Quantum deleting and signalling

Pati, Arun Kumar; Braunstein, Samuel L.

doi:10.1016/s0375-9601(03)01047-8

Cited by 31 publications

(27 citation statements)

References 26 publications

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“…There have also been attempts to derive them by applying some other fundamental principles of nature, like no-faster-thanlight communication between spatially separated parties [8,9,10,11] and preservation of entanglement for closed systems under local operations [12]. Another important no-go theorem is the non-existence of a universal flipping operator (NOT gate) [13] which would take an arbitrary qubit state to its orthogonal complement.…”

Section: Introductionmentioning

confidence: 99%

On the Non-Existence of a Universal Hadamard Gate

Parashar

2007

Int. J. Quantum Inform.

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We establish the non-existence of a universal Hadamard gate for arbitrary unknown qubits, by considering two different principles; namely, no-superluminal signalling and non-increase of entanglement under LOCC. It is also shown that these principles are not violated if and only if the qubit states are of the special form obtained in our recent work [IJQI (in press), quant-ph/0505068].Keywords: Hadamard gate, no-signalling, non-increase of entanglement. PACS: 03.67.Mn; 03.67.HkThe Hadamard gate is a one qubit unitary operator which has played a very crucial role in the developement of quantum algorithms [1,2,3]. It rotates the computational basis (CB) states |0 and |1 by creating an equal superposition of the amplitudes, and the new states so obtained are orthogonal to each other. Can this action of the Hadamard gate be generalized to any arbitrary unknown qubit state? It was proved in [4] using unitarity of the evolution, that there does not exist a Hadamard gate for a completely arbitrary qubit state. In other words, the Hadamard gate is not universal. Then the natural question to ask is: What is the most general class of states for which it is possible to design this gate in a universal manner? We endeavored to address this aspect in our recent investigation [5] where a special ensemble of qubits was obtained. The Hadamard gate holds for each and every state belonging to this ensemble.The restrictions imposed by the structure of quantum mechanics has led to several other impossible operations in quantum information theory; for example, the famous nocloning theorem [6] and the no-deleting principle [7]. There have also been attempts to derive them by applying some other fundamental principles of nature, like no-faster-thanlight communication between spatially separated parties [8,9,10,11] and preservation of entanglement for closed systems under local operations [12]. Another important no-go theorem is the non-existence of a universal flipping operator (NOT gate) [13] which would take an arbitrary qubit state to its orthogonal complement. An alternative proof of this was

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Section: Introductionmentioning

confidence: 99%

On the Non-Existence of a Universal Hadamard Gate

Parashar

2007

Int. J. Quantum Inform.

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“…03.65.Wj Quantum bits (qubits) are fundamentally different from classical bits in that unknown qubits cannot be copied with unit efficiency [1,2,3] (no-cloning theorem). Another related property of qubits is that nonorthogonal qubits cannot be distinguished with certainty [4].Interestingly, however, it has been found that the nosignaling condition is entangled with other impossibility proofs [5,6,7,8]. In particular, it has been shown that no-signaling condition gives the same tight bound on probability of conclusive measurement as obtained by quantum mechanical formula [7].…”

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Helstrom theorem from the no-signaling condition

Hwang

2005

Phys. Rev. A

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We prove a special case of Helstrom theorem by using no-signaling condition in the special theory of relativity that faster-than-light communication is impossible. 03.65.Wj Quantum bits (qubits) are fundamentally different from classical bits in that unknown qubits cannot be copied with unit efficiency [1,2,3] (no-cloning theorem). Another related property of qubits is that nonorthogonal qubits cannot be distinguished with certainty [4].Interestingly, however, it has been found that the nosignaling condition is entangled with other impossibility proofs [5,6,7,8]. In particular, it has been shown that no-signaling condition gives the same tight bound on probability of conclusive measurement as obtained by quantum mechanical formula [7].In this paper, we add one in the list of theorems that can be proven by the no-signaling condition. We prove a special case of Helstrom theorem [9]. Result in this paper is closely related to other's works but different. In particular, our argument is quite similar to the one in Ref.[5]. Our contribution is an observation that violation of Helstrom theorem implies that two appropriately chosen (different) decompositions of the same density operator can be discriminated. This paper is organized as follows. We describe the proposition that we will prove. We prove it by no-signaling condition and then we conclude.Roughly speaking, Helstrom theorem means that the more non-orthogonal two qubits are, the more difficult it is to discriminate them by positive-operator-valuedmeasurement [4]. Let us consider a special case of Helstrom theorem.Proposition-1: Consider two non-orthogonal qubits, |α and |β , whose overlap, | α|β | 2 , is between 0 and 1. We are given a qubit that is either |α or |β with equal a priori probability, 1/2. We want to identify the qubit quantum mechanically. Identifier of the qubit gives either an output, 0, or the other output, 1. P E is the probability of making error in the identification. Minimal value of P E is given by. ♦ (Proposition-1 has interesting applications in quantum cryptography. For example, Bennett 1992 quantum key distribution protocol [10] and quantum remote gambling protocol [11].) Before we prove Proposition-1, let us introduce the followings. Any pure qubit |i i| can be represented by a three-dimensional Euclidean Bloch vector r i as |i i| = (1/2)(1 +r i · σ) [12]. Here 1 is identity operator, σ = (σ x , σ y , σ z ), and σ x , σ y , σ z are Pauli operators. Two Bloch vectors corresponding to |α and |β arer α andr β , respectively. We define an angle between r α andr β to be 2θ. That is, | α|β | 2 = cos 2 θ. A pure state |γ is defined as that its Bloch vectorr γ bisects the two Bloch vectorsr α andr β in the same plane, namelŷ r γ = C(r α +r β ) where C is a constant for normalization.A pure state |δ is defined as its Bloch vectorr δ makes an angle π/2 and π/2 + θ with the Bloch vectorr γ and r α , in the same plane, respectively. A pure state | − δ is defined as its Bloch vectorr −δ is the negative of that of |δ , namely −r δ . Note that al...

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“…This is called no-deleting. Again the above statement of no-deleting can be proven as a theorem, by assuming a unitary evolution and was called the no-deleting theorem [21,22]. Note that from a formal point of view, it is not clear whether there is any common physical origin of the above theorems or a relationship between them.…”

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confidence: 99%

“…One may also see deletion in the following way (which is again not possible under unitary dynamics [22]):…”

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Common Origin of No-Cloning and No-Deleting Principles Conservation of Information

Horodecki

Sen

et al. 2005

Found Phys

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We discuss the role of the notion of information in the description of physical reality. We consider theories for which dynamics is linear with respect to stochastic mixing. We point out that the nocloning and no-deleting principles emerge in any such theory, if law of conservation of information is valid, and two copies contain more information than one copy. We then describe the quantum case from this point of view. This paper is dedicated to Asher Peres on the occasion of his seventieth birthday.The fact that Nature allows us to describe itself mathematically, seems even more astonishing than the laws of Nature themselves. In particular, quantum formalism is enigma, which in spirit of Gödel's theorem can not be explained by itself. The lack of clear relation between the description and reality, brings about difficulties in the interpretation of quantum formalism. As one knows, any attempt of objectivisation of the latter leads to a number of paradoxes [1]. This gap between formalism and reality has, in particular, its reflection in Asher Peres's phrase: "the physics is what physicists do in laboratories" [1] and "entanglement is a trick the quantum magicians use to produce phenomena, that can not be imitated by clasical magicians" [2]. But again, why quantum magicians are better than their classical counterparts? This question forces us to adopt the primitive notions which are autonomic with respect to the formalism. In this context two notions seem to be relevant: information and informational isomorphism [3].The recent discoveries [4,5,6,7,8] concerning processing of information in the quantum regime [9] convince us more and more that Landauer's slogan, "Information is physical!" [10], is not empty [11]. The idea that the notion of information should be regarded as a fundamental ingredient in a physical theory was proposed in different contexts [12,13,14,15,16]. However it is not quite clear, what the term "physical" means, in the above context. In fact, there are two opposing pictures of information: i) subjective, according to which information represents knowledge, ii) objective, which treats information (just like energy) as a property of the physical system. This cognitive duality can be surmounted by postulating that any consistent description of Nature, is a sort of isomorphism between the laws of Nature and their mathematical representation. According to this view [3] (called informational isomorphism), although no notion itself is reality, yet it reflects physical reality. Then any theoretical structure, although is not a real thing, is an isomorphic image of the existing reality. In this sense, information can be treated as physical, and it is natural to ask: What are the fundamental consequences of such statement? The problem is by no means trivial, as it concerns the properties of the quantity that we regard

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Quantum deleting and signalling

Cited by 31 publications

References 26 publications

On the Non-Existence of a Universal Hadamard Gate

On the Non-Existence of a Universal Hadamard Gate

Helstrom theorem from the no-signaling condition

Common Origin of No-Cloning and No-Deleting Principles Conservation of Information

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