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Measurement Values in Quantum Measurement Theory

誠司 小杉, Seiji Kosugi

Published Feb 25, 2016• 0 citations
OpenAlex

Abstract

測定対象の相互作用前のオブザーバブルと相互作用後のオブザーバブルの測定値を,測定誤差が最小になるように定義し,理論的測定値を得た.偏りのある測定では,測定値が奇妙な振舞いをすることがあるが,理論的測定値を用いた測定は常に偏りのない測定となる.しかし理論的測定値は測定対象の初期状態に依存しているので,初期状態がわからなければ実際に理論的測定値を求めることはできない.そこで初期状態についての情報が全くない場合に使用すべき測定値を,理論的測定値から求め,それを実践的測定値と名付けた.プローブの初期状態を式(1)の(固有値が連続の場合)に,あるいは式(25)の(固有値が離散的な場合)に設定すると,測定誤差を0にすることができる.このとき理論的測定値及び実践的測定値を用いたときには,測定対象の任意の初期状態に対して,Bornの確率則を再現するが,Ozawaの測定値を用いたときには,任意の初期状態に対してBornの確率則を再現しない.

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ÒÑÊÐÂÍÑÏËÕßÔâ Ô ÆÇÕÂ-ÎâÏË, ÆÑÎÉÇÐ ÑÃÓÂÕËÕßÔâ Í ÙËÕËÓÑÄÂÐÐÑÌ ÎËÕÇÓÂÕÖÓÇ.£ ÚÂÔÕÐÑÔÕË, ÑÕÐÑÛÇÐËÇ ÏÇÉÆÖ ÒÑÔÕÖÎÂÕÑÏ ÓÇÆÖÍÙËË Ë ÆÇÍÑ-ÅÇÓÇÐÙËÇÌ ÆÇÕÂÎßÐÑ ÑÃÔÖÉÆÂÇÕÔâ Ä [3] Ë [4] (ÒÇÓÄÂâ ËÊ àÕËØ ÓÂÃÑÕ ÒÓÑÙËÕËÓÑÄÂÐÂ Ä ÒËÔßÏÇ ËÒÍËÐÂ, ÐÑ ÕÂÍÉÇ, ÒÑ-ÄËÆËÏÑÏÖ, ÒÑÐâÕ ÒÓÇÄÓÂÕÐÑ).•ÑÕâ ÕÇÑÓËâ ÆÇÍÑÅÇÓÇÐÙËË ÆÂÇÕ ×ËÊËÚÇÔÍÑÇ ÑÃÝâÔÐÇÐËÇ ÕÑÅÑ, ÚÕÑ ×ÇÐÑÏÇÐÑÎÑÅËÚÇÔÍË ÑÒËÔÞÄÂÇÕÔâ ÓÇÆÖÍÙËÇÌ, ÔÂÏ ÒÑÔÕÖÎÂÕ ÓÇÆÖÍÙËË ÒÓË àÕÑÏ ÐÇ ÎËÛÂÇÕÔâ ÔÏÞÔÎÂ, ÐÑ ÏÇÐâÇÕÔâ ÇÅÑ ÔÕÂÕÖÔ.²ÇÆÖÍÙËâ ÑÔÕÂÇÕÔâ ÒÓÑÔÕÞÏ Ë ËÊâÜ-ÐÞÏ ÄÞÚËÔÎËÕÇÎßÐÞÏ ÒÓËÇÏÑÏ Ä ÕÑÏ ÔÎÖÚÂÇ, ÇÔÎË ÕÓÇÃÖÇÕÔâ ÓÂÔÔÚËÕÂÕß ÒÑÄÇÆÇÐËÇ ÔËÔÕÇÏÞ ÒÑÔÎÇ ÕÑÅÑ, ÍÂÍ ÒÓÑËÊÑÛÎÑ ËÊÏÇÓÇÐËÇ, Ë ÒÓË àÕÑÏ ËÊÄÇÔÕÐÑ, ÍÂÍÑÌ ÓÇÊÖÎßÕÂÕ ÆÂÎÑ àÕÑ ËÊÏÇÓÇÐËÇ.£ ÚÂÔÕÐÑÔÕË, ÍÂÓÕËРÓÇÆÖÍÙËË ÒÑÎÇÊРÆÎâ ÓÂÔÚÇÕ ÓÇÊÖÎßÕÂÕÑÄ ÆÄÖØ ËÎË ÐÇÔÍÑÎßÍËØ ËÊÏÇÓÇÐËÌ, ÔÎÇ-ÆÖáÜËØ ÆÓÖÅ Ê ÆÓÖÅÑÏ.´ÂÍËÏ ÑÃÓÂÊÑÏ, ÄÑÊÐËÍÂÇÕ ÄÑÊÏÑÉÐÑÔÕß ÑÒËÔÞÄÂÕß ÍÄÂÐÕÑÄÑÇ ËÊÏÇÓÇÐËÇ Ð ÓÂÊÎËÚÐÞØ ÖÓÑÄÐâØ: ÒÑÔÕÖÎÂÕ ÓÇÆÖÍÙËË ÆÂÇÕ ×ÇÐÑÏÇÐÑÎÑÅËÚÇÔÍËÌ

Uspekhi Fizicheskih Nauk•Jan 1, 2001•13 citations

Does a classical probability space for two-dimensional quantum measurement theory exist?

Koji Nagata, Tadao Nakamura

Recently, a new measurement theory based on the truth values is proposed \cite{NN1}. The results of measurements are either 0 or 1. The measurement theory accepts a hidden variables model for a single Pauli observable. Therefore we can introduce a classical probability space for the measurement theory in this case. On the other hand, we discuss the fact that the projective measurement theory (the results of measurements are either $+1$ or $-1$) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.

•Jun 1, 2016•0 citations

Theory of measurements and the collapse of the wave function

Gordey B. Lesovik

®ÑÆÇÎß ÏÑÉÐÑ ÒÑÒÞÕÂÕßÔâ ÖÎÖÚÛËÕß, ÑÃÝâÄËÄ ×ÑÓÏÂ-ÎßÐÑ, ÚÕÑ ÑÍÓÖÉÇÐËÇ E ÒÓÇÆÔÕÂÄÎâÇÕ ÔÑÃÑÌ ÄÇÔß ÑÔÕÂÕÑÍ £ÔÇÎÇÐÐÑÌ.´ÑÅÆ ÔËÔÕÇÏ U ÔÕÂÐÑÄËÕÔâ ÆÇÌÔÕÄËÕÇÎßÐÑ ÊÂÏÍÐÖÕÑÌ, ÕÂÍ ÍÂÍ £ÔÇÎÇÐÐÂâ ÐÇ ËÏÇÇÕ ÑÍÓÖÉÇÐËâ.°ÆÐÂÍÑ àÕÑ ÐÇÆÑÒÖÔÕËÏÑ ÒÑ ÔÎÇÆÖáÜÇÌ ÒÓËÚËÐÇ.¦ÔÎË ÏÞ ÑÃÝâÄËÎË ÔËÔÕÇÏÖ E ÑÔÕÂÕÍÑÏ £ÔÇÎÇÐÐÑÌ, ÕÑ ÔËÔÕÇÏ U ÒÓÇÆÔÕÂÄÎâÇÕ ÖÉÇ £ÔÇÎÇÐÐÖá ÙÇÎËÍÑÏ.°ÆÐÂÍÑ ØÑÓÑÛÑ ËÊÄÇÔÕÐÑ [4], ÚÕÑ ÆÎâ ÕÑÚÐÑÅÑ ÍÄÂÐÕÑÄÑÅÑ ÔÑÔÕÑâÐËâ £ÔÇÎÇÐÐÑÌ ÐÇ ÔÖÜÇÔÕÄÖÇÕ ÒÑÐâÕËâ ÄÐÇÛÐÇÅÑ ÄÓÇÏÇÐË, Ë àÄÑÎáÙËâ ÍÄÂÐÕÑÄÑÅÑ ÔÑÔÕÑâ-ÐËâ £ÔÇÎÇÐÐÑÌ ÐÇ ÇÔÕß ÖÐËÕÂÓÐÂâ àÄÑÎáÙËâ ÄÑ ÄÓÇÏÇÐË.´ÂÍËÏ ÑÃÓÂÊÑÏ, ÑÒâÕß ÖÐËÕÂÓÐÑÇ ÑÒËÔÂÐËÇ àÄÑÎáÙËË ÔËÔÕÇÏÞ U ÍÂÍ àÄÑÎáÙËË ÄÑ ÄÓÇÏÇÐË ÔÕÂÐÑÄËÕÔâ ÐÇÄÑÊÏÑÉ-ÐÞÏ.®ÑÉÐÑ ÒÓÇÆÔÕÂÄËÕß ÔÇÃÇ ÆÄ ÒÖÕË (ÆÄÇ ÒÓÑÅÓÂÏÏÞ), РÍÑÕÑÓÞØ ÖÍÂÊÂÐÐÂâ ÕÓÖÆÐÑÔÕß ÏÑÉÇÕ ÃÞÕß ÖÔÕÓÂÐÇÐÂ.¿ÕÑ ÎËÃÑ A ÔÑÊÆÂÐËÇ ÒÑÔÎÇÆÑÄÂÕÇÎßÐÑ ÍÄÂÐÕÑÄÑÅÑ ÑÒËÔÂÐËâ £ÔÇÎÇÐÐÑÌ ÄÏÇÔÕÇ Ô âÄÐÞÏ ÑÒËÔÂÐËÇÏ ÅÇÐÇÓÂÙËË ÄÐÖÕÓÇÐ-ÐÇÅÑ ×ÇÐÑÏÇÐÑÎÑÅËÚÇÔÍÑÅÑ ÄÓÇÏÇÐË, ÖÍÂÊÂÐËÇÏ âÄÐÑÅÑ ÔÒÑÔÑàÑÒËÔÂÐËâ ÒÑÆÔËÔÕÇÏ £ÔÇÎÇÐÐÑÌ Ë ÖÍÂÊÂÐËÇÏ ÔÒÑ-ÔÑàÔÄâÊË àÕËØ ÒÑÆÔËÔÕÇÏ Ô ÄÐÖÕÓÇÐÐËÏ ÄÓÇÏÇÐÇÏ, Õ.Ç.ÔÑÊÆÂÐËÇ ÒÑÎÐÑÌ ÍÄÂÐÕÑÄÑÌ ÍÑÔÏÑÎÑÅËË.ËÃÑ B ÖÚÇÕ ÄÔÇÌ ÄÐÇÛÐÇÌ ÒÑ ÑÕÐÑÛÇÐËá Í ÓÂÔÔÏÂÕÓËÄÂÇÏÑÌ ÑÃÝÇÆËÐÇÐ-ÐÑÌ ÍÄÂÐÕÑÄÑÌ ÔËÔÕÇÏÇ U £ÔÇÎÇÐÐÑÌ ×ÇÐÑÏÇÐÑÎÑÅËÚÇÔÍË: Ô ËÔÒÑÎßÊÑÄÂÐËÇÏ ÔÒÑÐÕÂÐÐÑÌ ÓÇÆÖÍÙËË ÄÑÎÐÑÄÑÌ ×ÖÐÍÙËË, Ô ÒÑÏÑÜßá ÒÑÎÑÉËÕÇÎßÐÑ-ÑÒÓÇÆÇÎÇÐÐÞØ ÑÒÇÓÂÕÑÓÑÄ, Ô ÒÑÏÑÜßá ÑÅÓÂÐËÚÇÐÐÑÅÑ ËÐÕÇÅÓÂΠÒÑ ÒÖÕâÏ ËÎË ÍÂÍËÏ-ÕÑ ÆÓÖÅËÏ ÔÒÑÔÑÃÑÏ.¯ÂËÃÑÎÇÇ ÒÑÔÎÇÆÑÄÂÕÇÎßÐÞÏ ÒÓÇÆÔÕÂ-ÄÎâÇÕÔâ ÒÖÕß A, ÑÆÐÂÍÑ Ä ÐÂÔÕÑâÜÇÇ ÄÓÇÏâ ÐÇâÔÐÑ, ÄÞÒÑÎÐËÏ ÎË ÕÂÍÂâ ÒÓÑÅÓÂÏÏ ØÑÕâ ÃÞ Ä ÒÓËÐÙËÒÇ.±ÖÕß B, ÍÂÍ ÊÂÏÇÚÂÇÕ ÖÄÂÉÂÇÏÞÌ ÂÄÕÑÓ ÔÕÂÕßË [1], ÐÇ ÒÓËÄÑÆËÕ Í ÎÑÅËÚÇÔÍËÏ ÕÓÖÆÐÑÔÕâÏ Ë ÒÂÓÂÆÑÍÔÂÏ.°ÆÐÂÍÑ, ÒÑ ÐÂÛÇÏÖ ÏÐÇÐËá, ÑÐ ÏÑÉÇÕ ÄÞÊÞÄÂÕß ÐÇÖÆÑÄÎÇÕÄÑÓÇÐÐÑÔÕß Ð ÕÑÏ ÑÔÐÑÄÂÐËË, ÚÕÑ ÐÂÓâÆÖ Ô ×ÖÐÆÂÏÇÐÕÂÎßÐÞÏ ÆËÐÂÏËÚÇÔÍËÏ ÊÂÍÑÐÑÏ (ÖÓÂÄÐÇÐËÇ ºÓÈÆËÐÅÇÓÂ) Ä ÕÇÑÓËá ÄÄÑÆËÕÔâ ×ÇÐÑ-ÏÇÐÑÎÑÅËâ, ÍÑÕÑÓÂâ ËÏÇÇÕ ÔÕÑÎß ÉÇ ×ÖÐÆÂÏÇÐÕÂÎßÐÞÌ ØÂÓÂÍÕÇÓ, ÍÂÍ Ë ÔÂÏË ÆËÐÂÏËÚÇÔÍËÇ ÊÂÍÑÐÞ.¿Õ ×ÇÐÑÏÇÐÑ-ÎÑÅËâ ×ÖÐÆÂÏÇÐÕÂÎßÐÂ Ä ÕÑÏ ÔÏÞÔÎÇ, ÚÕÑ ÑРÆÑÎÉРÔÚËÕÂÕßÔâ Ä ÒÓËÐÙËÒÇ ÐËÑÕÍÖÆ ÐÇÄÞÄÑÆËÏÑÌ (Ä ÑÕÎËÚËÇ, ÐÂÒÓËÏÇÓ, ÑÕ ÕÇÓÏÑÆËÐÂÏËÍË, ÍÑÕÑÓÂâ ÄÞÄÑÆËÏ ËÊ ÔÕÂÕË-ÔÕËÚÇÔÍÑÌ ×ËÊËÍË).°Ð ÆÑÎÉРÃÞΠÃÞ ÄÞÄÑÆËÕßÔâ ËÊ ÍÄÂÐÕÑÄÑÌ ÍÑÔÏÑÎÑÅËË, ÐÑ Ä ÒÓÑÅÓÂÏÏÇ B ÍÄÂÐÕÑÄÂâ ÍÑÔÏÑÎÑÅËâ ÑÍÂÊÞÄÂÇÕÔâ ÔÖÜÇÔÕÄÇÐÐÑ Ê ÓÂÏÍÂÏË ÕÇÑÓËË.±ÓÑÃÎÇÏ ÖÔÖÅÖÃÎâÇÕÔâ ÇÜÇ Ë ÕÇÏ (Ë àÕÑ, ÒÑ ÐÂÛÇÏÖ ÏÐÇÐËá, âÄÎâÇÕÔâ ÎÑÅËÚÇÔÍÑÌ ÕÓÖÆÐÑÔÕßá ÒÓÑÅÓÂÏÏÞ B, ÚÕÑ ÄÑÊÏÑÉÐÞ ÓÂÊÐÞÇ ÔÒÑÔÑÃÞ ÄÄÇÆÇÐËâ ÕÂÍÑÌ ×ÇÐÑ-ÏÇÐÑÎÑÅËË, ÒÓË àÕÑÏ ÐÇ ÆÑÍÂÊÂÐÑ, ÚÕÑ ÄÔÇ ÔÒÑÔÑÃÞ àÍÄËÄÂ-ÎÇÐÕÐÞ.¯Â ÐÂÛ ÄÊÅÎâÆ, ÒÓÑÃÎÇÏ ÍÄÂÐÕÑÄÑÅÑ ËÊÏÇÓÇÐËâ ÍÂÍ ÓÂÊ Ë ÊÂÍÎáÚÂÕÔâ Ä ÔÖÜÇÔÕÄÑÄÂÐËË ÆËÎÇÏÏÞ: ÎËÃÑ ÍÄÂÐÕÑÄÂâ ÕÇÑÓËâ ËÊÏÇÓÇÐËâ ÇÔÕß Ð ÔÂÏÑÏ ÆÇÎÇ ÍÄÂÐÕÑÄÂâ ÍÑÔÏÑÎÑÅËâ, ÎËÃÑ ÑРÔÑÆÇÓÉËÕ ÐÇÖÔÕÓÂÐËÏÖá Ë ÐÇ ÄÒÑÎÐÇ ÑÆÐÑÊÐÂÚÐÖá ×ÇÐÑÏÇÐÑÎÑÅËá.®Þ ÒÑÍÂÊÂÎË, ÚÕÑ ÆËÐÂÏËÚÇÔÍÑÇ ÑÒËÔÂÐËÇ ÄÞÃÑÓ ÂÎßÕÇÓÐÂÕËÄÞ ÔÑÊÐÂÐËÇÏ ÄÑÊÏÑÉÐÑ Ä ÓÂÏÍÂØ ÓÂÔÔÏÑÕ-ÓÇÐÐÑÌ ÄÞÛÇ ÖÐËÕÂÓÐÑÌ ÏÑÆÇÎË.•ÑÕâ ÏÑÆÇÎß, ÍÂÍ ÃÞÎÑ ÑÕÏÇÚÇÐÑ, Ë ÏÑÉÇÕ ÄÞÊÞÄÂÕß ÔÇÓßÇÊÐÞÇ ÄÑÊÓÂÉÇÐËâ, ÄÑÊ-ÏÑÉÐÑÔÕß ÕÂÍÑÅÑ ÑÒËÔÂÐËâ, ÒÑ ÐÂÛÇÏÖ ÏÇÐËá, âÄÎâÇÕÔâ ÑÒÓÇÆÇÎÇÐÐÞÏ ÖÍÂÊÂÐËÇÏ Ð ÕÑ, ÚÕÑ ÔÑÊÐÂÐËÇ ÐÂÃÎáÆÂÕÇÎâ a priori ÐÇ ÔÎÇÆÖÇÕ ÄÞÄÑÆËÕß Ê ÓÂÏÍË ÖÐËÕÂÓÐÑÌ ÍÄÂÐÕÑÄÑÌ ÆËÐÂÏËÍË Ë ÒÑÆÆÇÓÉËÄÂÇÕ ÒÓËÐÙËÒ ÒÔËØÑ×ËÊËÚÇÔÍÑÅÑ ÒÂÓÂÎÎÇÎËÊϠРÍÄÂÐÕÑÄÑÏ ÖÓÑÄÐÇ.

Uspekhi Fizicheskih Nauk•Jan 1, 2001•3 citations

Weak values in a classical theory with an epistemic restriction

Angela Karanjai, Eric G. Cavalcanti, Stephen D. Bartlett +1 more

Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.

arXiv:1503.05203•New Journal of Physics•Jul 14, 2015•15 citations
測定対象の相互作用前のオブザーバブルと相互作用後のオブザーバブルの測定値を,測定誤差が最小になるように定義し,理論的測定値を得た.偏りのある測定では,測定値が奇妙な振舞いをすることがあるが,理論的測定値を用いた測定は常に偏りのない測定となる.しかし理論的測定値は測定対象の初期状態に依存しているので,初期状態がわからなければ実際に理論的測定値を求めることはできない.そこで初期状態についての情報が全くない場合に使用すべき測定値を,理論的測定値から求め,それを実践的測定値と名付けた.プローブの初期状態を式(1)の(固有値が連続の場合)に,あるいは式(25)の(固有値が離散的な場合)に設定すると,測定誤差を0にすることができる.このとき理論的測定値及び実践的測定値を用いたときには,測定対象の任意の初期状態に対して,Bornの確率則を再現するが,Ozawaの測定値を用いたときには,任意の初期状態に対してBornの確率則を再現しない.
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Recently, a new measurement theory based on the truth values is proposed \cite{NN1}. The results of measurements are either 0 or 1. The measurement theory accepts a hidden variables model for a single Pauli observable. Therefore we can introduce a classical probability space for the measurement theory in this case. On the other hand, we discuss the fact that the projective measurement theory (the results of measurements are either +1+1+1 or −1-1−1) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.
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Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.