Schrödinger's Register: Foundational Issues and Physical Realization Stephen Pink School of Computing and Communication Lancaster University Lancaster, UK pink@comp.lancs.ac.uk Stanley Martens School of Accountancy and Management Information Systems DePaul University (retired) Chicago, IL USA smartens00@gmail.com This paper consists of four results which relate to the foundations and physical realization of quantum computing. The first result is that the qubit can not be taken as the basic unit for quantum computing, because not every superposition of bit-strings of length n can be factored into a string of nqubits. The second result is that the "No-cloning" theorem does not apply to the copying of one quantum register into another register, because the mathematical representation of this copying is the identity operator, which is manifestly linear. The third result is that quantum parallelism is not destroyed only by environmental decoherence. There are two other forms of decoherence, which we call measurement decoherence and internal decoherence, that can also destroy quantum parallelism. The fourth result is that processing the contents of a quantum register "one qubit at a time" destroys entanglement. Keywords-qubit; entanglement; decoherence; no-cloning theorem; quantum register; I. INTRODUCTION This paper will make four points. Points (A) and (B) are foundational. Points (B), (C), and (D) relate to the physical realization of quantum computing. We will state the points and then elaborate on them. A. The basic element of quantum computing is not the qubit but the q-string. The qubit is not basic because not every q-string can be factored into a string of qubits. B. A processing step in quantum computing is defined as the application of a unitary linear operator on q-strings [4]. For physical realization purposes, this definition is incomplete. In a real quantum computer, q-strings of length n "live" on n-bit registers in superposed states. To specify a processing step, one must specify the input and output registers. Let ψ be a q-string. A "cloning" function f12 can be defined as f(ψ on register 1) = ψ on register 2; the function copies ψ from register 1 to register 2. The existence of such a processing step does not violate the "No-cloning" theorem. C. The power of quantum computing depends on quantum parallelism. Quantum parallelism is destroyed if q-strings environmentally decohere during processing. Experimental results indicate that the time to environmental decoherence is inversely related to the size of the physical system considered. If this is 0, then the longer the q-string, the shorter the time to its environmental decoherence. This rules out quantum parallelism for q-strings of arbitrary length. Besides environmental decoherence, two other notions of decoherence are introduced and discussed. D. If a q-string is processed "one qubit at a time", then the resulting q-string is a string of qubits. So, any entanglement in the original q-string is destroyed. II. ELABORATIONS OF THE FOUR POINTS A. Qubits and Q-Strings A bit = bi is 0 or 1. A string of bits of length n = | bi . . . bn〉. The number of all strings of bits of length n = 2n. A q-string of length n is a sum of the bit strings of length n weighted by complex numbers. So, a q-string of length n = ψ = m=2n∑ m=1 cmφm where the φn are the bit strings of length n and the complex numbers cm satisfy the condition ∑ n | cm |2= 1. Some of the cn may be 0, so it may be that not every bit string of length n is a nonzero-weighted component of the q-string. A qubit qi = a q-string of length 1. So, qi = αi | 0〉+βi | 1〉, when | αi |2 + | βi |2= 1. A string of qubits is a product of qubits = q1 ⊗ q2 ⊗ . . . qi. Consider a string of two qubits: q1 ⊗ q2 = (α1 | 0〉 + β1 | 1〉) ⊗ (α2 | 0〉 + β2 | 1〉 = α1α2 | 0〉 | 0〉 + α1β2 | 0〉 + β1α2 | 1〉 | 0〉 + β1β2 | 1〉 | 1〉 = α1α2 | 00〉 + α1β2 | 01〉 + β1α2 | 10〉 + β1β2 | 11〉. | α1 |2 + | β1 |2 = 1 and | α2 |2 + | β2 |2 = 1. Now consider the q-string of length 2: 0 | 00〉 + 1 √ 2 | 10〉 + 1 √ 2 | 10〉 + 0 | 11〉. If this q-string = q1 ⊗ q2, then α1α2 = 0, α1β2 = 1√ 2 , β1α2 = 1√2 , and β1β2 = 0 6= 1√ 2 . So, either α1 = 0 or α2 = 1√2 . But if α1 = 0, then α1β = 0 6= 1√ 2 . If α2 = 0, then β1α2 = 0 6= 1√2 . So the above q-string of length 2 is not a string of 2 qubits. This argument can be generalized. So, not every q-string of length n is a string of n qubits. The above q-string of length 2 is said to have "entangled qubits", because it cannot be factored into a string of 2 qubits. B. Q-Strings and the No-cloning Theorem The physical realization of a q-string of length n in a real quantum computer will be the state of a register with n bit positions. Suppose a real quantum computer contains two n-bit registers. Suppose register 1 contains a q-string of length n, i.e., suppose register 1 is in a superposition of states where each of the states is a bit-string of length n. Suppose register 2 contains a string of n zeroes. Nothing in what has been described so far rules out that the computer can execute the command: Store the content of register 1 in register 2. The result of this processing will be two registers, each containing the same q-string. Wouldn't this violate the "No-cloning theorem" [2]? A processing step in quantum computing is defined as the application of unitary linear operator f on q-strings of length n [4]. Suppose, ψ = m=2n∑ m cmφm. Since f is unitary, the action of f on a bit string yields a bit string of the same length. So, f(φm) = φm′ = φf(m). Thus, f(ψ) = m=2n∑ m cmφf(m). As far as it goes, this definition of processing step is correct. For physical realization purposes, however, it is incomplete. In a real quantum computer, quantum strings live on n-bit registers. So, the mathematical representation of a processing step must specify the input and output registers. Thus, the "cloning" function C must be identified as C12. So, C(ψ on register 1) = ψ on register 2, and C is simply the Identity Transformation, Iψ = Ψ, which is unitary and linear. The "No-cloning theorem" does not apply here. What the No-cloning theorem states is as follows: Let ψ be a q-string of length n. Let 0 be a string of n zeroes. No-cloning result: there is no unitary, linear function g, or q-strings of length 2n, such that g(ψ ⊗ 0) = g(ψ ⊗ ψ). C. Quantum Parallelism and Decoherence What is quantum parallelism? Suppose ψ is a q-string and ψ has m different bit strings appearing as nonzeroweighted components. Then, quantum parallelism is the idea that one processing step on ψ is, in a sense, equivalent to m processing steps on the bit-string of components of ψ [2]. Since processing takes time, quantum parallelism is lost if the q-string decoheres during processing. What is decoherence? The only kind of decoherence discussed as such in quantum computing is environmental decoherence. We believe that there are two other forms of decoherence, measurement decoherence and internal decoherence, and that these other forms may pose obstacles for quantum parallelism as well. Let us start with environmental decoherence. Let ψ = ∑ m cmφm be the state of a physical system where the φm are the base states of the system and cm are the complex numbers satisfying the usual condition. Let E0 represent the initial state of the environment. Environmental decoherence is the idea that after a time (decoherence time) the physical system interacts enough with the environment so that the state of the system plus environment evolves to the following: | ψ,E0〉 −→ ∑ m cm | φm, Em〉, where −→ means decoherence time and the Em are states of the environment that do not mutually "interfere". What the "non-interference" means practically is that the evolved state immediately collapses: ∑ m cm | φm, Em〉 −→ one of the | φm, Em〉 states with a probability of | cm |2. Q-strings live on the register of the quantum computer. So, Ψ above is the state of the register(s) of the computer, and E above is the state of the environment of the register(s); i.e., the rest of the computer plus the external world. So, if decoherence time is less than processing time, a q-string will collapse into one of its component bit-strings, and quantum parallelism will be destroyed. Erich Joos [1] states that experimental results seem to indicate that decoherence time is related inversely to size; he even says (p. 13): "..macroscopic objects are extremely sensitive and immediately decohered." If what Joos says is true, then the longer the q-string, the shorter the time to its decoherence. This rules out quantum parallelism for qstrings of arbitrary length. Joos says (p. 14): ...(decoherence) represents a major obstacle for people trying to construct a quantum computer. Building a really big one may well turn out to be as difficult as detecting other Everett worlds! Many think that detecting other Everett worlds is impossible [3]. Measurement decoherence can be explained as follows. Let ψ = ∑ m cmφm be the state of a physical system, and suppose at t0 (the initial time), ψ is coupled with a measuring device in state M0. Let "measurement time" be the amount of time required for the measuring device to measure the physical system, i.e., the amount of time for the measuring device to evolve from M0 to a superposition of indicator states Mi. The picture of the evolution is as follows: ∑ m cmφmM0 −→ ∑ m cmφmMm (where −→ denotes "measurement time"). If we make the assumption that the Mi(i ≥ 1) do not mutually interfere, then ψ immediately collapses: ∑ m cmφmM0 −→ −→ one of the φmMm with a probability of | cm |2 . Measurement decoherence (called quantum measurement by quantum computer scientists) is a resource of, and not an obstacle to, quantum computing if it occurs after processing is complete. Measuring the output q-string is the way to read information contained in that q-string. No one will intentionally apply a measuring device to a register or registers before processing is complete. So, how can measurement decoherence be an obstacle to quantum computing? A physical quantum computer will contain a register or registers, but will also contain other devices (for processing, etc.) besides registers. If the "innards" of a physical quantum computer exclusive of the registers act like a measuring device during processing, then there will be an unintentional measurement of a register or registers during processing, and quantum parallelism will be lost. Thus, it is a challenge not only to build registers that can exist in superposed states, but also to build the rest of the quantum computer so that it does not act like a measuring device on registers during processing. The third form of decoherence is internal decoherence. Suppose we have a physical system in an initial state ψ0. Suppose also that in some time interval (evolution time), the physical system evolves to a superposition of base states: ψ −→ ∑ m cmφm (where −→ denotes evolution time). In the standard two-slit experiment, we have evolution to: α | particle travels through slit 1〉 + β | particle travels through slit 2〉, but these two states mutually interfere, as is evidenced by the interference pattern built up on the photographic backstop as the experiment is repeated. So the standard twoslit experiment is not an example of internal decoherence. We can get internal decoherence if we modify the two-slit experiment. Put a light source near slit 1, so that a particle traveling through slit 1 produces a light flash because a photon from the source bounces off the particle. Then we have evolution to: α | particle travels through slit 1 + flash of light〉 + β | particle travels through slit 2 + no flash of light〉. These two states do not mutually interfere, as is evidenced by the lack of interference pattern on the photographic backstop. (Remember that observation of the light flash is not necessary to destroy interference; only existence of the flash is necessary. Another physical system that internally decoheres is Schrödinger's Cat Box, consisting of a box occupied by a radioactive source, Geiger counter, trip hammer, vial of cyanide, and live cat. The box evolves into: α | dead cat, smashed vial, tripped hammer, etc.〉 + β | live cat, un-smashed vial, un-tripped hammer, etc.〉 Based on all available observational evidence, these two states do not mutually interfere. No one has ever observed a superposition of α | dead cat〉 + β | live cat〉, let alone: α | smashed vial〉 + β | un-smashed vial〉, or α | tripped hammer〉 + β | un-tripped hammer〉, etc. So, the solution to Schrödinger's Paradox is internal decoherence. We can talk of Schrödinger's Register instead of Schrödinger's Cat and mean by this an n-bit register that can exist in a superposition of bit-strings of length n such that those bit strings do interfere. (We want the bit-strings to interfere, or else we would have a collapse to a single bit-string and no quantum parallelism.) So, the challenge for quantum computer scientists is to build Schrödinger's Register. Good luck! D. Qubits, Q-Strings, and Entanglement Consider a q-string of length n, ψ. Suppose ψ is "entangled." Then it is not equal to a string of n qubits, but a string of n qubits can be constructed from it in the following way: Survey the bit-string components of ψ. Let m be a position from 1 to n in the bit-string. Add the amplitudes for all components with a 0 in position m. Call the sum αm. Add the amplitudes for all components with a 1 in position m. Call the sum βm. Construct the qubit (αm | 0〉). Take the product of such qubits for all positions. This is the string of qubits to be constructed: m=n⊗ m=1 (αm | 0〉 + βm | 1〉) = m=n⊗ m=1 9m . The result of processing ψ "one qubit at a time" = the product of the results of applying a processing step f on qubits to each qubit in the string constructed from: ψ = m=n⊗ m=1 f(qm) = m=n⊗ m=1 q′m . A string of m qubits has no entanglement among qubits. So, processing ψ "one qubit at a time" destroys entanglement. III. CONCLUSION We believe that the four points above will all be necessary to the progress in understanding how to realize a quantum computer. In particular, we think that the role of the basic notion of quantum computation, the qubit, as well as the "No-cloning" theorem and entanglement need to be rethought in their relationship to quantum parallelism. REFERENCES [1] E. Joos, Elements of Environmental Decoherence, (1999), eprint available at http://xxx.lanl.gov/abs/quant-ph/9908008v1. [2] J. Gruska, Quantum Computing Challenges, In Mathematics Unlimited, 2001 and Beyond, Vol. 1. Berlin-Heidelberg : Springer-Verlag, 2000. ISBN 3-540-66913-2, pp. 529-563. [3] D. Albert and B. Loewer Interpreting the Many Worlds Interpretation. SYNTHESE, Volume 77, Number 2, 195-213, DOI: 10.1007/BF00869434. [4] J. Stolze and D. Suter, Quantum Computing, Wiley-VCH Verlag GMBH & Co., 2nd Edition, 2008, ISBN 978-3-52740787-3, pp.6-7.