The “transition probability” in the state space of a ∗-algebra

Uhlmann, Armin

doi:10.1016/0034-4877(76)90060-4

Cited by 1,348 publications

(895 citation statements)

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“…Furthermore, it follows from Uhlmann's theorem [Uhl76] that this decoupling is also sufficient. More formally, choose K and L such that log K − log L = H ε 2 /13 max (A|B) ρ + 4 log(1/ε) + 2 log 13,…”

Section: One-shot State Mergingmentioning

confidence: 93%

“…We have by Uhlmann's theorem [Uhl76] that ω AB andσ B E R are related by an isometry V A→E R , and hence by the invariance of the smooth conditional max-entropy under local isometries (Lemma 2.6) that…”

Section: Proof Of the Converse Theorem (Theorem 41) Let ρ Ae R Be Amentioning

confidence: 99%

“…Such a state exists by Uhlmann's theorem [Uhl76], and can be shown to satisfy P(σ , σ ) √ 6ε + 2ε. The latter bound is obtained from…”

Section: F(σ B B E R σ B B E Rmentioning

confidence: 99%

“…In the second step of the protocol, Bob decodes the system to the state ρ B B E A 1 B 1 . A suitable decoder can be shown to exist using Uhlmann's theorem [Uhl76]. There exists an isometry V B B 0 X B →B B B 1 X B such that for…”

Section: One-shot State Mergingmentioning

confidence: 99%

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One-Shot Decoupling

Dupuis

Berta

Wullschleger

et al. 2014

Commun. Math. Phys.

108

198

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If a quantum system A, which is initially correlated to another system, E, undergoes an evolution separated from E, then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears (almost) completely, resulting in a decoupling of A from E. We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E, and the other characterizing the mapping that describes the evolution of A. The criterion applies to arbitrary such mappings in the general one-shot setting. Furthermore, the criterion is tight for mappings that satisfy certain natural conditions. One-shot decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.

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Section: One-shot State Mergingmentioning

confidence: 93%

Section: Proof Of the Converse Theorem (Theorem 41) Let ρ Ae R Be Amentioning

confidence: 99%

“…Such a state exists by Uhlmann's theorem [Uhl76], and can be shown to satisfy P(σ , σ ) √ 6ε + 2ε. The latter bound is obtained from…”

Section: F(σ B B E R σ B B E Rmentioning

confidence: 99%

Section: One-shot State Mergingmentioning

confidence: 99%

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One-Shot Decoupling

Dupuis

Berta

Wullschleger

et al. 2014

Commun. Math. Phys.

108

198

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“…(We note that one may prefer normalized density operators; see the first remark at the end of the paper. )According to Uhlmann [6,7], for any A, B ∈ C + 1 (H) we define the fidelity of A and B byThis is in fact the square-root of the transition probability introduced by Uhlmann in [5] for density operators which later Jozsa called fidelity and showed its use in quantum information theory [3]. The reason that Uhlmann defined the fidelity in the way above is that after taking square-root the function F behaves significantly better.…”

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confidence: 99%

Fidelity preserving maps on density operators

Molnár¹

2001

Reports on Mathematical Physics

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Abstract. We prove that any bijective fidelity preserving transformation on the set of all density operators on a Hilbert space is implemented by an either unitary or antiunitary operator on the underlying Hilbert space.Let H be a Hilbert space. The set of all density operators on H, that is, the set of all positive self-adjoint operators on H with finite trace is denoted by C + 1 (H). (We note that one may prefer normalized density operators; see the first remark at the end of the paper.)According to Uhlmann [6,7], for any A, B ∈ C + 1 (H) we define the fidelity of A and B byThis is in fact the square-root of the transition probability introduced by Uhlmann in [5] for density operators which later Jozsa called fidelity and showed its use in quantum information theory [3]. The reason that Uhlmann defined the fidelity in the way above is that after taking square-root the function F behaves significantly better.Clearly, the fidelity is in intimate connection with the transition probability between pure states. Wigner's theorem describing the form of all bijective transformations on the set of all pure states which preserve the transition probability plays fundamental role in the theory of quantum systems. By analogy, it seems to be of some importance to describe all bijective transformations on the density operators which preserve the fidelity. This is exactly what we are performing in the present paper. We show that the fidelity preserving transformations on C + 1 (H) are implemented by an either unitary or antiunitary operator of the underlying Hilbert space.The main result of the paper reads as follows.

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Information Gain vs. State Disturbance in Quantum Theory

Fuchs

1998

Fortschr. Phys.

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The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the "Heisenberg uncertainty principle." A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for isolated systems are unitary. In this paper we shall explore several aspects of the information/disturbance principle in an attempt to make it firmly quantitative and flesh out its significance for quantum theory as a whole.

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The “transition probability” in the state space of a ∗-algebra

Cited by 1,348 publications

References 1 publication

One-Shot Decoupling

One-Shot Decoupling

Fidelity preserving maps on density operators

Information Gain vs. State Disturbance in Quantum Theory

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