Quantum marginal problem and incompatibility

Haapasalo, Erkka; Kraft, Tristan; Miklin, Nikolai; Uola, Roope

doi:10.22331/q-2021-06-15-476

Cited by 19 publications

(18 citation statements)

References 53 publications

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“…In particular, every quantum state marginal problem can be reformulated as the problem of determining compatibility of some pair of channels. This has already been established in [33] in the case when the marginals of the states have full rank. In this section, we prove this reduction holds even in the more general case when the marginals do not necessarily have full rank.…”

Section: Equivalence Of the Quantum State And Quantum Channel Version...mentioning

confidence: 55%

“…To see how the channel and state versions of the marginal problem are generalizations of each other, we may consider the Choi representations of the channels. It is not hard to see [29,33] that the conditions for the compatibility of Φ 1 and Φ 2 is equivalent to the following conditions on the Choi matrices:…”

Section: Connections Between Marginal Problemsmentioning

confidence: 99%

“…We show that the quantum channel marginal problem also generalizes the state marginal problem in the following sense. Recently, a somewhat weaker version of this result was proved in [33], where it was shown that the marginal problem for quantum states with invertible marginals is equivalent to the compatibility of channels. By using a different method to prove the result, we bypass the need for invertible marginal.…”

Section: The Quantum Channel Marginal Problem Generalizes the State V...mentioning

confidence: 99%

“…It is straightforward to see that the marginal problem for quantum states generalizes the problem of determining compatibility for quantum channels. Indeed, two channels are compatible precisely when their corresponding normalized Choi representations are compatible as states [29,33].…”

Section: Equivalence Of the Quantum State And Quantum Channel Version...mentioning

confidence: 99%

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Jordan products of quantum channels and their compatibility

Girard,

Plávala,

Sikora

2020

Preprint

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Given two quantum channels, we examine the task of determining whether they are compatible-meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). We show several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-andprepare channels (i.e., entanglement-breaking channels) do not necessarily have a measureand-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolaring channels are compatible.

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Section: Equivalence Of the Quantum State And Quantum Channel Version...mentioning

confidence: 55%

Section: Connections Between Marginal Problemsmentioning

confidence: 99%

Section: The Quantum Channel Marginal Problem Generalizes the State V...mentioning

confidence: 99%

Section: Equivalence Of the Quantum State And Quantum Channel Version...mentioning

confidence: 99%

See 2 more Smart Citations

Jordan products of quantum channels and their compatibility

Girard,

Plávala,

Sikora

2020

Preprint

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“…Interestingly, a unified framework covering problems such as channel extendibility [96,97], broadcasting channels [98][99][100][101][102][103][104][105] and causal channels [74][75][76][77]106] has recently been introduced under the name of quantum channel marginal problems [65]. Utilizing this framework, we present the dynamical generalization of Definition 1 and that of Definition 2, as well as the dynamical counterpart of Theorem 1, Theorem 3, and Theorem 4.…”

Section: From States To Dynamics: Channel Resource Marginal Problemsmentioning

confidence: 99%

Resource Marginal Problems

Hsieh¹,

Tabia²,

Yin³

et al. 2022

Preprint

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We introduce the resource marginal problems, which concern the possibility of having a resource-free target subsystem compatible with a given collection of marginal density matrices. By identifying an appropriate choice of resource 𝑅 and target subsystem T, our problems reduce, respectively, to the well-known marginal problems for quantum states and the problem of determining if a given quantum system is a resource. More generally, we define a set of marginal states to be resource-free incompatible with a target subsystem T if all global states compatible with this set must result in a resourceful state in T. We show that this incompatibility induces a resource theory that can be quantified by a monotone. We derive necessary and sufficient conditions for this monotone to be a finite conic program and further show, via the corresponding witnesses, that resourcefree incompatibility is equivalent to an operational advantage in some subchannel discrimination task. Via our framework, a clear connection can be established between any marginal problem (that involves some notion of incompatibility) for quantum states and a resource theory for quantum states. In addition, the universality of our framework leads, for example, to an immediate, quantitative understanding of the incompatibility associated with the recently-proposed entanglement marginal problems. As further applications, we provide the first example showing a form of transitivity of nonlocality (steerability) for quantum states. We discuss also the analogous framework in the dynamical regime and establish a set of theoretical results that closely mirror those obtained for a resource theory for quantum states. Again, they allow us to recover a wide variety of results related to the studies of channel marginal problems, channel resource theories, etc.

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Relating Compatibility and Divisibility of Quantum Channels

2022

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We connect two key concepts in quantum information: compatibility and divisibility of quantum channels. Two channels are compatible if they can be both obtained via marginalization from a third channel. A channel divides another channel if it reproduces its action by sequential composition with a third channel. (In)compatibility is of central importance for studying the difference between classical and quantum dynamics. The relevance of divisibility stands in its close relationship with the onset of Markovianity. We emphasize the simulability character of compatibility and divisibility, and, despite their structural difference, we find a set of channels – self-degradable channels – for which the two notions coincide. We also show that, for degradable channels, compatibility implies divisibility, and that, for anti-degradable channels, divisibility implies compatibility. These results provide physical insights and motivate further research on these classes of channels and shed new light on the meaning of these two largely studied notions.

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Quantum marginal problem and incompatibility

Cited by 19 publications

References 53 publications

Jordan products of quantum channels and their compatibility

Jordan products of quantum channels and their compatibility

Resource Marginal Problems

Relating Compatibility and Divisibility of Quantum Channels

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