Weedbrook +4 more

Although quantum computers are capable of solving problems like factoring exponentially faster than the best-known classical algorithms, determining the resources responsible for their computational power remains unclear. An important class of problems where quantum computers possess an advantage is phase estimation, which includes applications like factoring. We introduce a computational model based on a single squeezed state resource that can perform phase estimation, which we call the power of one qumode. This model is inspired by an interesting computational model known as deterministic quantum computing with one quantum bit (DQC1). Using the power of one qumode, we identify that the amount of squeezing is sufficient to quantify the resource requirements of different computational problems based on phase estimation. In particular, we can use the amount of squeezing to quantitatively relate the resource requirements of DQC1 and factoring. Furthermore, we can connect the squeezing to other known resources like precision, energy, qudit dimensionality, and qubit number. We show the circumstances under which they can likewise be considered good resources.

arXiv:1510.04758•Physical Review A•May 3, 2016•36 citations

Power of One Bit of Quantum Information

Emanuel H. Knill, Raymond Laflamme

In standard quantum computation, the initial state is pure and the answer is determined by making a measurement of some of the bits in the computational basis. What can be accomplished if the initial state is a highly mixed state and the answer is determined by measuring the expectation of ${\ensuremath{\sigma}}_{z}$ on the first bit with bounded sensitivity? This is the situation in high temperature ensemble quantum computation. We show that in this model it is possible to perform interesting physics simulations that have no known efficient classical algorithms, even though the model is less powerful than standard quantum computation in the presence of oracles.

arXiv:quant-ph/9802037•Physical Review Letters•Dec 21, 1998•886 citations

Universal Quantum Computation with Little Entanglement

Maarten Van den Nest

We show that universal quantum computation can be achieved in the standard pure-state circuit model while the entanglement entropy of every bipartition is small in each step of the computation. The entanglement entropy required for large-scale quantum computation even tends to zero. Moreover we show that the same conclusion applies to many entanglement measures commonly used in the literature. This includes e.g., the geometric measure, localizable entanglement, multipartite concurrence, squashed entanglement, witness-based measures, and more generally any entanglement measure which is continuous in a certain natural sense. These results demonstrate that many entanglement measures are unsuitable tools to assess the power of quantum computers.

arXiv:1204.3107•Physical Review Letters•Feb 7, 2013•117 citations

Entanglement-changing power of two-qubit unitary operations

Ming-Yong Ye, Dong Sun, Yong-Sheng Zhang +1 more

Entanglement-changing power of an arbitrary two-qubit operation, including increasing and decreasing power, is investigated in this paper. We consider the maximal entanglement ${C}_{\mathrm{max}}$ and the minimal entanglement ${C}_{\mathrm{min}}$ of the states obtained by a given two-qubit unitary operation ${U}_{d}$ acting on arbitrary pure states with fixed entanglement ${C}_{0}$. We give the condition that the maximal entanglement ${C}_{\mathrm{max}}$ of the obtained states can be $1$ and the minimal entanglement ${C}_{\mathrm{min}}$ can be $0$. When the maximal entanglement ${C}_{\mathrm{max}}$ cannot be $1$, we give the maximal value it can reach. When the minimal entanglement ${C}_{\mathrm{min}}$ cannot be $0$, we give the minimal value it can reach. We think ${C}_{\mathrm{max}}$ and ${C}_{\mathrm{min}}$ represent the entanglement-changing power of two-qubit unitary operations.

arXiv:quant-ph/0403163•Physical Review A•Aug 30, 2004•21 citations
The ``power of one qubit'' refers to a computational model that has access to only one pure bit of quantum information, along with nnn qubits in the totally mixed state. This model, though not as powerful as a pure-state quantum computer, is capable of performing some computational tasks exponentially faster than any known classical algorithm. One such task is to estimate with fixed accuracy the normalized trace of a unitary operator that can be implemented efficiently in a quantum circuit. We show that circuits of this type generally lead to entangled states, and we investigate the amount of entanglement possible in such circuits, as measured by the multiplicative negativity. We show that the multiplicative negativity is bounded by a constant, independent of nnn, for all bipartite divisions of the n+1n+1n+1 qubits, and so becomes, when nnn is large, a vanishingly small fraction of the maximum possible multiplicative negativity for roughly equal divisions. This suggests that the global nature of entanglement is a more important resource for quantum computation than the magnitude of the entanglement.
In this thesis, I look at the role of quantum entanglement in mixed-state quantum computation. The model we consider is the DQC1 or `power of one qubit' model. I show that there is minimal bipartite entanglement in a typical instance of the DQC1 circuit and even put an upper bound on the possible amount of entanglement. This limited amount of entanglement, however, does not imply that the system is classically simulatable. A matrix product state (MPS) algorithm for a typical instance of the DQC1 system requires exponential classical resources. This exposes a gap between the amount of entanglement and the amount of purely nonclassical correlations in a quantum system. This gap, I suggest, can be filled by quantum discord. I calculate it in a typical instance of the DQC1 circuit and find that the amount of discord is a constant fraction of the maximum possible discord for a system of that size. This allows an interpretation of quantum discord as the resource that drives mixed-state quantum computation. I also study quantum discord as a quantity of independent interest. Its role in the phenomenon of entanglement distribution is studied through an easily comprehensible example. This thesis also contains discussions on the relation between the complexity classes P, BQP and DQC1. Additional material is presented on the connections between the DQC1 model, Jones polynomials and statistical mechanics. The thesis concludes with a discussion of a few open problems related to the DQC1 model, the quantum discord and their scope in quantum information science.
Deterministic quantum computing with one qubit (DQC1) is a mixed-state quantum computation algorithm that evaluates the normalized trace of a unitary matrix and is more powerful than the classical counterpart. We find that the normalized trace of the unitary matrix can be directly described by the entangling power of the quantum circuit of DQC1, so the nontrivial DQC1 is always accompanied by nonvanishing entangling power. In addition, it is shown that the entangling power also determines the intrinsic complexity of this quantum computation algorithm, i.e., larger entangling power corresponds to higher complexity. In addition, it is also shown that nonvanishing entangling power always exists in other similar tasks of DQC1.
Although quantum computers are capable of solving problems like factoring exponentially faster than the best-known classical algorithms, determining the resources responsible for their computational power remains unclear. An important class of problems where quantum computers possess an advantage is phase estimation, which includes applications like factoring. We introduce a computational model based on a single squeezed state resource that can perform phase estimation, which we call the power of one qumode. This model is inspired by an interesting computational model known as deterministic quantum computing with one quantum bit (DQC1). Using the power of one qumode, we identify that the amount of squeezing is sufficient to quantify the resource requirements of different computational problems based on phase estimation. In particular, we can use the amount of squeezing to quantitatively relate the resource requirements of DQC1 and factoring. Furthermore, we can connect the squeezing to other known resources like precision, energy, qudit dimensionality, and qubit number. We show the circumstances under which they can likewise be considered good resources.
In standard quantum computation, the initial state is pure and the answer is determined by making a measurement of some of the bits in the computational basis. What can be accomplished if the initial state is a highly mixed state and the answer is determined by measuring the expectation of ${\ensuremath{\sigma}}_{z}$ on the first bit with bounded sensitivity? This is the situation in high temperature ensemble quantum computation. We show that in this model it is possible to perform interesting physics simulations that have no known efficient classical algorithms, even though the model is less powerful than standard quantum computation in the presence of oracles.
We show that universal quantum computation can be achieved in the standard pure-state circuit model while the entanglement entropy of every bipartition is small in each step of the computation. The entanglement entropy required for large-scale quantum computation even tends to zero. Moreover we show that the same conclusion applies to many entanglement measures commonly used in the literature. This includes e.g., the geometric measure, localizable entanglement, multipartite concurrence, squashed entanglement, witness-based measures, and more generally any entanglement measure which is continuous in a certain natural sense. These results demonstrate that many entanglement measures are unsuitable tools to assess the power of quantum computers.
Entanglement-changing power of an arbitrary two-qubit operation, including increasing and decreasing power, is investigated in this paper. We consider the maximal entanglement Cmax{C}_{\mathrm{max}}Cmax​ and the minimal entanglement Cmin{C}_{\mathrm{min}}Cmin​ of the states obtained by a given two-qubit unitary operation Ud{U}_{d}Ud​ acting on arbitrary pure states with fixed entanglement C0{C}_{0}C0​. We give the condition that the maximal entanglement Cmax{C}_{\mathrm{max}}Cmax​ of the obtained states can be 111 and the minimal entanglement Cmin{C}_{\mathrm{min}}Cmin​ can be 000. When the maximal entanglement Cmax{C}_{\mathrm{max}}Cmax​ cannot be 111, we give the maximal value it can reach. When the minimal entanglement Cmin{C}_{\mathrm{min}}Cmin​ cannot be 000, we give the minimal value it can reach. We think Cmax{C}_{\mathrm{max}}Cmax​ and Cmin{C}_{\mathrm{min}}Cmin​ represent the entanglement-changing power of two-qubit unitary operations.