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/0612190 · Physical Review A

Pooling quantum states obtained by indirect measurements

Robert W. Spekkens, Howard M. Wiseman

Published Apr 11, 2007• 9 citations
arXivHTMLPDF

Abstract

We consider the pooling of quantum states when Alice and Bob both have one part of a tripartite system and, on the basis of measurements on their respective parts, each infers a quantum state for the third part $S$. We denote the conditioned states which Alice and Bob assign to $S$ by $\ensuremath{\alpha}$ and $\ensuremath{\beta}$, respectively, while the unconditioned state of $S$ is $\ensuremath{\rho}$. The state assigned by an overseer, who has all the data available to Alice and Bob, is $\ensuremath{\omega}$. The pooler is told only $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\rho}$. We show that for certain classes of tripartite states, this information is enough for her to reconstruct $\ensuremath{\omega}$ by the formula $\ensuremath{\omega}\ensuremath{\propto}\ensuremath{\alpha}{\ensuremath{\rho}}^{\ensuremath{-}1}\ensuremath{\beta}$. Specifically, we identify two classes of states for which this pooling formula works: (i) all pure states for which the rank of $\ensuremath{\rho}$ is equal to the product of the ranks of the states of Alice's and Bob's subsystems; (ii) all mixtures of tripartite product states that are mutually orthogonal on $S$.

DOI
10.1103/physreva.75.042104

Related papers

Simple formula for pooling knowledge about a quantum system

Kurt A. Jacobs

When various observers obtain information in an independent fashion about a classical system, there is a simple rule which allows them to pool their knowledge, and this requires only the states-of-knowledge of the respective observers. Here we derive an equivalent quantum formula. While its realm of applicability is necessarily more limited, it does apply to a large class of measurements, and we show explicitly for a single qubit that it satisfies the intuitive notions of what it means to pool knowledge about a quantum system. This analysis also provides a physical interpretation for the trace of the product of two density matrices.

arXiv:quant-ph/0503193•Physical Review A•Oct 4, 2005•12 citations

Conditional Quantum One-Time Pad

Kunal Sharma, Eyuri Wakakuwa, Mark M. Wilde

Suppose that Alice and Bob are located in distant laboratories, which are connected by an ideal quantum channel. Suppose further that they share many copies of a quantum state ρ_{ABE}, such that Alice possesses the A systems and Bob the BE systems. In our model, there is an identifiable part of Bob's laboratory that is insecure: a third party named Eve has infiltrated Bob's laboratory and gained control of the E systems. Alice, knowing this, would like use their shared state and the ideal quantum channel to communicate a message in such a way that Bob, who has access to the whole of his laboratory (BE systems), can decode it, while Eve, who has access only to a sector of Bob's laboratory (E systems) and the ideal quantum channel connecting Alice to Bob, cannot learn anything about Alice's transmitted message. We call this task the conditional one-time pad, and in this Letter, we prove that the optimal rate of secret communication for this task is equal to the conditional quantum mutual information I(A;B|E) of their shared state. We thus give the conditional quantum mutual information an operational meaning that is different from those given in prior works, via state redistribution, conditional erasure, or state deconstruction. We also generalize the model and method in several ways, one of which is a secret-sharing task, i.e., the case in which Alice's message should be secure from someone possessing only the AB or AE systems, but should be decodable by someone possessing all systems A, B, and E.

arXiv:1703.02903•Physical Review Letters•Feb 7, 2020•24 citations

Precoding-based protocols for entanglement assisted linear computation over a quantum many-to-one network

Ruoyu Meng, Aditya Ramamoorthy

In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are $k$ senders, Alice$_1$, $\ldots$, Alice$_k$, and a single receiver, Bob. Each Alice$_i$ has a data vector $W_i \in \mathbb{F}^{m_i}$, where $\mathbb{F}$ is a finite field. Bob wants to compute the linear combination $Y = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^m$, where $V_i$ is an $m \times m_i$ matrix over $\mathbb{F}$. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information $W_i$, $i=1,\ldots,k$, into their local subsystems and transmit them to Bob so that he can recover $Y$ through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders. We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.

cs.ITCross-listedarXiv:2607.13756•Jul 16, 2026•0 citations

Three-party pure quantum states are determined by two two-party reduced states

Lajos Diósi

We can uniquely calculate almost all entangled state vectors of tripartite systems $ABC$ if we know the reduced states of any two bipartite subsystems, e.g., of $AB$ and of $BC$. We construct the explicit solution.

arXiv:quant-ph/0403200•Physical Review A•Jul 19, 2004•45 citations

Exact Cost of Redistributing Multipartite Quantum States

Igor Devetak, Jon Yard

How correlated are two quantum systems from the perspective of a third? We answer this by providing an optimal "quantum state redistribution" protocol for multipartite product sources. Specifically, given an arbitrary quantum state of three systems, where Alice holds two and Bob holds one, we identify the cost, in terms of quantum communication and entanglement, for Alice to give one of her parts to Bob. The communication cost gives the first known operational interpretation to quantum conditional mutual information. The optimal procedure is self-dual under time reversal and is perfectly composable. This generalizes known protocols such as the state merging and fully quantum Slepian-Wolf protocols, from which almost every known protocol in quantum Shannon theory can be derived.

arXiv:quant-ph/0612050•Physical Review Letters•Jun 9, 2008•150 citations

Compatibility of state assignments and pooling of information

Todd A. Brun, Min-Hsiu Hsieh, Christopher Perry

A measure for the compatibility of state assignments of independent observers of the same quantum system is proposed, and the possibility of arriving at a unique state assignment by pooling of information is investigated.

arXiv:1310.5325•Physical Review A•Jul 13, 2015•4 citations
We consider the pooling of quantum states when Alice and Bob both have one part of a tripartite system and, on the basis of measurements on their respective parts, each infers a quantum state for the third part SSS. We denote the conditioned states which Alice and Bob assign to SSS by \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathβ\ensuremath{\beta}\ensuremathβ, respectively, while the unconditioned state of SSS is \ensuremathρ\ensuremath{\rho}\ensuremathρ. The state assigned by an overseer, who has all the data available to Alice and Bob, is \ensuremathω\ensuremath{\omega}\ensuremathω. The pooler is told only \ensuremathα\ensuremath{\alpha}\ensuremathα, \ensuremathβ\ensuremath{\beta}\ensuremathβ, and \ensuremathρ\ensuremath{\rho}\ensuremathρ. We show that for certain classes of tripartite states, this information is enough for her to reconstruct \ensuremathω\ensuremath{\omega}\ensuremathω by the formula \ensuremathω\ensuremath∝\ensuremathα\ensuremathρ\ensuremath−1\ensuremathβ\ensuremath{\omega}\ensuremath{\propto}\ensuremath{\alpha}{\ensuremath{\rho}}^{\ensuremath{-}1}\ensuremath{\beta}\ensuremathω\ensuremath∝\ensuremathα\ensuremathρ\ensuremath−1\ensuremathβ. Specifically, we identify two classes of states for which this pooling formula works: (i) all pure states for which the rank of \ensuremathρ\ensuremath{\rho}\ensuremathρ is equal to the product of the ranks of the states of Alice's and Bob's subsystems; (ii) all mixtures of tripartite product states that are mutually orthogonal on SSS.
When various observers obtain information in an independent fashion about a classical system, there is a simple rule which allows them to pool their knowledge, and this requires only the states-of-knowledge of the respective observers. Here we derive an equivalent quantum formula. While its realm of applicability is necessarily more limited, it does apply to a large class of measurements, and we show explicitly for a single qubit that it satisfies the intuitive notions of what it means to pool knowledge about a quantum system. This analysis also provides a physical interpretation for the trace of the product of two density matrices.
Suppose that Alice and Bob are located in distant laboratories, which are connected by an ideal quantum channel. Suppose further that they share many copies of a quantum state ρ_{ABE}, such that Alice possesses the A systems and Bob the BE systems. In our model, there is an identifiable part of Bob's laboratory that is insecure: a third party named Eve has infiltrated Bob's laboratory and gained control of the E systems. Alice, knowing this, would like use their shared state and the ideal quantum channel to communicate a message in such a way that Bob, who has access to the whole of his laboratory (BE systems), can decode it, while Eve, who has access only to a sector of Bob's laboratory (E systems) and the ideal quantum channel connecting Alice to Bob, cannot learn anything about Alice's transmitted message. We call this task the conditional one-time pad, and in this Letter, we prove that the optimal rate of secret communication for this task is equal to the conditional quantum mutual information I(A;B|E) of their shared state. We thus give the conditional quantum mutual information an operational meaning that is different from those given in prior works, via state redistribution, conditional erasure, or state deconstruction. We also generalize the model and method in several ways, one of which is a secret-sharing task, i.e., the case in which Alice's message should be secure from someone possessing only the AB or AE systems, but should be decodable by someone possessing all systems A, B, and E.
In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are kkk senders, Alice1_11​, …\ldots…, Alicek_kk​, and a single receiver, Bob. Each Alicei_ii​ has a data vector Wi∈FmiW_i \in \mathbb{F}^{m_i}Wi​∈Fmi​, where F\mathbb{F}F is a finite field. Bob wants to compute the linear combination Y=V1W1+V2W2+⋯+VkWk∈FmY = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^mY=V1​W1​+V2​W2​+⋯+Vk​Wk​∈Fm, where ViV_iVi​ is an m×mim \times m_im×mi​ matrix over F\mathbb{F}F. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information WiW_iWi​, i=1,…,ki=1,\ldots,ki=1,…,k, into their local subsystems and transmit them to Bob so that he can recover YYY through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders. We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.
We can uniquely calculate almost all entangled state vectors of tripartite systems ABCABCABC if we know the reduced states of any two bipartite subsystems, e.g., of ABABAB and of BCBCBC. We construct the explicit solution.
How correlated are two quantum systems from the perspective of a third? We answer this by providing an optimal "quantum state redistribution" protocol for multipartite product sources. Specifically, given an arbitrary quantum state of three systems, where Alice holds two and Bob holds one, we identify the cost, in terms of quantum communication and entanglement, for Alice to give one of her parts to Bob. The communication cost gives the first known operational interpretation to quantum conditional mutual information. The optimal procedure is self-dual under time reversal and is perfectly composable. This generalizes known protocols such as the state merging and fully quantum Slepian-Wolf protocols, from which almost every known protocol in quantum Shannon theory can be derived.