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:1410.1441 · Physical Review A

Fidelity of recovery, squashed entanglement, and measurement recoverability

Kaushik Parasuram Seshadreesan, Mark M. Wilde

Published Oct 19, 2015• 60 citations
arXivHTMLPDF

Abstract

This paper defines the fidelity of recovery of a tripartite quantum state on systems $A,\phantom{\rule{0.16em}{0ex}}B$, and $C$ as a measure of how well one can recover the full state on all three systems if system $A$ is lost and a recovery operation is performed on system $C$ alone. The surprisal of the fidelity of recovery (its negative logarithm) is an information quantity which obeys nearly all of the properties of the conditional quantum mutual information $I(A;B|C)$, including non-negativity, monotonicity with respect to local operations, duality, invariance with respect to local isometries, a dimension bound, and continuity. We then define a (pseudo) entanglement measure based on this quantity, which we call the ``geometric squashed entanglement.'' We prove that the geometric squashed entanglement is a 1-LOCC monotone (i.e., monotone nonincreasing with respect to local operations and classical communication from Bob to Alice), that it vanishes if and only if the state on which it is evaluated is unentangled, and that it reduces to the geometric measure of entanglement if the state is pure. We also show that it is invariant with respect to local isometries, subadditive, continuous, and normalized on maximally entangled states. We next define the surprisal of measurement recoverability, which is an information quantity in the spirit of quantum discord, characterizing how well one can recover a share of a bipartite state if it is measured. We prove that this discordlike quantity satisfies several properties, including non-negativity, faithfulness on classical-quantum states, invariance with respect to local isometries, a dimension bound, and normalization on maximally entangled states. This quantity, combined with a recent breakthrough of Fawzi and Renner, makes it possible to characterize states with discord nearly equal to zero as being approximate fixed points of entanglement-breaking channels (equivalently, they are recoverable from the state of a measuring apparatus). Finally, we discuss a multipartite fidelity of recovery and several of its properties.

DOI
10.1103/physreva.92.042321

Related papers

Multipartite quantum correlations and local recoverability

Mark M. Wilde

Characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory. More recently, however, the total correlation or multipartite information measure has been helpful in accomplishing this goal, especially with the multipartite symmetric quantum (MSQ) discord (Piani et al. 2008 Phys. Rev. Lett. 100, 090502. ( doi:10.1103/PhysRevLett.100.090502 )) and the conditional entanglement of multipartite information (CEMI) (Yang et al. 2008 Phys. Rev. Lett. 101, 140501. ( doi:10.1103/PhysRevLett.101.140501 )). Here, we apply a recent and significant improvement of strong subadditivity of quantum entropy (Fawzi & Renner 2014 ( http://arxiv.org/abs/1410.0664 )) in order to develop these quantities further. In particular, we prove that the MSQ discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems. Furthermore, we prove that the CEMI is a faithful entanglement measure, i.e. it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state. Along the way, we provide an operational interpretation of the MSQ discord in terms of the partial state distribution protocol, which in turn, as a special case, gives an interpretation for the original discord quantity. Finally, we prove an inequality that could potentially improve upon the Fawzi–Renner inequality in the multipartite context, but it remains an open question to determine whether this is so.

arXiv:1412.0333•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Apr 15, 2015•28 citations

R�nyi Quantum Conditional Mutual Information and Parity Quantum Optical Metrology

Kaushik Parasuram Seshadreesan

This is a two-part thesis strung together by a common underlying theme—quantum correlations. We present some new characterizations and quantifications of quantum correlations and an application of one such correlation—entanglement—for quantum technology. In Part I of the thesis, we use a Rényi generalization of the quantum conditional mutual information (QCMI) to define and study new measures of quantum entanglement and quantum discord. In particular, using a quantity derived from a Rényi QCMI, we introduce: a) the geometric squashed entanglement, a faithful entanglement measure, which is a lower bound on the squashed entanglement and which reduces to the geometric measure of entanglement for pure quantum states, b) the surprisal of measurement recoverability, a discord-like measure, which is similarly a lower bound on the quantum discord. The surprisal of measurement recoverability enhances our understanding of quantum discord in terms of the ability to recover one share of a bipartite quantum system after it has been measured. In Part II, we discuss entanglement-enhanced quantum sensing. In particular, we consider optical interferometric sensors that use photon-number parity detection. Using the quantum and classical Cramér-Rao bounds (QCRB and CCRB) on phase precision as the figures of merit, we characterize a class of two-mode pure states for which photon-number parity measurement is optimal for phase estimation. These states turn out to be a subset of the class of path-symmetric states—a class for which photon-number counting-based measurements are known to be optimal. Further, we gauge the performance of the particular interferometry based on coherent light mixed with squeezed vacuum light and photon-number parity measurement. We show that photon-number parity is an optimal measurement for the above state in the sense that the detection scheme is capable of achieving the best phase precision offered by the state (given by its QCRB). The state by itself is also known to be capable of optimal phase precision for any state in linear interferometry for a given photon budget, called the Heisenberg limit. Thus, we demonstrate Heisenberg-limited phase estimation for the state with photon-number parity detection.

Civil War Book Review•Mar 23, 2015•0 citations

Rényi squashed entanglement, discord, and relative entropy differences

Kaushik Parasuram Seshadreesan, Mario Berta, Mark M. Wilde

The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205), we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333), we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set . Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.

Civil War Book Review•Sep 14, 2015•22 citations

Rényi squashed entanglement, discord, and relative entropy differences

Kaushik P Seshadreesan, Mario Berta, Mark M. Wilde

The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205 ), we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A ). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B , we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333 ), we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set . Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.

arXiv:1410.1443•Journal of Physics A Mathematical and Theoretical•Sep 14, 2015•36 citations

The Fidelity of Recovery Is Multiplicative

Mario Berta, Marco Tomamichel

Fawzi and Renner recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states ABC in terms of the fidelity of recovery (FoR), i.e., the maximal fidelity of the state ABC with a state reconstructed from its marginal BC by acting only on the C system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here, we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandão et al. of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.

arXiv:1502.07973•IEEE Transactions on Information Theory•Feb 10, 2016•45 citations

Operational meaning of quantum measures of recovery

Tom Cooney, Christoph Hirche, Ciara Morgan +4 more

Several information measures have recently been defined that capture the notion of recoverability. In particular, the fidelity of recovery quantifies how well one can recover a system $A$ of a tripartite quantum state, defined on systems $ABC$, by acting on system $C$ alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the type I error is constrained to be no larger than a constant.

arXiv:1512.05324•Physical Review A•Aug 10, 2016•17 citations
This paper defines the fidelity of recovery of a tripartite quantum state on systems A,BA,\phantom{\rule{0.16em}{0ex}}BA,B, and CCC as a measure of how well one can recover the full state on all three systems if system AAA is lost and a recovery operation is performed on system CCC alone. The surprisal of the fidelity of recovery (its negative logarithm) is an information quantity which obeys nearly all of the properties of the conditional quantum mutual information I(A;B∣C)I(A;B|C)I(A;B∣C), including non-negativity, monotonicity with respect to local operations, duality, invariance with respect to local isometries, a dimension bound, and continuity. We then define a (pseudo) entanglement measure based on this quantity, which we call the ``geometric squashed entanglement.'' We prove that the geometric squashed entanglement is a 1-LOCC monotone (i.e., monotone nonincreasing with respect to local operations and classical communication from Bob to Alice), that it vanishes if and only if the state on which it is evaluated is unentangled, and that it reduces to the geometric measure of entanglement if the state is pure. We also show that it is invariant with respect to local isometries, subadditive, continuous, and normalized on maximally entangled states. We next define the surprisal of measurement recoverability, which is an information quantity in the spirit of quantum discord, characterizing how well one can recover a share of a bipartite state if it is measured. We prove that this discordlike quantity satisfies several properties, including non-negativity, faithfulness on classical-quantum states, invariance with respect to local isometries, a dimension bound, and normalization on maximally entangled states. This quantity, combined with a recent breakthrough of Fawzi and Renner, makes it possible to characterize states with discord nearly equal to zero as being approximate fixed points of entanglement-breaking channels (equivalently, they are recoverable from the state of a measuring apparatus). Finally, we discuss a multipartite fidelity of recovery and several of its properties.
Characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory. More recently, however, the total correlation or multipartite information measure has been helpful in accomplishing this goal, especially with the multipartite symmetric quantum (MSQ) discord (Piani et al. 2008 Phys. Rev. Lett. 100, 090502. ( doi:10.1103/PhysRevLett.100.090502 )) and the conditional entanglement of multipartite information (CEMI) (Yang et al. 2008 Phys. Rev. Lett. 101, 140501. ( doi:10.1103/PhysRevLett.101.140501 )). Here, we apply a recent and significant improvement of strong subadditivity of quantum entropy (Fawzi & Renner 2014 ( http://arxiv.org/abs/1410.0664 )) in order to develop these quantities further. In particular, we prove that the MSQ discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems. Furthermore, we prove that the CEMI is a faithful entanglement measure, i.e. it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state. Along the way, we provide an operational interpretation of the MSQ discord in terms of the partial state distribution protocol, which in turn, as a special case, gives an interpretation for the original discord quantity. Finally, we prove an inequality that could potentially improve upon the Fawzi–Renner inequality in the multipartite context, but it remains an open question to determine whether this is so.
This is a two-part thesis strung together by a common underlying theme—quantum correlations. We present some new characterizations and quantifications of quantum correlations and an application of one such correlation—entanglement—for quantum technology. In Part I of the thesis, we use a Rényi generalization of the quantum conditional mutual information (QCMI) to define and study new measures of quantum entanglement and quantum discord. In particular, using a quantity derived from a Rényi QCMI, we introduce: a) the geometric squashed entanglement, a faithful entanglement measure, which is a lower bound on the squashed entanglement and which reduces to the geometric measure of entanglement for pure quantum states, b) the surprisal of measurement recoverability, a discord-like measure, which is similarly a lower bound on the quantum discord. The surprisal of measurement recoverability enhances our understanding of quantum discord in terms of the ability to recover one share of a bipartite quantum system after it has been measured. In Part II, we discuss entanglement-enhanced quantum sensing. In particular, we consider optical interferometric sensors that use photon-number parity detection. Using the quantum and classical Cramér-Rao bounds (QCRB and CCRB) on phase precision as the figures of merit, we characterize a class of two-mode pure states for which photon-number parity measurement is optimal for phase estimation. These states turn out to be a subset of the class of path-symmetric states—a class for which photon-number counting-based measurements are known to be optimal. Further, we gauge the performance of the particular interferometry based on coherent light mixed with squeezed vacuum light and photon-number parity measurement. We show that photon-number parity is an optimal measurement for the above state in the sense that the detection scheme is capable of achieving the best phase precision offered by the state (given by its QCRB). The state by itself is also known to be capable of optimal phase precision for any state in linear interferometry for a given photon budget, called the Heisenberg limit. Thus, we demonstrate Heisenberg-limited phase estimation for the state with photon-number parity detection.
The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205), we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333), we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set . Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.
The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205 ), we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A ). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B , we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333 ), we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set . Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.
Fawzi and Renner recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states ABC in terms of the fidelity of recovery (FoR), i.e., the maximal fidelity of the state ABC with a state reconstructed from its marginal BC by acting only on the C system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here, we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandão et al. of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.
Several information measures have recently been defined that capture the notion of recoverability. In particular, the fidelity of recovery quantifies how well one can recover a system AAA of a tripartite quantum state, defined on systems ABCABCABC, by acting on system CCC alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the type I error is constrained to be no larger than a constant.