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/0305104 · Physical Review A

Estimation of unitary quantum operations

Manuel A. Ballester

Published Feb 5, 2004• 71 citations
arXivHTMLPDF

Abstract

The problem of optimally estimating an unknown unitary quantum operation with the aid of entanglement is addressed. The idea is to prepare an entangled pair, apply the unknown unitary to one of the two parts, and then measure the joint output state. This measurement could be an entangled one or it could be separable (e.g., measurements which can be implemented with local operations and classical communication or LOCC). A comparison is made between these possibilities and it is shown that by using nonseparable measurements one can improve the accuracy of the estimation by a factor of $2(d+1)/d$ where d is the dimension of the Hilbert space on which U acts.

DOI
10.1103/physreva.69.022303

Related papers

Estimation of Quantum States and Operations

Ballester Sánchez, Manuel Alberto

The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it. A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties. Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement. Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output. In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.

•Jan 1, 2001•3 citations

Optimal estimation of an unknown maximally entangled state with local operations and classical communication

ShengLi Zhang, Xu‐Bo Zou, Ke Li +2 more

We consider the optimal measurement procedures for estimating an unknown maximally entangled state with local operations and classical communications (LOCC). By the ``optimal,'' we mean that there is no other LOCC measurements that can provide a higher estimation fidelity. In fact, we first use the positive partial transpose constraint to obtain an upper bound of the fidelity; then, we show there exists a kind of LOCC measurement that can attain such a bound. The estimation fidelity of the optimal LOCC estimation procedures is $\frac{D+2}{(D+1){D}^{2}}$ with $D$ denoting the dimension of the relevant quantum system.

Physical Review A•Oct 18, 2007•3 citations

Optimal estimation of SU(d) using exact and approximate 2-designs

Manuel A. Ballester

We consider the problem of estimating an SU(d) quantum operation when n copies of it are available at the same time. It is well known that, if one uses a separable state as the input for the unitaries, the optimal mean square error will decrease as 1/n. However it is shown here that, if a proper entangled state is used, the optimal mean square error will decrease at a 1/n^2 rate. It is also shown that spherical 2-designs (e.g. complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures) can be used to design optimal input states. Although 2-designs are believed to exist for every dimension, this has not yet been proven. Therefore, we give an alternative input state based on approximate 2-designs which can be made arbitrarily close to optimal. It is shown that measurement strategies which are based on local operations and classical communication between the ancilla and the rest of the system can be optimal.

arXiv:quant-ph/0507073•Jul 7, 2005•6 citations

Unambiguous comparison of unitary channels

Michal Sedlák, Mário Ziman

We address the problem of an unambiguous comparison of a pair of unknown qudit unitary channels. Using the framework of process positive operator valued measures we characterize all solutions and identify the optimal ones. We prove that the entanglement is the key ingredient in designing the optimal experiment for comparison of unitary channels. Without entanglement the optimality cannot be achieved. The proposed scheme is also experimentally feasible.

arXiv:0809.4401•Physical Review A•Jan 7, 2009•15 citations

Towards a quantum algorithm deciding the separability problem

Paweł Horodecki, Michał Banacki, Michał Eckstein +1 more

Abstract Determining whether an unknown quantum state is entangled can be approached through quantum tomography and related algorithms. However, these methods are generally inefficient due to the NP-hardness of the problem. An alternative strategy is to treat it as a decision (or estimation) problem with a quantum data structure (Horodecki and Ekert 2002 Phys. Rev. Lett. 89 127902). Over the years, significant progress has been made: the two-qubit case is now fully understood, but for higher-dimensional systems only entanglement detection via negativity has been established. In this work, we propose schemes for measuring upper bounds on bipartite biconcurrence and concurrence, expressed as functions of a fixed (but arbitrary) unitary operation acting on a Hilbert space that is quadratically larger than the original state space. Independently, we provide a scheme for measuring a lower bound for concurrence. In the case of the states proportional to the projectors the schemes are getting significantly simpler and the upper bound on the biconcurrence, as well as the lower bound on the concurrence, become exact (up to experimental and numerical errors). This framework opens the door to tackling the separability problem using methods inspired by the variational quantum eigensolver.

Quantum Science and Technology•May 15, 2026•0 citations

Local entanglement is not necessary for perfect discrimination between unitary operations acting on two qudits by local operations and classical communication

Lvzhou Li, Daowen Qiu

Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state held by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this paper, we show that a perfect discrimination between two unitary operations acting on a two-qudit system can always be achieved without exploiting any entanglement. As a result, we conclude that local entanglement is not necessary for perfect discrimination between unitary operations acting on a two-qudit system by LOCC. Our result, in general, is based on the assumption made by X. F. Zhou, Y. S. Zhang, and G. C. Guo [Phys. Rev. Lett. 99, 170401 (2007)] that for any unitary operation $U$, we can obtain its reverse transformation ${U}^{\ifmmode\dagger\else\textdagger\fi{}}$ by exchanging the input and output ports of the whole setup. Thus, we further consider whether we can avoid using this assumption, and we find that for unitary operations acting on two qubits, our result can be obtained without using this assumption. However, in higher dimension how to do that seems to be unclear.

arXiv:0706.0941•Physical Review A•Mar 24, 2008•18 citations
The problem of optimally estimating an unknown unitary quantum operation with the aid of entanglement is addressed. The idea is to prepare an entangled pair, apply the unknown unitary to one of the two parts, and then measure the joint output state. This measurement could be an entangled one or it could be separable (e.g., measurements which can be implemented with local operations and classical communication or LOCC). A comparison is made between these possibilities and it is shown that by using nonseparable measurements one can improve the accuracy of the estimation by a factor of 2(d+1)/d2(d+1)/d2(d+1)/d where d is the dimension of the Hilbert space on which U acts.
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it. A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties. Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement. Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output. In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
We consider the optimal measurement procedures for estimating an unknown maximally entangled state with local operations and classical communications (LOCC). By the ``optimal,'' we mean that there is no other LOCC measurements that can provide a higher estimation fidelity. In fact, we first use the positive partial transpose constraint to obtain an upper bound of the fidelity; then, we show there exists a kind of LOCC measurement that can attain such a bound. The estimation fidelity of the optimal LOCC estimation procedures is D+2(D+1)D2\frac{D+2}{(D+1){D}^{2}}(D+1)D2D+2​ with DDD denoting the dimension of the relevant quantum system.
We consider the problem of estimating an SU(d) quantum operation when n copies of it are available at the same time. It is well known that, if one uses a separable state as the input for the unitaries, the optimal mean square error will decrease as 1/n. However it is shown here that, if a proper entangled state is used, the optimal mean square error will decrease at a 1/n^2 rate. It is also shown that spherical 2-designs (e.g. complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures) can be used to design optimal input states. Although 2-designs are believed to exist for every dimension, this has not yet been proven. Therefore, we give an alternative input state based on approximate 2-designs which can be made arbitrarily close to optimal. It is shown that measurement strategies which are based on local operations and classical communication between the ancilla and the rest of the system can be optimal.
Abstract Determining whether an unknown quantum state is entangled can be approached through quantum tomography and related algorithms. However, these methods are generally inefficient due to the NP-hardness of the problem. An alternative strategy is to treat it as a decision (or estimation) problem with a quantum data structure (Horodecki and Ekert 2002 Phys. Rev. Lett. 89 127902). Over the years, significant progress has been made: the two-qubit case is now fully understood, but for higher-dimensional systems only entanglement detection via negativity has been established. In this work, we propose schemes for measuring upper bounds on bipartite biconcurrence and concurrence, expressed as functions of a fixed (but arbitrary) unitary operation acting on a Hilbert space that is quadratically larger than the original state space. Independently, we provide a scheme for measuring a lower bound for concurrence. In the case of the states proportional to the projectors the schemes are getting significantly simpler and the upper bound on the biconcurrence, as well as the lower bound on the concurrence, become exact (up to experimental and numerical errors). This framework opens the door to tackling the separability problem using methods inspired by the variational quantum eigensolver.
Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state held by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this paper, we show that a perfect discrimination between two unitary operations acting on a two-qudit system can always be achieved without exploiting any entanglement. As a result, we conclude that local entanglement is not necessary for perfect discrimination between unitary operations acting on a two-qudit system by LOCC. Our result, in general, is based on the assumption made by X. F. Zhou, Y. S. Zhang, and G. C. Guo [Phys. Rev. Lett. 99, 170401 (2007)] that for any unitary operation UUU, we can obtain its reverse transformation ${U}^{\ifmmode\dagger\else\textdagger\fi{}}$ by exchanging the input and output ports of the whole setup. Thus, we further consider whether we can avoid using this assumption, and we find that for unitary operations acting on two qubits, our result can be obtained without using this assumption. However, in higher dimension how to do that seems to be unclear.