## Characterizing quantum theory in terms of informationtheoretic constraints (2003)

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Venue: | Foundations of Physics |

Citations: | 28 - 3 self |

### BibTeX

@ARTICLE{Clifton03characterizingquantum,

author = {Rob Clifton and Jeffrey Bub and Hans Halvorson},

title = {Characterizing quantum theory in terms of informationtheoretic constraints},

journal = {Foundations of Physics},

year = {2003},

pages = {1561--1592}

}

### OpenURL

### Abstract

We show that three fundamental information-theoretic constraints—the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibility of unconditionally secure bit commitment—suffice to entail that the observables and state space of a physical theory are quantum-mechanical. We demonstrate the converse derivation in part, and consider the implications of alternative answers to a remaining open question about nonlocality and bit commitment. KEY WORDS: quantum theory; information-theoretic constraints. Of John Wheeler’s ‘‘Really Big Questions,’ ’ the one on which most progress has been made is It from Bit?—does information play a significant role at the foundations of physics? It is perhaps less ambitious than some of the other Questions, such as How Come Existence?, because it does not necessarily require a metaphysical answer. And unlike, say, Why the Quantum?, it does not require the discovery of new laws of nature: there was room for hope that it might be answered through a better understanding of the laws as we currently know them, particularly those of quantum physics. And this is what has happened: the better understanding is the quantum theory of information and computation. 1

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Citation Context ...es simply to observe that all physical theories that have been found empirically successful—not just phase space and Hilbert space theories (Landsman [34]), but also theories based a manifold (Connes =-=[13]-=-)—fall under this framework (whereas, for example, so-called ‘nondistributive’ Segal algebras permit violations of the Bell inequality far in excess of that permitted by standard quantum theory and ob... |

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Citation Context ...such that d(AB)=A(dB)+(dA) B. Proof. Suppose that TE(B)=B where E is an effect in A and B is a self-adjoint operator in B. Then a tedious but elementary calculation shows that [E 1/2 ,[E 1/2 , B]]=0. =-=(8)-=- Clearly, the map X W i[E1/2 ,X]defines a derivation d of AKB. Moreover, since d(dB)=0 and B is self-adjoint, it follows that i[E1/2 , B]=dB=0 (Ref. 10, Appendix A). Thus, [E, B]=0. Finally, since a C... |

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Citation Context ...endent states. Schrödinger’s result here anticipates the later result by Hughston, Jozsa, and Wootters (28) that underlies the ‘‘no go’’ bit commitment theorem. (Similar results were proved by Jaynes =-=(29)-=- and Gisin (27) .) What Schrödinger found problematic—indeed, objectionable—about entanglement was this possibility of remote steering (Ref. 43, p. 556): It is rather discomforting that the theory sho... |

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Citation Context ...ue of the bit, for Bob to be convinced that the protocol does not allow Alice to cheat by encoding the bit in a way that leaves her free to reveal either 0 or 1 at will. In 1984, Bennett and Brassard =-=[5]-=- proposed a quantum bit commitment protocol now referred to as BB84. The basic idea was to associate the 0 and 1 commitments with two equivalent quantum mechanical mixtures represented by the same den... |

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Citation Context ...cient unless something further is said about the presence of entangled states. In 1935 and 1936, Schrödinger published an extended two-part commentary [43, 44] on the Einstein-Podolsky-Rosen argument =-=[19]-=-, where he introduced the term ‘entanglement’ to describe the peculiar correlations of the EPR-state as [43, p. 555]: ‘the characteristic trait of quantum mechanics, the one that enforces its entire d... |

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Citation Context ...ρ. 3 In the context of elementary quantum mechanics, neither cloning nor broadcasting is generally possible: A pair of pure states can be cloned if and only if they are orthogonal (Wootters and Zurek =-=[48]-=-, Dieks [15]), and (more generally) an arbitrary pair of states can be broadcast if and only if they are represented by mutually commuting density matrices (Barnum et al [4]). Thus, one might suspect ... |

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134 | Unconditionally Secure Quantum Bit Commitment is Impossible
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Citation Context ...y secure bit commitment from kinematic independence, noncommutativity, and nonlocality in the theory-neutral C ∗ -algebraic framework. The proof of this 4sresult in standard quantum mechanics (Mayers =-=[37, 38]-=-, Lo and Chau [35]) depends on the biorthogonal decomposition theorem, which is not available in the more general framework. If the derivation goes through, then we have a characterization theorem for... |

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Citation Context ...hall now explain, we do not believe this is sufficient unless something further is said about the presence of entangled states. In 1935 and 1936, Schrödinger published an extended two-part commentary =-=[43, 44]-=- on the Einstein-Podolsky-Rosen argument [19], where he introduced the term ‘entanglement’ to describe the peculiar correlations of the EPR-state as [43, p. 555]: ‘the characteristic trait of quantum ... |

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Citation Context ...t from kinematic independence, noncommutativity, and nonlocality in the theory-neutral C ∗ -algebraic framework. The proof of this 4sresult in standard quantum mechanics (Mayers [37, 38], Lo and Chau =-=[35]-=-) depends on the biorthogonal decomposition theorem, which is not available in the more general framework. If the derivation goes through, then we have a characterization theorem for quantum theory in... |

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Citation Context ..., there is a unitary operator U ¥ A such that r 1=(r 0) U and p(r 0, r 1)=|r 0(U)| 2 . Thus r 1 é s=(r 0 é s) (U é I), and therefore p(r 0 é s, r 1 é s)=|(r 0 é s)(U é I)| 2 =|r 0(U)| 2 =p(r 0, r 1). =-=(20)-=- Similarly, r 0 é r 0=(r 1 é r 1) (U é U), and therefore p(r 0 é r 0, r 0 é r 1)=|(r 0 é r 0)(U é U)| 2 =|r 0(U)| 4 =p(r 0, r 1) 2 i (21) Lemma 3. Suppose that A and B are kinematically independent. I... |

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Citation Context ...context of elementary quantum mechanics, neither cloning nor broadcasting is generally possible: A pair of pure states can be cloned if and only if they are orthogonal (Wootters and Zurek [48], Dieks =-=[15]-=-), and (more generally) an arbitrary pair of states can be broadcast if and only if they are represented by mutually commuting density matrices (Barnum et al [4]). Thus, one might suspect that in a cl... |

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Citation Context ...gonal (Wootters and Zurek [48], Dieks [15]), and (more generally) an arbitrary pair of states can be broadcast if and only if they are represented by mutually commuting density matrices (Barnum et al =-=[4]-=-). Thus, one might suspect that in a classical theory (in which all operators commute), all states can be broadcast. In this section, we show that this is indeed the case; and, in fact, the ability to... |

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Citation Context ...ent [23, 24, 25, 7]. 3s• that the algebras of observables pertaining to distinct physical systems must commute, usually called microcausality or (a term we prefer) kinematic independence (see Summers =-=[46]-=-); • that any individual system’s algebra of observables must be nonabelian, i.e., noncommutative; • that the physical world must be nonlocal, in that spacelike separated systems must at least sometim... |

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Citation Context ...ependent states. Schrödinger’s result here anticipates the later result by Hughston, Jozsa, and Wootters [28] that underlies the ‘no go’ bit commitment theorem. (Similar results were proved by Jaynes =-=[29]-=- and Gisin [27].) What Schrödinger found problematic—indeed, objectionable—about entanglement was this possibility of remote steering [43, p. 556]: It is rather discomforting that the theory should al... |

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Citation Context ... i of A such that T g 0 (ai é bi)=a − i é bi and T g 1 (ai é bi)=a ' i é bi. Moreover, since T g 0 and T g 1 are affine, r0=T g n 0 s= C li(a − i é bi), (39) i=1 r1=T g n 1 s= C i=1 l i(a ' i é b i). =-=(40)-=- Let m denote the mixed state (1/2)(w1+w2)=(1/2)(w++w−) of B. Then ; n i=1 libi=m, so that each bi is quasi-equivalent to m. Let (p, H) be the representation of B defined in Lemma 4, let P1 denote the... |

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Citation Context ...text of elementary quantum mechanics, neither cloning nor broadcasting is generally possible: A pair of pure states can be cloned if and only if they are orthogonal (Wootters and Zurek (48) and Dieks =-=(15)-=-), and (more generally) an arbitrary pair of states can be broadcast if and only if they are represented by mutually commuting density matrices (Barnum et al. (4)). Thus, one might suspect that in a c... |

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Citation Context ...of theories whose algebra of observables falls short of being isomorphic to the self-adjoint part of a C ∗ -algebra and, instead, only instantiates some weaker mathematical structure, such as a Segal =-=[45]-=- algebra. To foreclose such possibilities, it could be of interest to pursue an axiomatic justification of the C ∗ -algebraic framework along lines similar to those provided by Emch [21, Ch. 1.2]. How... |

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Citation Context ....) Since each of the states w 1, 2, w + is represented by a vector in H, ultraweak continuity of normal states entails that: lim i r 1(A i é (I−A i))=0, (41) lim i lim i lim i r 1((I − A i) é A i)=0, =-=(42)-=- r 0(B i é (I−B i))=0, (43) r 0((I − B i) é B i)=0. (44) Furthermore, since limi m(Ai)=1/2, there exists some j ¥ [1, n] such that limi bj(Ai)>0. Let b=bj and let aŒ=a − j. Then, combining the previou... |

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Citation Context ...o the space C(X), where X is some compact Hausdorff space, and therefore A é B 5 C(X) é C(X) 5 C(X × X) (Ref. 30, p. 849). Define a mapping g from X×X into X×X by setting g((x, y))=(x, x) (x, y ¥ X). =-=(11)-=- Since g is continuous, we can define a linear mapping T on C(X × X) by setting Tf=f p g. Since the range of Tf is a subset of the range of f, the mapping T is positive, and T(I)=I. Furthermore, every... |

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Citation Context ...culation (Brassard’s preferred term) that quantum mechanics can be derived from two cryptographic principles: the possibility of secure key distribution and the impossibility of secure bit commitment =-=[23, 24, 25, 7]-=-. 3s• that the algebras of observables pertaining to distinct physical systems must commute, usually called microcausality or (a term we prefer) kinematic independence (see Summers [46]); • that any i... |

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Citation Context ... allow that A = B, which obviously fails to capture the situation we are intending to describe.) Various notions of independence for a pair A, B of C ∗ -algebras have been developed in the literature =-=[22, 46]-=-. We are particularly interested in the notion of C ∗ -independence developed in [22], because it does not presuppose that A and B are kinematically independent (i.e., that [A, B] = 0 for all A ∈ A an... |

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Citation Context ...we have seen, B(R 6n )), it follows from the Gelfand-Naimark theorem that classical mechanics can be done in Hilbert space! Yet this really is nothing new, having been pointed out long ago by Koopman =-=[31]-=- and von Neumann [47] (see Mauro [36] for an up-to-date discussion). 7sOf course, nothing we have said proves that all physical theories admit a C ∗ - algebraic formulation. Indeed, that would be absu... |

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Citation Context ...theoretic perspective. That is, we are suggesting that quantum theory be viewed, not as first and foremost a mechanical theory of waves and particles (cf. Bohr’s infamous dictum, reported in Petersen =-=[39]-=-: ‘There is no quantum world.’), but as a theory about the possibilities and impossibilities of information transfer. We begin, in section 2, by laying out the mathematical framework within which our ... |

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Citation Context ...ebra A ∨ B they generate (see Landau [32], who shows that there is a state ρ on A ∨ B that violates Bell’s inequality and hence is nonlocally entangled; also Summers and Werner [46] and Bacciagaluppi =-=[3]-=-). So, at least mathematically, the presence of nonlocal entangled states in the formalism is guaranteed, once we know that the algebras of observables are nonabelian. What does not follow is that the... |

11 |
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Citation Context ...culation (Brassard’s preferred term) that quantum mechanics can be derived from two cryptographic principles: the possibility of secure key distribution and the impossibility of secure bit commitment =-=[23, 24, 25, 7]-=-. 3s• that the algebras of observables pertaining to distinct physical systems must commute, usually called microcausality or (a term we prefer) kinematic independence (see Summers [46]); • that any i... |

10 |
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Citation Context ...cloned, then they must be orthogonal. Two pure states ρ, ω of a C ∗ -algebra are said to be orthogonal just in case �ρ−ω� = 2. More generally, the transition probability p(ρ, ω) is defined to be (see =-=[41]-=-): � p(ρ, ω) = 1 − 1 4 �ρ − ω�2 (14) If ρ is a state of A, and U is a unitary operator in A, then we let ρU denote the state defined by ρU(A) = ρ(U ∗ AU) (A ∈ A) (15) In this case, ρ and ρU are said t... |

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Citation Context ...in her bit commitment protocol with Bob. It is easy enough to see this for the original BB84 protocol. Suprisingly, this is also the case for any conceivable quantum bit commitment protocol. (See Bub =-=[9]-=- for a discussion.) Now, unconditionally secure bit commitment is impossible for classical systems, in which the algebras of observables are abelian. It might seem inappropriate, then, that we propose... |

9 |
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Citation Context .... Now, if A and B are nonabelian and mutually commuting (and C ∗ - independent 2 ), it follows immediately that there are nonlocal entangled states on the C ∗ -algebra A ∨ B they generate (see Landau =-=[32]-=-, who shows that there is a state ρ on A ∨ B that violates Bell’s inequality and hence is nonlocally entangled; also Summers and Werner [46] and Bacciagaluppi [3]). So, at least mathematically, the pr... |

7 |
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Citation Context ...ics will not be implementable in a quantum field theory on a curved space-time, for example, which might be a preliminary semi-classical step towards a quantum theory of gravity (see Arageorgis et al =-=[2]-=-). The foundational significance of our derivation, as we see it, is that quantum mechanics should be interpreted as a principle theory, where the principles at issue are information-theoretic. The di... |

7 |
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Citation Context ...algebra generated by A and B. Suppose then that s0 and s1 are states of A K B such that s0(AB)=w(A) r(B)=s1(AB) for all A ¥ A and B ¥ B. Then s 0 1 n C i=1 n AiBi2 = C w(Ai) r(Bi)=s 1 1 i=1 n C AiBi2 =-=(10)-=- i=1 for all A i ¥ A and B i ¥ B. Since A K B is the closure of S in the norm topology, and since states are continuous in the norm topology, s 0=s 1. i When it will not cause confusion, we will hence... |

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Citation Context ...a (which turns out to be important not to do in quantum field theory, where one needs to allow for inequivalent representations of the canonical commutation relations—see, e.g., Clifton and Halvorson =-=[12]-=-). 2.2 Physical Generality of the C ∗ -Algebraic Language If C∗-algebras supply little more than a way of talking abstractly about operator algebras, and the latter are characteristic of quantum theor... |

6 |
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Citation Context ...a (which turns out to be important not to do in quantum field theory, where one needs to allow for inequivalent representations of the canonical commutation relations—see, e.g., Clifton and Halvorson =-=[12]-=-). 2.2 Physical Generality of the C ∗ -Algebraic Language If C ∗ -algebras supply little more than a way of talking abstractly about operator algebras, and the latter are characteristic of quantum the... |

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Citation Context ...ependent states. Schrödinger’s result here anticipates the later result by Hughston, Jozsa, and Wootters [28] that underlies the ‘no go’ bit commitment theorem. (Similar results were proved by Jaynes =-=[29]-=- and Gisin [27].) What Schrödinger found problematic—indeed, objectionable—about entanglement was this possibility of remote steering [43, p. 556]: It is rather discomforting that the theory should al... |

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Citation Context ...culation (Brassard’s preferred term) that quantum mechanics can be derived from two cryptographic principles: the possibility of secure key distribution and the impossibility of secure bit commitment =-=[23, 24, 25, 7]-=-. 3s• that the algebras of observables pertaining to distinct physical systems must commute, usually called microcausality or (a term we prefer) kinematic independence (see Summers [46]); • that any i... |

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Citation Context ...tion-theoretic. The distinction between principle and constructive theories is 23sintroduced by Einstein in his discussion of the significance of the transition from Newtonian to relativistic physics =-=[18]-=-. As Einstein puts it, most theories in physics are constructive, with the aim of representing complex phenomena as constructed out of the elements of a simple formal scheme. So, for example, the kine... |

2 |
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Citation Context ..., for example, so-called ‘nondistributive’ Segal algebras permit violations of the Bell inequality far in excess of that permitted by standard quantum theory and observed in the laboratory—see Landau =-=[33]-=-). 2.3 Classical versus Quantum Theories We must mention one final important representation theorem: every (unital) abelian C∗-algebra A is isomorphic to the set C(X) of all continuous, complex-valued... |

2 |
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Citation Context ...imply to observe that all physical theories that have been found empirically successful—not just phase space and Hilbert space theories (Landsman (34) ), but also theories based on a manifold (Connes =-=(13)-=- )— fall under this framework (whereas, for example, so-called ‘‘nondistributive’’ Segal algebras permit violations of the Bell inequality far in excess of that permitted by standard quantum theory an... |

2 |
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Citation Context ..., and Halvorson Naimark theorem that classical mechanics can be done in Hilbert space! Yet this really is nothing new, having been pointed out long ago by Koopman (31) and von Neumann (47) (see Mauro =-=(36)-=- for an up-to-date discussion). Of course, nothing we have said proves that all physical theories admit a C*-algebraic formulation. Indeed, that would be absurd to claim: one can certainly conceive of... |

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1 |
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Citation Context |

1 | Recurrence in quantum mechanics,”quant-ph/0202023 - Duvenhage |

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Citation Context ...e, r and r U are said to be unitarily equivalent. Furthermore, if U is a unitary operator in A and V is a unitary operator in B, then (w é r) U é V (A é B)=(w é r)(U*AU é V*BV) (16) =(wU é rV)(A é B) =-=(17)-=- for all A ¥ A and B ¥ B. Thus, the uniqueness of product states (Lemma 1) entails that (w é r) U é V=w U é r V. For the following lemma, we will need to make use of the fact that p(r, r U)=|r(U)| 2 f... |